3.7.0 ∫ (d+ex)7∕2 ______________

Integrand size = 19, antiderivative size = 887 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=-\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

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

/c^2+1/8*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^4-4*a*c*d^2*e^2-5*a^2*e^4+d*(13*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/c^(9/4)*

2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/8*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^4-4*a*c*d^2*e^2-5*a^2*e^

4+d*(13*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/c^(9/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)-(a*e^2+c*d^

2)^(1/2))^(1/2)-1/16*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^4-4*a*c*d^2*e^2-5*a^2*e^4-d*(13*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/c^(9/4

)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)+1/16*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^4-4*a*c*d^2*e^2-5*a^2*e^4-d*(1

3*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a/c^(9/4)*2^(1/2)/(a*e^2+c*d^2)^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/

Rubi [A] (veriﬁed)

Time = 3.76 (sec) , antiderivative size = 887, 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 = {753, 839, 841, 1183, 648, 632, 212, 642} \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (c x^2+a\right )}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}+\frac {e \left (c^2 d^4-4 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c e^2 d^2+\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c e^2 d^2-\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^2 d^4-4 a c e^2 d^2-\sqrt {c} \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right ) d-5 a^2 e^4\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

-1/2*(e*(c*d^2 - 5*a*e^2)*Sqrt[d + e*x])/(a*c^2) - (d*e*(d + e*x)^(3/2))/(2*a*c) - ((a*e - c*d*x)*(d + e*x)^(5

/2))/(2*a*c*(a + c*x^2)) + (e*(c^2*d^4 - 4*a*c*d^2*e^2 - 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13

*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^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^2*d

^4 - 4*a*c*d^2*e^2 - 5*a^2*e^4 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*a*e^2))*ArcTanh[(Sqrt[Sqrt[c]*d + S

qrt[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^(9

/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c^2*d^4 - 4*a*c*d^2*e^2 - 5*a^2*e^4 - Sqr

t[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*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^(9/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d +

Sqrt[c*d^2 + a*e^2]]) + (e*(c^2*d^4 - 4*a*c*d^2*e^2 - 5*a^2*e^4 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 13*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^(9/4)*Sqrt[c*d^2 + a*e^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*c*(p + 1))), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -

c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^

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

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

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

; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]

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_)^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) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (2 c d^2+5 a e^2\right )-\frac {3}{2} c d e x\right )}{a+c x^2} \, dx}{2 a c} \\ & = -\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {\sqrt {d+e x} \left (c d \left (c d^2+4 a e^2\right )-\frac {1}{2} c e \left (c d^2-5 a e^2\right ) x\right )}{a+c x^2} \, dx}{2 a c^2} \\ & = -\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {\frac {1}{2} c \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )+\frac {1}{2} c^2 d e \left (c d^2+13 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{2 a c^3} \\ & = -\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} c e \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )-\frac {1}{2} c^2 d^2 e \left (c d^2+13 a e^2\right )+\frac {1}{2} c^2 d e \left (c d^2+13 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 c^3} \\ & = -\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \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 e \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )-\frac {1}{2} c^2 d^2 e \left (c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}-\left (\frac {1}{2} c e \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )-\frac {1}{2} c^2 d^2 e \left (c d^2+13 a e^2\right )-\frac {1}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\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 c^{13/4} \sqrt {c d^2+a e^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 e \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )-\frac {1}{2} c^2 d^2 e \left (c d^2+13 a e^2\right )\right )}{\sqrt [4]{c}}+\left (\frac {1}{2} c e \left (2 c d^2-a e^2\right ) \left (c d^2+5 a e^2\right )-\frac {1}{2} c^2 d^2 e \left (c d^2+13 a e^2\right )-\frac {1}{2} c^{3/2} d e \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\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 c^{13/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = -\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}-\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{5/2} \sqrt {c d^2+a e^2}}+\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{5/2} \sqrt {c d^2+a e^2}} \\ & = -\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}-\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{5/2} \sqrt {c d^2+a e^2}}-\frac {\left (e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{5/2} \sqrt {c d^2+a e^2}} \\ & = -\frac {e \left (c d^2-5 a e^2\right ) \sqrt {d+e x}}{2 a c^2}-\frac {d e (d+e x)^{3/2}}{2 a c}-\frac {(a e-c d x) (d+e x)^{5/2}}{2 a c \left (a+c x^2\right )}+\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4+\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c^2 d^4-4 a c d^2 e^2-5 a^2 e^4-\sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+13 a e^2\right )\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 c^{9/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ \end{align*}

