3.7.0 ∫ 1 ____________ (d+ex)5∕2(a+cx2)2 dx [637]

Integrand size = 19, antiderivative size = 930 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

1/6*e*(-7*a*e^2+3*c*d^2)/a/(a*e^2+c*d^2)^2/(e*x+d)^(3/2)+1/2*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(e*x+d)^(3/2)/(c*x^2+

a)+1/2*c*d*e*(-19*a*e^2+c*d^2)/a/(a*e^2+c*d^2)^3/(e*x+d)^(1/2)+1/8*c^(3/4)*e*arctanh((-c^(1/4)*2^(1/2)*(e*x+d)

^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(c^2*d^4+34*a*c*d^2*e^2-7

*a^2*e^4+d*(-19*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(7/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^

2)^(1/2))^(1/2)-1/8*c^(3/4)*e*arctanh((c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d

*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(c^2*d^4+34*a*c*d^2*e^2-7*a^2*e^4+d*(-19*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^

2)^(1/2))/a/(a*e^2+c*d^2)^(7/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/16*c^(3/4)*e*ln((e*x+d)*c^(1/2

)+(a*e^2+c*d^2)^(1/2)-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(c^2*d^4+34*a*c*d^2

*e^2-7*a^2*e^4-d*(-19*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/(a*e^2+c*d^2)^(7/2)*2^(1/2)/(d*c^(1/2)+(a*e^

2+c*d^2)^(1/2))^(1/2)+1/16*c^(3/4)*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)+c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c

^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(c^2*d^4+34*a*c*d^2*e^2-7*a^2*e^4-d*(-19*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)

^(1/2))/a/(a*e^2+c*d^2)^(7/2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)

Rubi [A] (veriﬁed)

Time = 4.20 (sec) , antiderivative size = 930, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {755, 843, 841, 1183, 648, 632, 212, 642} \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2+\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c e^2 d^2-\sqrt {c} \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2} d-7 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (c x^2+a\right )}+\frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}} \]

(e*(3*c*d^2 - 7*a*e^2))/(6*a*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) + (c*d*e*(c*d^2 - 19*a*e^2))/(2*a*(c*d^2 + a*e

^2)^3*Sqrt[d + e*x]) + (a*e + c*d*x)/(2*a*(c*d^2 + a*e^2)*(d + e*x)^(3/2)*(a + c*x^2)) + (c^(3/4)*e*(c^2*d^4 +

34*a*c*d^2*e^2 - 7*a^2*e^4 + Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt

[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*(c*d^2

+ a*e^2)^(7/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(3/4)*e*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 + Sqr

t[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4

)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*(c*d^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d - S

qrt[c*d^2 + a*e^2]]) - (c^(3/4)*e*(c^2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 - Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[c*

d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] +

Sqrt[c]*(d + e*x)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(3/4)*e*(c^

2*d^4 + 34*a*c*d^2*e^2 - 7*a^2*e^4 - Sqrt[c]*d*(c*d^2 - 19*a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2]

+ Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(8*Sqrt[2]*a*(c*d

^2 + a*e^2)^(7/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]

/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]

, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +

c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +

b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &

& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*

((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^

m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[

{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f

- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d

+ e*x)^(m + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(Simp[c*d*f + a*

e*g - c*(e*f - d*g)*x, x]/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =

Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(

d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2

Rubi steps \begin{align*} \text {integral}& = \frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-2 c d^2-7 a e^2\right )-\frac {5}{2} c d e x}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )} \\ & = \frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\int \frac {-c d \left (c d^2+6 a e^2\right )-\frac {1}{2} c e \left (3 c d^2-7 a e^2\right ) x}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )^2} \\ & = \frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\int \frac {-\frac {1}{2} c \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )-\frac {1}{2} c^2 d e \left (c d^2-19 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )^3} \\ & = \frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {1}{2} c^2 d^2 e \left (c d^2-19 a e^2\right )-\frac {1}{2} c e \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )-\frac {1}{2} c^2 d e \left (c d^2-19 a e^2\right ) x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{a \left (c d^2+a e^2\right )^3} \\ & = \frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c^2 d^2 e \left (c d^2-19 a e^2\right )-\frac {1}{2} c e \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )\right )}{\sqrt [4]{c}}-\left (\frac {1}{2} c^2 d^2 e \left (c d^2-19 a e^2\right )+\frac {1}{2} c^{3/2} d e \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}-\frac {1}{2} c e \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c^2 d^2 e \left (c d^2-19 a e^2\right )-\frac {1}{2} c e \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )\right )}{\sqrt [4]{c}}+\left (\frac {1}{2} c^2 d^2 e \left (c d^2-19 a e^2\right )+\frac {1}{2} c^{3/2} d e \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}-\frac {1}{2} c e \left (2 c^2 d^4+15 a c d^2 e^2-7 a^2 e^4\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = \frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {\left (c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (\sqrt {c} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 a \left (c d^2+a e^2\right )^{7/2}}+\frac {\left (\sqrt {c} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 a \left (c d^2+a e^2\right )^{7/2}} \\ & = \frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (\sqrt {c} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{4 a \left (c d^2+a e^2\right )^{7/2}}-\frac {\left (\sqrt {c} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{4 a \left (c d^2+a e^2\right )^{7/2}} \\ & = \frac {e \left (3 c d^2-7 a e^2\right )}{6 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (c d^2-19 a e^2\right )}{2 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \left (a+c x^2\right )}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4+\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (c^2 d^4+34 a c d^2 e^2-7 a^2 e^4-\sqrt {c} d \left (c d^2-19 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{7/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ \end{align*}

Mathematica [C] (veriﬁed)

Time = 3.13 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.41 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\frac {-\frac {2 \sqrt {a} \left (4 a^3 e^5-3 c^3 d^3 x (d+e x)^2+a^2 c e^3 \left (55 d^2+54 d e x+7 e^2 x^2\right )+a c^2 d e \left (-9 d^3-9 d^2 e x+61 d e^2 x^2+57 e^3 x^3\right )\right )}{\left (c d^2+a e^2\right )^3 (d+e x)^{3/2} \left (a+c x^2\right )}+\frac {3 \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \left (-2 i c d+7 \sqrt {a} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\left (\sqrt {c} d+i \sqrt {a} e\right )^4}+\frac {3 \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \left (2 i c d+7 \sqrt {a} \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right )^4}}{12 a^{3/2}} \]

Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)^2),x]

((-2*Sqrt[a]*(4*a^3*e^5 - 3*c^3*d^3*x*(d + e*x)^2 + a^2*c*e^3*(55*d^2 + 54*d*e*x + 7*e^2*x^2) + a*c^2*d*e*(-9*

d^3 - 9*d^2*e*x + 61*d*e^2*x^2 + 57*e^3*x^3)))/((c*d^2 + a*e^2)^3*(d + e*x)^(3/2)*(a + c*x^2)) + (3*Sqrt[-(c*d

) - I*Sqrt[a]*Sqrt[c]*e]*((-2*I)*c*d + 7*Sqrt[a]*Sqrt[c]*e)*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d

+ e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(Sqrt[c]*d + I*Sqrt[a]*e)^4 + (3*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*((2*I)

*c*d + 7*Sqrt[a]*Sqrt[c]*e)*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e

)])/(Sqrt[c]*d - I*Sqrt[a]*e)^4)/(12*a^(3/2))

Maple [A] (veriﬁed)

Time = 4.36 (sec) , antiderivative size = 1162, normalized size of antiderivative = 1.25


method	result	size

pseudoelliptic	\(\text {Expression too large to display}\)	\(1162\)
derivativedivides	\(\text {Expression too large to display}\)	\(3541\)
default	\(\text {Expression too large to display}\)	\(3541\)

int(1/(e*x+d)^(5/2)/(c*x^2+a)^2,x,method=_RETURNVERBOSE)

7/4/(a*e^2+c*d^2)^(7/2)/(e*x+d)^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*(1

/4*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*(e*x+d)^(3/2)*(2*((a*e^2+c*d^2)*c)^(1

/2)+2*c*d)^(1/2)*((-19/7*(-1/19*a*(-19*e^2*x^2+d^2)*c^(3/2)+c^(1/2)*a^2*e^2-1/19*x^2*c^(5/2)*d^2)*d*(a*e^2+c*d

