3.7.0 ∫ (d+ex)3∕2 ______________

Integrand size = 19, antiderivative size = 769 \[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^3} \, dx=-\frac {(a e-c d x) \sqrt {d+e x}}{4 a c \left (a+c x^2\right )^2}+\frac {(a e+6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a+c x^2\right )}+\frac {3 e \left (2 c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{32 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]

-1/4*(-c*d*x+a*e)*(e*x+d)^(1/2)/a/c/(c*x^2+a)^2+1/16*(6*c*d*x+a*e)*(e*x+d)^(1/2)/a^2/c/(c*x^2+a)+3/64*e*arctan

h((-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)

)*(2*c*d^2+a*e^2+2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(5/4)*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)-3/64*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))*(2*c*d^2+a*e^2+2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(5/4)*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)-3/128*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))*(2*c*d^2+a*e^2-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(5

/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)+3/128*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))*(2*c*d^2+a*e^2-2*d*c^(1/2)*(a*e^2+

c*d^2)^(1/2))/a^2/c^(5/4)*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)

Rubi [A] (veriﬁed)

Time = 1.35 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {753, 837, 841, 1183, 648, 632, 212, 642} \[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^3} \, dx=\frac {3 e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+2 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{32 \sqrt {2} a^2 c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {3 e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+2 c d^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{32 \sqrt {2} a^2 c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {3 e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+2 c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {3 e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+2 c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\sqrt {d+e x} (a e+6 c d x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]

-1/4*((a*e - c*d*x)*Sqrt[d + e*x])/(a*c*(a + c*x^2)^2) + ((a*e + 6*c*d*x)*Sqrt[d + e*x])/(16*a^2*c*(a + c*x^2)

) + (3*e*(2*c*d^2 + a*e^2 + 2*Sqrt[c]*d*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]]])/(32*Sqrt[2]*a^2*c^(5/4)*Sqrt[c*d^2 + a*

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

])/(32*Sqrt[2]*a^2*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c*d^2 + a*e^2

- 2*Sqrt[c]*d*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)])/(64*Sqrt[2]*a^2*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d

^2 + a*e^2]]) + (3*e*(2*c*d^2 + a*e^2 - 2*Sqrt[c]*d*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)])/(64*Sqrt[2]*a^2*c^(5/4)*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)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(

m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +

Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^

2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},

x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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) \sqrt {d+e x}}{4 a c \left (a+c x^2\right )^2}+\frac {\int \frac {\frac {1}{2} \left (6 c d^2+a e^2\right )+\frac {5}{2} c d e x}{\sqrt {d+e x} \left (a+c x^2\right )^2} \, dx}{4 a c} \\ & = -\frac {(a e-c d x) \sqrt {d+e x}}{4 a c \left (a+c x^2\right )^2}+\frac {(a e+6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a+c x^2\right )}-\frac {\int \frac {-\frac {3}{4} c \left (c d^2+a e^2\right ) \left (4 c d^2+a e^2\right )-\frac {3}{2} c^2 d e \left (c d^2+a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{8 a^2 c^2 \left (c d^2+a e^2\right )} \\ & = -\frac {(a e-c d x) \sqrt {d+e x}}{4 a c \left (a+c x^2\right )^2}+\frac {(a e+6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a+c x^2\right )}-\frac {\text {Subst}\left (\int \frac {\frac {3}{2} c^2 d^2 e \left (c d^2+a e^2\right )-\frac {3}{4} c e \left (c d^2+a e^2\right ) \left (4 c d^2+a e^2\right )-\frac {3}{2} c^2 d e \left (c d^2+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 )}{4 a^2 c^2 \left (c d^2+a e^2\right )} \\ & = -\frac {(a e-c d x) \sqrt {d+e x}}{4 a c \left (a+c x^2\right )^2}+\frac {(a e+6 c d x) \sqrt {d+e x}}{16 a^2 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 {3}{2} c^2 d^2 e \left (c d^2+a e^2\right )-\frac {3}{4} c e \left (c d^2+a e^2\right ) \left (4 c d^2+a e^2\right )\right )}{\sqrt [4]{c}}-\left (\frac {3}{2} c^2 d^2 e \left (c d^2+a e^2\right )+\frac {3}{2} c^{3/2} d e \left (c d^2+a e^2\right )^{3/2}-\frac {3}{4} c e \left (c d^2+a e^2\right ) \left (4 c d^2+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 )}{8 \sqrt {2} a^2 c^{9/4} \left (c d^2+a e^2\right )^{3/2} \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 {3}{2} c^2 d^2 e \left (c d^2+a e^2\right )-\frac {3}{4} c e \left (c d^2+a e^2\right ) \left (4 c d^2+a e^2\right )\right )}{\sqrt [4]{c}}+\left (\frac {3}{2} c^2 d^2 e \left (c d^2+a e^2\right )+\frac {3}{2} c^{3/2} d e \left (c d^2+a e^2\right )^{3/2}-\frac {3}{4} c e \left (c d^2+a e^2\right ) \left (4 c d^2+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 )}{8 \sqrt {2} a^2 c^{9/4} \left (c d^2+a e^2\right )^{3/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ & = -\frac {(a e-c d x) \sqrt {d+e x}}{4 a c \left (a+c x^2\right )^2}+\frac {(a e+6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a+c x^2\right )}-\frac {\left (3 e \left (2 c d^2+a e^2-2 \sqrt {c} d \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 )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (3 e \left (2 c d^2+a e^2-2 \sqrt {c} d \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 )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (3 e \left (2 c d^2+a e^2+2 \sqrt {c} d \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 )}{64 a^2 c^{3/2} \sqrt {c d^2+a e^2}}+\frac {\left (3 e \left (2 c d^2+a e^2+2 \sqrt {c} d \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 )}{64 a^2 c^{3/2} \sqrt {c d^2+a e^2}} \\ & = -\frac {(a e-c d x) \sqrt {d+e x}}{4 a c \left (a+c x^2\right )^2}+\frac {(a e+6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a+c x^2\right )}-\frac {3 e \left (2 c d^2+a e^2-2 \sqrt {c} d \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 )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c d^2+a e^2-2 \sqrt {c} d \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 )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (3 e \left (2 c d^2+a e^2+2 \sqrt {c} d \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 )}{32 a^2 c^{3/2} \sqrt {c d^2+a e^2}}-\frac {\left (3 e \left (2 c d^2+a e^2+2 \sqrt {c} d \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 )}{32 a^2 c^{3/2} \sqrt {c d^2+a e^2}} \\ & = -\frac {(a e-c d x) \sqrt {d+e x}}{4 a c \left (a+c x^2\right )^2}+\frac {(a e+6 c d x) \sqrt {d+e x}}{16 a^2 c \left (a+c x^2\right )}+\frac {3 e \left (2 c d^2+a e^2+2 \sqrt {c} d \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 )}{32 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c d^2+a e^2+2 \sqrt {c} d \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 )}{32 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {3 e \left (2 c d^2+a e^2-2 \sqrt {c} d \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 )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {3 e \left (2 c d^2+a e^2-2 \sqrt {c} d \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 )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \\ \end{align*}