Mathematica [C] (veriﬁed)

Time = 1.75 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.36 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {d+e x} \left (5 a^2 e^3+c^2 d^3 x+a c e \left (-3 d^2-3 d e x+4 e^2 x^2\right )\right )}{a+c x^2}+\frac {\left (\sqrt {c} d+i \sqrt {a} e\right )^3 \left (2 i \sqrt {c} d+5 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-c d-i \sqrt {a} \sqrt {c} e}}+\frac {\left (\sqrt {c} d-i \sqrt {a} e\right )^3 \left (-2 i \sqrt {c} d+5 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{4 a^{3/2} c^2} \]

((2*Sqrt[a]*Sqrt[d + e*x]*(5*a^2*e^3 + c^2*d^3*x + a*c*e*(-3*d^2 - 3*d*e*x + 4*e^2*x^2)))/(a + c*x^2) + ((Sqrt

[c]*d + I*Sqrt[a]*e)^3*((2*I)*Sqrt[c]*d + 5*Sqrt[a]*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]*Sqrt[c]*e] + ((Sqrt[c]*d - I*Sqrt[a]*e)^3*((-2*I)*Sqrt[

c]*d + 5*Sqrt[a]*e)*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/Sqrt

Maple [A] (veriﬁed)

Time = 3.24 (sec) , antiderivative size = 1015, normalized size of antiderivative = 1.14


method	result	size

pseudoelliptic	\(\text {Expression too large to display}\)	\(1015\)
derivativedivides	\(\text {Expression too large to display}\)	\(2224\)
default	\(\text {Expression too large to display}\)	\(2224\)
risch	\(\text {Expression too large to display}\)	\(2226\)

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)/c^(11/2)*(1/4*(((-x^2*c^(9/2)*d^2-a*

(-5*e^2*x^2+d^2)*c^(7/2)+5*c^(5/2)*a^2*e^2)*(a*e^2+c*d^2)^(1/2)+13*c^3*(e^2*a+1/13*c*d^2)*d*(c*x^2+a))*((a*e^2

+c*d^2)*c)^(1/2)+((a*(-5*e^2*x^2+d^2)*c^(9/2)+c^(11/2)*d^2*x^2-5*a^2*e^2*c^(7/2))*(a*e^2+c*d^2)^(1/2)-13*c^4*(

e^2*a+1/13*c*d^2)*d*(c*x^2+a))*d)*(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)*(2*((a

*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)*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*(((-x^2*c^(9/2)*d^2-a*(-5*e^2*x^2+d^2)*c^(7/2)+5*c^(5/2)*a^2*e^2)*(a*e^2+c*d^2)^(1/2)+13

*c^3*(e^2*a+1/13*c*d^2)*d*(c*x^2+a))*((a*e^2+c*d^2)*c)^(1/2)+((a*(-5*e^2*x^2+d^2)*c^(9/2)+c^(11/2)*d^2*x^2-5*a

^2*e^2*c^(7/2))*(a*e^2+c*d^2)^(1/2)-13*c^4*(e^2*a+1/13*c*d^2)*d*(c*x^2+a))*d)*(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)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)*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))+e*(2*(-3*(-4/3*x^2*e^2+d*e*x+d^2)*e*a*c^(9/2)+c^

(11/2)*d^3*x+5*a^2*e^3*c^(7/2))*(e*x+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))-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*((a*(-5*e^2*x^2+d^2)*c^(9/2)+c^

(11/2)*d^2*x^2-5*a^2*e^2*c^(7/2))*(a*e^2+c*d^2)^(1/2)+13*c^4*(e^2*a+1/13*c*d^2)*d*(c*x^2+a)))*a)/e/(c*x^2+a)/a

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2091 vs. \(2 (737) = 1474\).