^2)^(1/2)+(a^2*e^4-34/7*a*c*d^2*e^2-1/7*c^2*d^4)*(c*x^2+a))*((a*e^2+c*d^2)*c)^(1/2)-d*(-19/7*d*(-1/19*a*(-19*e

^2*x^2+d^2)*c^(5/2)+a^2*e^2*c^(3/2)-1/19*c^(7/2)*d^2*x^2)*(a*e^2+c*d^2)^(1/2)+(a^2*e^4-34/7*a*c*d^2*e^2-1/7*c^

2*d^4)*c*(c*x^2+a)))*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1

/2))-1/4*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*(e*x+d)^(3/2)*(2*((a*e^2+c*d^2)

*c)^(1/2)+2*c*d)^(1/2)*((-19/7*(-1/19*a*(-19*e^2*x^2+d^2)*c^(3/2)+c^(1/2)*a^2*e^2-1/19*x^2*c^(5/2)*d^2)*d*(a*e

^2+c*d^2)^(1/2)+(a^2*e^4-34/7*a*c*d^2*e^2-1/7*c^2*d^4)*(c*x^2+a))*((a*e^2+c*d^2)*c)^(1/2)-d*(-19/7*d*(-1/19*a*

(-19*e^2*x^2+d^2)*c^(5/2)+a^2*e^2*c^(3/2)-1/19*c^(7/2)*d^2*x^2)*(a*e^2+c*d^2)^(1/2)+(a^2*e^4-34/7*a*c*d^2*e^2-

1/7*c^2*d^4)*c*(c*x^2+a)))*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d

^2)^(1/2))+(-8/21*(-3/4*d^3*x*(e*x+d)^2*c^3-9/4*e*d*a*(-19/3*e^3*x^3-61/9*d*e^2*x^2+d^2*e*x+d^3)*c^2+55/4*(7/5

5*x^2*e^2+54/55*d*e*x+d^2)*e^3*a^2*c+a^3*e^5)*(a*e^2+c*d^2)^(1/2)*(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d

^2)*c)^(1/2)-2*c*d)^(1/2)+(e*x+d)^(3/2)*(19/7*d*(-1/19*a*(-19*e^2*x^2+d^2)*c^(5/2)+a^2*e^2*c^(3/2)-1/19*c^(7/2

)*d^2*x^2)*(a*e^2+c*d^2)^(1/2)+(a^2*e^4-34/7*a*c*d^2*e^2-1/7*c^2*d^4)*c*(c*x^2+a))*(arctan((-2*c^(1/2)*(e*x+d)

^(1/2)+(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d

)^(1/2))-arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/

2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)))*e)*e*a)/a^2/e/(c*x^2+a)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8281 vs. \(2 (780) = 1560\).

Time = 3.29 (sec) , antiderivative size = 8281, normalized size of antiderivative = 8.90 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a)^2,x, algorithm="fricas")

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\text {Timed out} \]

integrate(1/(e*x+d)**(5/2)/(c*x**2+a)**2,x)

Maxima [F]

\[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{2} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a)^2,x, algorithm="maxima")

integrate(1/((c*x^2 + a)^2*(e*x + d)^(5/2)), x)

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2014 vs. \(2 (780) = 1560\).

Time = 0.51 (sec) , antiderivative size = 2014, normalized size of antiderivative = 2.17 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a)^2,x, algorithm="giac")

-1/4*((a*c^3*d^6*e + 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 + a^4*e^7)^2*(c^2*d^3*e - 19*a*c*d*e^3)*abs(c) - (sqr

t(-a*c)*c^5*d^10*e + 37*sqrt(-a*c)*a*c^4*d^8*e^3 + 98*sqrt(-a*c)*a^2*c^3*d^6*e^5 + 82*sqrt(-a*c)*a^3*c^2*d^4*e

^7 + 13*sqrt(-a*c)*a^4*c*d^2*e^9 - 7*sqrt(-a*c)*a^5*e^11)*abs(-a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 - 3*a^3*c*d^2*e