Mathematica [C] (veriﬁed)

Time = 2.24 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.37 \[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {d+e x} \left (-3 a^2 e+6 c^2 d x^3+a c x (10 d+e x)\right )}{\left (a+c x^2\right )^2}+\frac {3 i \left (4 c d^2+2 i \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {3 i \left (4 c d^2-2 i \sqrt {a} \sqrt {c} d e+a e^2\right ) \arctan \left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{32 a^{5/2} c} \]

((2*Sqrt[a]*Sqrt[d + e*x]*(-3*a^2*e + 6*c^2*d*x^3 + a*c*x*(10*d + e*x)))/(a + c*x^2)^2 + ((3*I)*(4*c*d^2 + (2*

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

Maple [A] (veriﬁed)

Time = 3.28 (sec) , antiderivative size = 1080, normalized size of antiderivative = 1.40


method	result	size

pseudoelliptic	\(\text {Expression too large to display}\)	\(1080\)
derivativedivides	\(\text {Expression too large to display}\)	\(2645\)
default	\(\text {Expression too large to display}\)	\(2645\)

int((e*x+d)^(3/2)/(c*x^2+a)^3,x,method=_RETURNVERBOSE)

3/32/(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^(5/2)*(-1/4*((-2*c*d*(c*x^2+a)^2*

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

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

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

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

a)^2*(a*e^2+c*d^2)^(1/2)+2*a^2*(e^2*x^2+d^2)*c^(5/2)+(e^2*x^4+4*d^2*x^2)*a*c^(7/2)+2*c^(9/2)*d^2*x^4+c^(3/2)*a

^3*e^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)*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))+(-2*(e*

x+d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(a^2*c^(3/2)*e-10/3*x*(a*(1/10*e*x+d)*c^(5/2)+3/5*c^(7/2)*d*x^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)+(2*c^2*d*(c*x^2+a)^2*(a*e^2+c*d^2)^(1/2)+2*a^2*(e^2*x

^2+d^2)*c^(5/2)+(e^2*x^4+4*d^2*x^2)*a*c^(7/2)+2*c^(9/2)*d^2*x^4+c^(3/2)*a^3*e^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)*e*a)/(a*e^2+c*d^2)^(1/2)/e/a^3/(c*x^2+a)^2

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1752 vs. \(2 (623) = 1246\).