Time = 0.47 (sec) , antiderivative size = 2091, normalized size of antiderivative = 2.36 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]

integrate((e*x+d)^(7/2)/(c*x^2+a)^2,x, algorithm="fricas")

-1/8*((a*c^3*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*s

qrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a

^3*c^9)))/(a^3*c^4))*log(-(140*c^5*d^10*e^3 + 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 + 8366*a^3*c^2*d^4*e^9

+ 2500*a^4*c*d^2*e^11 - 625*a^5*e^13)*sqrt(e*x + d) + (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*

d^2*e^8 + 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 + 4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 +

21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2

+ 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4

*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - (a*c^3*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 +

35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 +

21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log(-(140*c^5*d^10*e^3 +

1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 + 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 - 625*a^5*e^13)*sqrt(e*

x + d) - (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 - 2*(a^3*c^8*d^3 +

4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12

+ 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 + a^3*c^4*s

qrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a

^3*c^9)))/(a^3*c^4))) + (a*c^3*x^2 + a^2*c^2)*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3

*d*e^6 - a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12

+ 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log(-(140*c^5*d^10*e^3 + 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 + 83

66*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 - 625*a^5*e^13)*sqrt(e*x + d) + (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*

e^6 - 795*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 + 2*(a^3*c^8*d^3 + 4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6 + 107

80*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 +

35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 +

21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - (a*c^3*x^2 + a^2*c^2)*

sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3*d*e^6 - a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 107

80*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))*log(-(1

40*c^5*d^10*e^3 + 1771*a*c^4*d^8*e^5 + 6872*a^2*c^3*d^6*e^7 + 8366*a^3*c^2*d^4*e^9 + 2500*a^4*c*d^2*e^11 - 625

*a^5*e^13)*sqrt(e*x + d) - (35*a^2*c^5*d^6*e^4 - 21*a^3*c^4*d^4*e^6 - 795*a^4*c^3*d^2*e^8 + 125*a^5*c^2*e^10 +

2*(a^3*c^8*d^3 + 4*a^4*c^7*d*e^2)*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 77

00*a^3*c*d^2*e^12 + 625*a^4*e^14)/(a^3*c^9)))*sqrt(-(4*c^3*d^7 + 35*a*c^2*d^5*e^2 + 70*a^2*c*d^3*e^4 - 105*a^3

*d*e^6 - a^3*c^4*sqrt(-(1225*c^4*d^8*e^6 + 10780*a*c^3*d^6*e^8 + 21966*a^2*c^2*d^4*e^10 - 7700*a^3*c*d^2*e^12

+ 625*a^4*e^14)/(a^3*c^9)))/(a^3*c^4))) - 4*(4*a*c*e^3*x^2 - 3*a*c*d^2*e + 5*a^2*e^3 + (c^2*d^3 - 3*a*c*d*e^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\text {Timed out} \]

Maxima [F]

\[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + a\right )}^{2}} \,d x } \]

integrate((e*x+d)^(7/2)/(c*x^2+a)^2,x, algorithm="maxima")

Giac [A] (veriﬁcation not implemented)