^5 - a^4*e^7)*abs(c) + (2*a*c^9*d^17*e + 27*a^2*c^8*d^15*e^3 + 113*a^3*c^7*d^13*e^5 + 223*a^4*c^6*d^11*e^7 + 2

25*a^5*c^5*d^9*e^9 + 97*a^6*c^4*d^7*e^11 - 13*a^7*c^3*d^5*e^13 - 27*a^8*c^2*d^3*e^15 - 7*a^9*c*d*e^17)*abs(c))

*arctan(sqrt(e*x + d)/sqrt(-(a*c^4*d^7 + 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6 + sqrt((a*c^4*d^7

+ 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)^2 - (a*c^4*d^8 + 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4

+ 4*a^4*c*d^2*e^6 + a^5*e^8)*(a*c^4*d^6 + 3*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 + a^4*c*e^6)))/(a*c^4*d^6 + 3

*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 + a^4*c*e^6)))/((a^2*c^6*d^12*e + 6*a^3*c^5*d^10*e^3 + 15*a^4*c^4*d^8*e^5

+ 20*a^5*c^3*d^6*e^7 + 15*a^6*c^2*d^4*e^9 + 6*a^7*c*d^2*e^11 + a^8*e^13 - sqrt(-a*c)*a*c^6*d^13 - 6*sqrt(-a*c

)*a^2*c^5*d^11*e^2 - 15*sqrt(-a*c)*a^3*c^4*d^9*e^4 - 20*sqrt(-a*c)*a^4*c^3*d^7*e^6 - 15*sqrt(-a*c)*a^5*c^2*d^5

*e^8 - 6*sqrt(-a*c)*a^6*c*d^3*e^10 - sqrt(-a*c)*a^7*d*e^12)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(-a*c^3*d^6*e - 3

*a^2*c^2*d^4*e^3 - 3*a^3*c*d^2*e^5 - a^4*e^7)) - 1/4*((a*c^3*d^6*e + 3*a^2*c^2*d^4*e^3 + 3*a^3*c*d^2*e^5 + a^4

*e^7)^2*(c^2*d^3*e - 19*a*c*d*e^3)*abs(c) + (sqrt(-a*c)*c^5*d^10*e + 37*sqrt(-a*c)*a*c^4*d^8*e^3 + 98*sqrt(-a*

c)*a^2*c^3*d^6*e^5 + 82*sqrt(-a*c)*a^3*c^2*d^4*e^7 + 13*sqrt(-a*c)*a^4*c*d^2*e^9 - 7*sqrt(-a*c)*a^5*e^11)*abs(

-a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 - 3*a^3*c*d^2*e^5 - a^4*e^7)*abs(c) + (2*a*c^9*d^17*e + 27*a^2*c^8*d^15*e^3 +

113*a^3*c^7*d^13*e^5 + 223*a^4*c^6*d^11*e^7 + 225*a^5*c^5*d^9*e^9 + 97*a^6*c^4*d^7*e^11 - 13*a^7*c^3*d^5*e^13

- 27*a^8*c^2*d^3*e^15 - 7*a^9*c*d*e^17)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a*c^4*d^7 + 3*a^2*c^3*d^5*e^2 + 3

*a^3*c^2*d^3*e^4 + a^4*c*d*e^6 - sqrt((a*c^4*d^7 + 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)^2 - (a

*c^4*d^8 + 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 + 4*a^4*c*d^2*e^6 + a^5*e^8)*(a*c^4*d^6 + 3*a^2*c^3*d^4*e^2 +

3*a^3*c^2*d^2*e^4 + a^4*c*e^6)))/(a*c^4*d^6 + 3*a^2*c^3*d^4*e^2 + 3*a^3*c^2*d^2*e^4 + a^4*c*e^6)))/((a^2*c^6*

d^12*e + 6*a^3*c^5*d^10*e^3 + 15*a^4*c^4*d^8*e^5 + 20*a^5*c^3*d^6*e^7 + 15*a^6*c^2*d^4*e^9 + 6*a^7*c*d^2*e^11