Time = 0.39 (sec) , antiderivative size = 1752, normalized size of antiderivative = 2.28 \[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]

integrate((e*x+d)^(3/2)/(c*x^2+a)^3,x, algorithm="fricas")

1/64*(3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2

+ a^6*c^2*e^2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))*log(2

7*(16*c^2*d^4*e^5 + 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d^2*e^6 + a^4*c*e^8 + (4*a^5*c^6*d

^5 + 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt(-(

16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 + a^6*c^2*e^2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^

2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))) - 3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(16*c^2*d

^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 + a^6*c^2*e^2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 +

a^7*c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 + 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) -

27*(2*a^3*c^2*d^2*e^6 + a^4*c*e^8 + (4*a^5*c^6*d^5 + 7*a^6*c^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*sqrt(-e^10/(a^5*c^7

*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt(-(16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4 + (a^5*c^3*d^2 + a^

6*c^2*e^2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))) + 3*(a^2

*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 + a^6*c^2*e^

2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))*log(27*(16*c^2*d^

4*e^5 + 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d^2*e^6 + a^4*c*e^8 - (4*a^5*c^6*d^5 + 7*a^6*c

^5*d^3*e^2 + 3*a^7*c^4*d*e^4)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt(-(16*c^2*d^5 +

20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 + a^6*c^2*e^2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*

c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))) - 3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(16*c^2*d^5 + 20*a*c*

d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 + a^6*c^2*e^2)*sqrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)

))/(a^5*c^3*d^2 + a^6*c^2*e^2))*log(27*(16*c^2*d^4*e^5 + 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) - 27*(2*a^3*c

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

*c^6*d^2*e^2 + a^7*c^5*e^4)))*sqrt(-(16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4 - (a^5*c^3*d^2 + a^6*c^2*e^2)*s

qrt(-e^10/(a^5*c^7*d^4 + 2*a^6*c^6*d^2*e^2 + a^7*c^5*e^4)))/(a^5*c^3*d^2 + a^6*c^2*e^2))) + 4*(6*c^2*d*x^3 + a

*c*e*x^2 + 10*a*c*d*x - 3*a^2*e)*sqrt(e*x + d))/(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^3} \, dx=\text {Timed out} \]

Maxima [F]

\[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^3} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{3}} \,d x } \]

integrate((e*x+d)^(3/2)/(c*x^2+a)^3,x, algorithm="maxima")

integrate((e*x + d)^(3/2)/(c*x^2 + a)^3, x)

Giac [A] (veriﬁcation not implemented)

Time = 0.43 (sec) , antiderivative size = 502, normalized size of antiderivative = 0.65 \[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, {\left (4 \, c^{3} d^{3} e + 3 \, a c^{2} d e^{3} - {\left (2 \, \sqrt {-a c} c d^{2} e + \sqrt {-a c} a e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d + \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{2} e + \sqrt {-a c} a^{2} c^{2} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | e \right |}} + \frac {3 \, {\left (4 \, c^{3} d^{3} e + 3 \, a c^{2} d e^{3} + {\left (2 \, \sqrt {-a c} c d^{2} e + \sqrt {-a c} a e^{3}\right )} {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d - \sqrt {a^{4} c^{4} d^{2} - {\left (a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} a^{2} c^{2}}}{a^{2} c^{2}}}}\right )}{32 \, {\left (a^{3} c^{2} e - \sqrt {-a c} a^{2} c^{2} d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | e \right |}} + \frac {6 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} d e - 18 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d^{2} e + 18 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{3} e - 6 \, \sqrt {e x + d} c^{2} d^{4} e + {\left (e x + d\right )}^{\frac {5}{2}} a c e^{3} + 8 \, {\left (e x + d\right )}^{\frac {3}{2}} a c d e^{3} - 9 \, \sqrt {e x + d} a c d^{2} e^{3} - 3 \, \sqrt {e x + d} a^{2} e^{5}}{16 \, {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + a e^{2}\right )}^{2} a^{2} c} \]

integrate((e*x+d)^(3/2)/(c*x^2+a)^3,x, algorithm="giac")

3/32*(4*c^3*d^3*e + 3*a*c^2*d*e^3 - (2*sqrt(-a*c)*c*d^2*e + sqrt(-a*c)*a*e^3)*abs(c)*abs(e))*arctan(sqrt(e*x +

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

a*c)*a^2*c^2*d)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(e)) + 3/32*(4*c^3*d^3*e + 3*a*c^2*d*e^3 + (2*sqrt(-a*c)*c*d^

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

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

+ 1/16*(6*(e*x + d)^(7/2)*c^2*d*e - 18*(e*x + d)^(5/2)*c^2*d^2*e + 18*(e*x + d)^(3/2)*c^2*d^3*e - 6*sqrt(e*x +

d)*c^2*d^4*e + (e*x + d)^(5/2)*a*c*e^3 + 8*(e*x + d)^(3/2)*a*c*d*e^3 - 9*sqrt(e*x + d)*a*c*d^2*e^3 - 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)^2*a^2*c)