Time = 0.38 (sec) , antiderivative size = 596, normalized size of antiderivative = 0.67 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\frac {2 \, \sqrt {e x + d} e^{3}}{c^{2}} + \frac {{\left ({\left (c^{2} d^{3} e + 13 \, a c d e^{3}\right )} a^{2} e^{2} {\left | c \right |} - {\left (\sqrt {-a c} c^{2} d^{4} e - 4 \, \sqrt {-a c} a c d^{2} e^{3} - 5 \, \sqrt {-a c} a^{2} e^{5}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |} + {\left (2 \, a c^{3} d^{5} e + 9 \, a^{2} c^{2} d^{3} e^{3} - 5 \, a^{3} c d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{3} d + \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} e + \sqrt {-a c} a c^{3} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | a \right |} {\left | e \right |}} + \frac {{\left ({\left (\sqrt {-a c} c d^{3} e + 13 \, \sqrt {-a c} a d e^{3}\right )} a^{2} e^{2} {\left | c \right |} - {\left (a c^{2} d^{4} e - 4 \, a^{2} c d^{2} e^{3} - 5 \, a^{3} e^{5}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |} + {\left (2 \, \sqrt {-a c} a c^{2} d^{5} e + 9 \, \sqrt {-a c} a^{2} c d^{3} e^{3} - 5 \, \sqrt {-a c} a^{3} d e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a c^{3} d - \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} d + \sqrt {-a c} a^{2} c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | a \right |} {\left | e \right |}} + \frac {{\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{3} e - \sqrt {e x + d} c^{2} d^{4} e - 3 \, {\left (e x + d\right )}^{\frac {3}{2}} a c d e^{3} + \sqrt {e x + d} a^{2} e^{5}}{2 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + a e^{2}\right )} a c^{2}} \]

integrate((e*x+d)^(7/2)/(c*x^2+a)^2,x, algorithm="giac")

2*sqrt(e*x + d)*e^3/c^2 + 1/4*((c^2*d^3*e + 13*a*c*d*e^3)*a^2*e^2*abs(c) - (sqrt(-a*c)*c^2*d^4*e - 4*sqrt(-a*c

)*a*c*d^2*e^3 - 5*sqrt(-a*c)*a^2*e^5)*abs(a)*abs(c)*abs(e) + (2*a*c^3*d^5*e + 9*a^2*c^2*d^3*e^3 - 5*a^3*c*d*e^

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

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

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

sqrt(-a*c)*a*c^2*d^5*e + 9*sqrt(-a*c)*a^2*c*d^3*e^3 - 5*sqrt(-a*c)*a^3*d*e^5)*abs(c))*arctan(sqrt(e*x + d)/sqr

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

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

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

Mupad [B] (veriﬁcation not implemented)

Time = 0.89 (sec) , antiderivative size = 4192, normalized size of antiderivative = 4.73 \[ \int \frac {(d+e x)^{7/2}}{\left (a+c x^2\right )^2} \, dx=\text {Too large to display} \]

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

e*x)^2 + c^3*d^2 + a*c^2*e^2 - 2*c^3*d*(d + e*x)) - atan((a^2*e^10*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7

/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) + (

77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((885*d^5*e

^9)/2 + (491*a*d^3*e^11)/(2*c) + (329*c*d^7*e^7)/(2*a) - (50*a^2*d*e^13)/c^2 + (35*c^2*d^9*e^5)/(2*a^2) + (125

*e^14*(-a^9*c^9)^(1/2))/(4*a^2*c^7) - (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*a^3*c^6) - (204*d^4*e^10*(-a^9*c^9)^(

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

*e^7*(-a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d

^5*e^2)/(64*a^2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (

35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((35*a^2*c^5*d^9*e^5)/2 + (329*a^3*c^4*d^7*e^7)/2 + (88

5*a^4*c^3*d^5*e^9)/2 + (491*a^5*c^2*d^3*e^11)/2 - 50*a^6*c*d*e^13 + (125*a^2*e^14*(-a^9*c^9)^(1/2))/(4*c^4) +

(35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^2) - (204*d^4*e^10*(-a^9*c^9)^(1/2))/c^2 - (335*a*d^2*e^12*(-a^9*c^9)^(1/2)

)/(2*c^3) + (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a*c)) + (d^5*e^5*(-a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64

*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a

^4*c^9) + (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/

((491*a^6*c*d^3*e^11)/2 - 50*a^7*d*e^13 + (35*a^3*c^4*d^9*e^5)/2 + (329*a^4*c^3*d^7*e^7)/2 + (885*a^5*c^2*d^5*

e^9)/2 + (125*a^3*e^14*(-a^9*c^9)^(1/2))/(4*c^5) + (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*c^2) - (204*a*d^4*e^10*(-a^