+ a^8*e^13 + sqrt(-a*c)*a*c^6*d^13 + 6*sqrt(-a*c)*a^2*c^5*d^11*e^2 + 15*sqrt(-a*c)*a^3*c^4*d^9*e^4 + 20*sqrt(-

a*c)*a^4*c^3*d^7*e^6 + 15*sqrt(-a*c)*a^5*c^2*d^5*e^8 + 6*sqrt(-a*c)*a^6*c*d^3*e^10 + sqrt(-a*c)*a^7*d*e^12)*sq

rt(-c^2*d - sqrt(-a*c)*c*e)*abs(-a*c^3*d^6*e - 3*a^2*c^2*d^4*e^3 - 3*a^3*c*d^2*e^5 - a^4*e^7)) + 1/2*((e*x + d

)^(3/2)*c^3*d^3*e - sqrt(e*x + d)*c^3*d^4*e - 3*(e*x + d)^(3/2)*a*c^2*d*e^3 + 6*sqrt(e*x + d)*a*c^2*d^2*e^3 -

sqrt(e*x + d)*a^2*c*e^5)/((a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*((e*x + d)^2*c - 2*(e*x

+ d)*c*d + c*d^2 + a*e^2)) - 2/3*(12*(e*x + d)*c*d*e^3 + c*d^2*e^3 + a*e^5)/((c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^

Mupad [B] (veriﬁcation not implemented)

Time = 13.90 (sec) , antiderivative size = 12390, normalized size of antiderivative = 13.32 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]

atan((((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d

^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d

^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13

*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d^2*e^26) + (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^

(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*

(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a

^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*

d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*

a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(

1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14

+ 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21

*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 215040*

a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 + 102

50240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d

^13*e^20 + 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6

*d^5*e^28 + 30720*a^20*c^5*d^3*e^30) - 1792*a^19*c^4*e^31 + 256*a^5*c^18*d^28*e^3 + 11776*a^6*c^17*d^26*e^5 +

119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13

+ 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*

d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^

2*e^29))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4

*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*

a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^

8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*1i + ((d + e*x)^(1/2

)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*

c^15*d^20*e^8 + 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^

10*c^11*d^12*e^16 - 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 10048

0*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d^2*e^26) - (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d

*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 81

9*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*

c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2

*d^4*e^10)))^(1/2)*(256*a^5*c^18*d^28*e^3 - 1792*a^19*c^4*e^31 - (d + e*x)^(1/2)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9

*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*

c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a

^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 +

35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7

*c^18*d^29*e^4 + 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a

^11*c^14*d^21*e^12 + 10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^1

8 + 10250240*a^15*c^10*d^13*e^20 + 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^

7*e^26 + 215040*a^19*c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^30) + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*

e^7 + 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*

d^16*e^15 + 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a

^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(4*a^3*c^6

*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c

^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c

^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a

^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*1i)/(((d + e*x)^(1/2)*(1568*a^16*c^5*e^28

+ 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 28256

0*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1

403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 -

4160*a^15*c^6*d^2*e^26) - (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7

*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9

*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^

6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2

56*a^5*c^18*d^28*e^3 - 1792*a^19*c^4*e^31 - (d + e*x)^(1/2)*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 3

15*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3

)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^

14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 +

21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 2150

40*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 +

10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^1

0*d^13*e^20 + 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a^19*

c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^30) + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15

*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825984*a

^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 18

3296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a

^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*

d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*

e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a

^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2) - ((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*d^26*e^2

+ 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 282560*a^7*c^14*d^18*e^10 + 2

42016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1403904*a^11*c^10*d^10*e^

18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d^2*e^26)

+ (-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e

^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*

d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*

d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*

c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^

6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^

9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 3

5*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*

d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d

^23*e^10 + 6150144*a^11*c^14*d^21*e^12 + 10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 1317888

0*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 + 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24

+ 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^30) - 1792*a^19*c^4*e^31 + 256*a^

5*c^18*d^28*e^3 + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*

c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^17 + 536

4480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6*e^25 -

13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*

c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2)

- 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*

a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^1

1*c^2*d^4*e^10)))^(1/2) + 7448*a^13*c^6*d*e^25 - 32*a^2*c^17*d^23*e^3 - 72*a^3*c^16*d^21*e^5 + 8240*a^4*c^15*d