Mupad [B] (veriﬁcation not implemented)

Time = 12.30 (sec) , antiderivative size = 3204, normalized size of antiderivative = 4.17 \[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^3} \, dx=\text {Too large to display} \]

(((4*a*d*e^3 + 9*c*d^3*e)*(d + e*x)^(3/2))/(8*a^2) + (e*(a*e^2 - 18*c*d^2)*(d + e*x)^(5/2))/(16*a^2) - (3*(d +

e*x)^(1/2)*(a^2*e^5 + 2*c^2*d^4*e + 3*a*c*d^2*e^3))/(16*a^2*c) + (3*c*d*e*(d + e*x)^(7/2))/(8*a^2))/(c^2*(d +

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

2*d*(d + e*x)^3 + 2*a*c*d^2*e^2) + atan(((((3*(2048*a^6*c^2*e^5 + 4096*a^5*c^3*d^2*e^3))/(2048*a^6) - 64*a*c^4

*d*e^2*(d + e*x)^(1/2)*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(

4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 + 2

0*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*

e^2 + 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 + 20*a^6*c^4

*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2)*1i - (((3*(2048*a^6*c^2*e^5 + 4096*a^5*c^3*d^2*e^3))/(2

048*a^6) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 + 20*

a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 +

5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*a^2*c

*e^6 + 144*c^3*d^4*e^2 + 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3

*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2)*1i)/((3*(9*a^2*d*e^7 + 144*c^2*d^5*e

^3 + 108*a*c*d^3*e^5))/(1024*a^6) + (((3*(2048*a^6*c^2*e^5 + 4096*a^5*c^3*d^2*e^3))/(2048*a^6) - 64*a*c^4*d*e^

2*(d + e*x)^(1/2)*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*

(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 + 20*a^6

*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 +

36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*

e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2) + (((3*(2048*a^6*c^2*e^5 + 4096*a^5*c^3*d^2*e^3))/(2048*a^6)

+ 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*

d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2))*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^

3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 1

44*c^3*d^4*e^2 + 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d^5 + 5*a^7*c^3*d*e^4 +

20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2)))*(-(9*(e^5*(-a^15*c^5)^(1/2) + 16*a^5*c^5*d

^5 + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2)*2i + atan(((((3*(2048*

a^6*c^2*e^5 + 4096*a^5*c^3*d^2*e^3))/(2048*a^6) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^5 - e^5*(-

a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2))*(-(9*(16

*a^5*c^5*d^5 - e^5*(-a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e

^2)))^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 + 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(16*a^5*c^5*

d^5 - e^5*(-a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/

2)*1i - (((3*(2048*a^6*c^2*e^5 + 4096*a^5*c^3*d^2*e^3))/(2048*a^6) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(16*a

^5*c^5*d^5 - e^5*(-a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2

)))^(1/2))*(-(9*(16*a^5*c^5*d^5 - e^5*(-a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c

^6*d^2 + a^11*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 + 36*a*c^2*d^2*e^4))/(64*a^4)

)*(-(9*(16*a^5*c^5*d^5 - e^5*(-a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 +

a^11*c^5*e^2)))^(1/2)*1i)/((3*(9*a^2*d*e^7 + 144*c^2*d^5*e^3 + 108*a*c*d^3*e^5))/(1024*a^6) + (((3*(2048*a^6*c

^2*e^5 + 4096*a^5*c^3*d^2*e^3))/(2048*a^6) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^5 - e^5*(-a^15*

c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2))*(-(9*(16*a^5*

c^5*d^5 - e^5*(-a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))

^(1/2) - ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 + 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(16*a^5*c^5*d^5 -

e^5*(-a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2) +

(((3*(2048*a^6*c^2*e^5 + 4096*a^5*c^3*d^2*e^3))/(2048*a^6) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d

^5 - e^5*(-a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5*e^2)))^(1/2

))*(-(9*(16*a^5*c^5*d^5 - e^5*(-a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 +

a^11*c^5*e^2)))^(1/2) + ((d + e*x)^(1/2)*(9*a^2*c*e^6 + 144*c^3*d^4*e^2 + 36*a*c^2*d^2*e^4))/(64*a^4))*(-(9*(

16*a^5*c^5*d^5 - e^5*(-a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a^10*c^6*d^2 + a^11*c^5

*e^2)))^(1/2)))*(-(9*(16*a^5*c^5*d^5 - e^5*(-a^15*c^5)^(1/2) + 5*a^7*c^3*d*e^4 + 20*a^6*c^4*d^3*e^2))/(4096*(a

\(\int \frac {(d+e x)^{3/2}}{(a+c x^2)^3} \, dx\) [645]

Optimal result