9*c^9)^(1/2))/c^3 + (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a*c) - (335*a^2*d^2*e^12*(-a^9*c^9)^(1/2))/(2*c^4)) - (a*

d^2*e^8*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^

2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(-a

^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((329*d^7*e^7)/(2*a) + (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2)

+ (35*c*d^9*e^5)/(2*a^2) - (50*a^2*d*e^13)/c^3 + (125*e^14*(-a^9*c^9)^(1/2))/(4*a^2*c^8) - (335*d^2*e^12*(-a^

9*c^9)^(1/2))/(2*a^3*c^7) - (204*d^4*e^10*(-a^9*c^9)^(1/2))/(a^4*c^6) + (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a^5*c^

5) + (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^6*c^4)) - (c*d^4*e^6*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a

^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2

*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/((329*d^7*e^7)/(2

*a) + (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2) + (35*c*d^9*e^5)/(2*a^2) - (50*a^2*d*e^13)/c^3 + (125*e^1

4*(-a^9*c^9)^(1/2))/(4*a^2*c^8) - (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*a^3*c^7) - (204*d^4*e^10*(-a^9*c^9)^(1/2)

)/(a^4*c^6) + (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a^5*c^5) + (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^6*c^4)) - (d*e^9*(

-a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2

)/(64*a^2*c^2) - (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) + (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) + (35*d^4

*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*50i)/((35*a*c^6*d^9*e^5)/2 - 50*a^5*c^2*d*e^13 + (329*a^2*c^5*d^7*e

^7)/2 + (885*a^3*c^4*d^5*e^9)/2 + (491*a^4*c^3*d^3*e^11)/2 + (125*a*e^14*(-a^9*c^9)^(1/2))/(4*c^3) + (7*d^6*e^

8*(-a^9*c^9)^(1/2))/(2*a^2) - (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*c^2) + (35*c*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^3

) - (204*d^4*e^10*(-a^9*c^9)^(1/2))/(a*c)))*(-(25*a^2*e^7*(-a^9*c^9)^(1/2) + 4*a^3*c^8*d^7 - 105*a^6*c^5*d*e^6

+ 35*a^4*c^7*d^5*e^2 + 70*a^5*c^6*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c^9)^(1/2) - 154*a*c*d^2*e^5*(-a^9*c^9)^(1/2

))/(64*a^6*c^9))^(1/2)*2i + atan((d^3*e^7*(-a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3

*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e

^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*308i)/((35*a^2*c^5*d^9*e

^5)/2 + (329*a^3*c^4*d^7*e^7)/2 + (885*a^4*c^3*d^5*e^9)/2 + (491*a^5*c^2*d^3*e^11)/2 - 50*a^6*c*d*e^13 - (125*

a^2*e^14*(-a^9*c^9)^(1/2))/(4*c^4) - (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^2) + (204*d^4*e^10*(-a^9*c^9)^(1/2))/c

^2 + (335*a*d^2*e^12*(-a^9*c^9)^(1/2))/(2*c^3) - (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a*c)) - (a^2*e^10*(d + e*x)^(

1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(-

a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*

a^6*c^7))^(1/2)*50i)/((885*d^5*e^9)/2 + (491*a*d^3*e^11)/(2*c) + (329*c*d^7*e^7)/(2*a) - (50*a^2*d*e^13)/c^2 +

(35*c^2*d^9*e^5)/(2*a^2) - (125*e^14*(-a^9*c^9)^(1/2))/(4*a^2*c^7) + (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*a^3*c

^6) + (204*d^4*e^10*(-a^9*c^9)^(1/2))/(a^4*c^5) - (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a^5*c^4) - (35*d^8*e^6*(-a^9

*c^9)^(1/2))/(4*a^6*c^3)) + (d^5*e^5*(-a^9*c^9)^(1/2)*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) -