^19*e^7 + 72120*a^5*c^14*d^17*e^9 + 282240*a^6*c^13*d^15*e^11 + 648816*a^7*c^12*d^13*e^13 + 962976*a^8*c^11*d^

11*e^15 + 955440*a^9*c^10*d^9*e^17 + 633120*a^10*c^9*d^7*e^19 + 270040*a^11*c^8*d^5*e^21 + 67248*a^12*c^7*d^3*

e^23))*(-(4*a^3*c^6*d^9 - 49*a^3*e^9*(-a^9*c^3)^(1/2) + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d

^5*e^4 - 1155*a^6*c^3*d^3*e^6 - 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) - 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) + 837*a^

2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*

c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*2i - ((2*e^3)/(3*(a*e^

2 + c*d^2)) + (20*c*d*e^3*(d + e*x))/(3*(a*e^2 + c*d^2)^2) + (c*e*(d + e*x)^2*(7*a^2*e^4 + 3*c^2*d^4 - 110*a*c

*d^2*e^2))/(6*a*(a*e^2 + c*d^2)^3) + (c^2*d*e*(19*a*e^2 - c*d^2)*(d + e*x)^3)/(2*a*(a*e^2 + c*d^2)^3))/(c*(d +

e*x)^(7/2) + (a*e^2 + c*d^2)*(d + e*x)^(3/2) - 2*c*d*(d + e*x)^(5/2)) + atan((((d + e*x)^(1/2)*(1568*a^16*c^5

*e^28 + 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 +

282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^1

6 - 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e

^24 - 4160*a^15*c^6*d^2*e^26) + (-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^

5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*

(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a

^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/

2)*((d + e*x)^(1/2)*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 +

189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(

1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*

e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^2

1*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d

^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 + 10250240*a^12*c^13*d^19*e^14 + 13178880*

a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 + 6150144*a^16*c^9*d^11*e^22

+ 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^30)

- 1792*a^19*c^4*e^31 + 256*a^5*c^18*d^28*e^3 + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8

*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825

984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23

+ 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(49*a^3*e^9*(-a^9*c^3)^(1/

2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105

*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(

a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 +

35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*1i + ((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*

d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 282560*a^7*c^14*d^18*

e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1403904*a^11*c^10

*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d

^2*e^26) - (-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c

^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 83

7*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*

a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(256*a^5*c^18*d^28

*e^3 - 1792*a^19*c^4*e^31 - (d + e*x)^(1/2)*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8

+ 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*

c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^

2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4

*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 215040*a^8*c^17*d^27

*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 + 10250240*a^12*c^

13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 + 61

50144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 + 3

0720*a^20*c^5*d^3*e^30) + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 + 1923

328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^

17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^

6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 +

315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^

3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d

^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8

+ 21*a^11*c^2*d^4*e^10)))^(1/2)*1i)/(((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*d^26*e^2 + 3040*a^4*c

^17*d^24*e^4 + 29120*a^5*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^1

3*d^16*e^12 - 282240*a^9*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1403904*a^11*c^10*d^10*e^18 - 1049440*

a^12*c^9*d^8*e^20 - 456512*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d^2*e^26) - (-(49*a^3*

e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6

*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9

*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35

*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(256*a^5*c^18*d^28*e^3 - 1792*a^19*c^4*

e^31 - (d + e*x)^(1/2)*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2

+ 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3

)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^

12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*

a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 + 215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^1

6*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^12 + 10250240*a^12*c^13*d^19*e^14 + 131788

80*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15*c^10*d^13*e^20 + 6150144*a^16*c^9*d^11*e

^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a^19*c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^3

0) + 11776*a^6*c^17*d^26*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^1

1 + 4116992*a^10*c^13*d^18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^1

0*d^12*e^19 + 2945536*a^14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c

^6*d^4*e^27 - 12800*a^18*c^5*d^2*e^29))*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 6

3*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*

d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^

12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^1

0)))^(1/2) - ((d + e*x)^(1/2)*(1568*a^16*c^5*e^28 + 128*a^3*c^18*d^26*e^2 + 3040*a^4*c^17*d^24*e^4 + 29120*a^5

*c^16*d^22*e^6 + 128128*a^6*c^15*d^20*e^8 + 282560*a^7*c^14*d^18*e^10 + 242016*a^8*c^13*d^16*e^12 - 282240*a^9

*c^12*d^14*e^14 - 1059840*a^10*c^11*d^12*e^16 - 1403904*a^11*c^10*d^10*e^18 - 1049440*a^12*c^9*d^8*e^20 - 4565

12*a^13*c^8*d^6*e^22 - 100480*a^14*c^7*d^4*e^24 - 4160*a^15*c^6*d^2*e^26) + (-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4

*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d

^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e

^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^

10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*((d + e*x)^(1/2)*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9

+ 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9

*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^

7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e

^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*(2048*a^21*c^4*d*e^32 + 2048*a^6*c^19*d^31*e^2 + 30720*a^7*c^18*d^29*e^4 +

215040*a^8*c^17*d^27*e^6 + 931840*a^9*c^16*d^25*e^8 + 2795520*a^10*c^15*d^23*e^10 + 6150144*a^11*c^14*d^21*e^1

2 + 10250240*a^12*c^13*d^19*e^14 + 13178880*a^13*c^12*d^17*e^16 + 13178880*a^14*c^11*d^15*e^18 + 10250240*a^15

*c^10*d^13*e^20 + 6150144*a^16*c^9*d^11*e^22 + 2795520*a^17*c^8*d^9*e^24 + 931840*a^18*c^7*d^7*e^26 + 215040*a

^19*c^6*d^5*e^28 + 30720*a^20*c^5*d^3*e^30) - 1792*a^19*c^4*e^31 + 256*a^5*c^18*d^28*e^3 + 11776*a^6*c^17*d^26

*e^5 + 119552*a^7*c^16*d^24*e^7 + 609280*a^8*c^15*d^22*e^9 + 1923328*a^9*c^14*d^20*e^11 + 4116992*a^10*c^13*d^

18*e^13 + 6243072*a^11*c^12*d^16*e^15 + 6825984*a^12*c^11*d^14*e^17 + 5364480*a^13*c^10*d^12*e^19 + 2945536*a^

14*c^9*d^10*e^21 + 1044736*a^15*c^8*d^8*e^23 + 183296*a^16*c^7*d^6*e^25 - 13568*a^17*c^6*d^4*e^27 - 12800*a^18

*c^5*d^2*e^29))*(-(49*a^3*e^9*(-a^9*c^3)^(1/2) + 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*

a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2)

- 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2

+ 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 35*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2) + 7448*a^13*c

^6*d*e^25 - 32*a^2*c^17*d^23*e^3 - 72*a^3*c^16*d^21*e^5 + 8240*a^4*c^15*d^19*e^7 + 72120*a^5*c^14*d^17*e^9 + 2

82240*a^6*c^13*d^15*e^11 + 648816*a^7*c^12*d^13*e^13 + 962976*a^8*c^11*d^11*e^15 + 955440*a^9*c^10*d^9*e^17 +

633120*a^10*c^9*d^7*e^19 + 270040*a^11*c^8*d^5*e^21 + 67248*a^12*c^7*d^3*e^23))*(-(49*a^3*e^9*(-a^9*c^3)^(1/2)

+ 4*a^3*c^6*d^9 + 315*a^7*c^2*d*e^8 + 63*a^4*c^5*d^7*e^2 + 189*a^5*c^4*d^5*e^4 - 1155*a^6*c^3*d^3*e^6 + 105*c

^3*d^6*e^3*(-a^9*c^3)^(1/2) + 819*a*c^2*d^4*e^5*(-a^9*c^3)^(1/2) - 837*a^2*c*d^2*e^7*(-a^9*c^3)^(1/2))/(64*(a^

13*e^14 + a^6*c^7*d^14 + 7*a^12*c*d^2*e^12 + 7*a^7*c^6*d^12*e^2 + 21*a^8*c^5*d^10*e^4 + 35*a^9*c^4*d^8*e^6 + 3

5*a^10*c^3*d^6*e^8 + 21*a^11*c^2*d^4*e^10)))^(1/2)*2i

\(\int \frac {1}{(d+e x)^{5/2} (a+c x^2)^2} \, dx\) [637]

Optimal result