(35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(-

a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/((491*a^6*c*d^3*e^11)/2

- 50*a^7*d*e^13 + (35*a^3*c^4*d^9*e^5)/2 + (329*a^4*c^3*d^7*e^7)/2 + (885*a^5*c^2*d^5*e^9)/2 - (125*a^3*e^14*(

-a^9*c^9)^(1/2))/(4*c^5) - (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*c^2) + (204*a*d^4*e^10*(-a^9*c^9)^(1/2))/c^3 - (35*

d^8*e^6*(-a^9*c^9)^(1/2))/(4*a*c) + (335*a^2*d^2*e^12*(-a^9*c^9)^(1/2))/(2*c^4)) + (a*d^2*e^8*(d + e*x)^(1/2)*

((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(-a^9*c

^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c

^7))^(1/2)*308i)/((329*d^7*e^7)/(2*a) + (885*d^5*e^9)/(2*c) + (491*a*d^3*e^11)/(2*c^2) + (35*c*d^9*e^5)/(2*a^2

) - (50*a^2*d*e^13)/c^3 - (125*e^14*(-a^9*c^9)^(1/2))/(4*a^2*c^8) + (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*a^3*c^7

) + (204*d^4*e^10*(-a^9*c^9)^(1/2))/(a^4*c^6) - (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a^5*c^5) - (35*d^8*e^6*(-a^9*c

^9)^(1/2))/(4*a^6*c^4)) + (c*d^4*e^6*(d + e*x)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32

*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7*(-a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(-a^9*c^9)^(1/2))/(

32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(64*a^6*c^7))^(1/2)*70i)/((329*d^7*e^7)/(2*a) + (885*d^5*e^9)/(2*c

) + (491*a*d^3*e^11)/(2*c^2) + (35*c*d^9*e^5)/(2*a^2) - (50*a^2*d*e^13)/c^3 - (125*e^14*(-a^9*c^9)^(1/2))/(4*a

^2*c^8) + (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*a^3*c^7) + (204*d^4*e^10*(-a^9*c^9)^(1/2))/(a^4*c^6) - (7*d^6*e^8

*(-a^9*c^9)^(1/2))/(2*a^5*c^5) - (35*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^6*c^4)) - (d*e^9*(-a^9*c^9)^(1/2)*(d + e*x

)^(1/2)*((105*d*e^6)/(64*c^4) - d^7/(16*a^3*c) - (35*d^3*e^4)/(32*a*c^3) - (35*d^5*e^2)/(64*a^2*c^2) + (25*e^7

*(-a^9*c^9)^(1/2))/(64*a^4*c^9) - (77*d^2*e^5*(-a^9*c^9)^(1/2))/(32*a^5*c^8) - (35*d^4*e^3*(-a^9*c^9)^(1/2))/(

64*a^6*c^7))^(1/2)*50i)/((35*a*c^6*d^9*e^5)/2 - 50*a^5*c^2*d*e^13 + (329*a^2*c^5*d^7*e^7)/2 + (885*a^3*c^4*d^5

*e^9)/2 + (491*a^4*c^3*d^3*e^11)/2 - (125*a*e^14*(-a^9*c^9)^(1/2))/(4*c^3) - (7*d^6*e^8*(-a^9*c^9)^(1/2))/(2*a

^2) + (335*d^2*e^12*(-a^9*c^9)^(1/2))/(2*c^2) - (35*c*d^8*e^6*(-a^9*c^9)^(1/2))/(4*a^3) + (204*d^4*e^10*(-a^9*

c^9)^(1/2))/(a*c)))*(-(4*a^3*c^8*d^7 - 25*a^2*e^7*(-a^9*c^9)^(1/2) - 105*a^6*c^5*d*e^6 + 35*a^4*c^7*d^5*e^2 +

70*a^5*c^6*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c^9)^(1/2) + 154*a*c*d^2*e^5*(-a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2)*2

\(\int \frac {(d+e x)^{7/2}}{(a+c x^2)^2} \, dx\) [631]

Optimal result