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

Optimal. Leaf size=769 \[ -\frac {3 e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+2 c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {3 e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+2 c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{64 \sqrt {2} a^2 c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {3 e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{32 \sqrt {2} a^2 c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {3 e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{32 \sqrt {2} a^2 c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}+\frac {\sqrt {d+e x} (a e+6 c d x)}{16 a^2 c \left (a+c x^2\right )}-\frac {\sqrt {d+e x} (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]

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Rubi [A]  time = 1.94, antiderivative size = 769, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {739, 823, 827, 1169, 634, 618, 206, 628} \begin {gather*} -\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 {3 e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+2 c d^2\right ) \tanh ^{-1}\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 ) \tanh ^{-1}\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 {\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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(a + c*x^2)^3,x]

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(4*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]] - Sq
rt[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]])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

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]

Rule 628

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]

Rule 634

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]

Rule 739

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]

Rule 823

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] + Di
st[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])

Rule 827

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]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1169

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
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \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 {\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 {\operatorname {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 {\operatorname {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 {\operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}

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Mathematica [A]  time = 1.16, size = 430, normalized size = 0.56 \begin {gather*} \frac {\frac {2 (d+e x)^{5/2} \left (3 a^2 e^3+a c d e (5 d+4 e x)+6 c^2 d^3 x\right )}{a+c x^2}+\frac {-2 \sqrt {-a} \sqrt [4]{c} e \sqrt {d+e x} \left (3 a^2 e^4+a c d e^2 (13 d+4 e x)+6 c^2 d^3 (2 d+e x)\right )+3 \sqrt {\sqrt {c} d-\sqrt {-a} e} \left (a e^2+c d^2\right ) \left (2 \sqrt {-a} c d^2 e+3 a \sqrt {c} d e^2+\sqrt {-a} a e^3+4 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )-3 \sqrt {\sqrt {-a} e+\sqrt {c} d} \left (a e^2+c d^2\right ) \left (-2 \sqrt {-a} c d^2 e+3 a \sqrt {c} d e^2+(-a)^{3/2} e^3+4 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} c^{5/4}}+\frac {8 a (d+e x)^{5/2} \left (a e^2+c d^2\right ) (a e+c d x)}{\left (a+c x^2\right )^2}}{32 a^2 \left (a e^2+c d^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(3/2)/(a + c*x^2)^3,x]

[Out]

((8*a*(c*d^2 + a*e^2)*(a*e + c*d*x)*(d + e*x)^(5/2))/(a + c*x^2)^2 + (2*(d + e*x)^(5/2)*(3*a^2*e^3 + 6*c^2*d^3
*x + a*c*d*e*(5*d + 4*e*x)))/(a + c*x^2) + (-2*Sqrt[-a]*c^(1/4)*e*Sqrt[d + e*x]*(3*a^2*e^4 + 6*c^2*d^3*(2*d +
e*x) + a*c*d*e^2*(13*d + 4*e*x)) + 3*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(c*d^2 + a*e^2)*(4*c^(3/2)*d^3 + 2*Sqrt[-a]*
c*d^2*e + 3*a*Sqrt[c]*d*e^2 + Sqrt[-a]*a*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] -
3*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(c*d^2 + a*e^2)*(4*c^(3/2)*d^3 - 2*Sqrt[-a]*c*d^2*e + 3*a*Sqrt[c]*d*e^2 + (-a)^
(3/2)*e^3)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(Sqrt[-a]*c^(5/4)))/(32*a^2*(c*d^2 +
a*e^2)^2)

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IntegrateAlgebraic [C]  time = 1.92, size = 403, normalized size = 0.52 \begin {gather*} \frac {3 i \left (2 i \sqrt {a} \sqrt {c} d e+a e^2+4 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}{\sqrt {c} d+i \sqrt {a} e}\right )}{32 a^{5/2} c \sqrt {-i \sqrt {c} \left (\sqrt {a} e-i \sqrt {c} d\right )}}-\frac {3 i \left (-2 i \sqrt {a} \sqrt {c} d e+a e^2+4 c d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}{\sqrt {c} d-i \sqrt {a} e}\right )}{32 a^{5/2} c \sqrt {i \sqrt {c} \left (\sqrt {a} e+i \sqrt {c} d\right )}}-\frac {e \sqrt {d+e x} \left (3 a^2 e^4+9 a c d^2 e^2-8 a c d e^2 (d+e x)-a c e^2 (d+e x)^2+6 c^2 d^4-18 c^2 d^3 (d+e x)+18 c^2 d^2 (d+e x)^2-6 c^2 d (d+e x)^3\right )}{16 a^2 c \left (a e^2+c d^2-2 c d (d+e x)+c (d+e x)^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d + e*x)^(3/2)/(a + c*x^2)^3,x]

[Out]

-1/16*(e*Sqrt[d + e*x]*(6*c^2*d^4 + 9*a*c*d^2*e^2 + 3*a^2*e^4 - 18*c^2*d^3*(d + e*x) - 8*a*c*d*e^2*(d + e*x) +
18*c^2*d^2*(d + e*x)^2 - a*c*e^2*(d + e*x)^2 - 6*c^2*d*(d + e*x)^3))/(a^2*c*(c*d^2 + a*e^2 - 2*c*d*(d + e*x)
+ c*(d + e*x)^2)^2) + (((3*I)/32)*(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)])/(a^(5/2)*c*Sqrt[(-I)*Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]
*e)]) - (((3*I)/32)*(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]*S
qrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/(a^(5/2)*c*Sqrt[I*Sqrt[c]*(I*Sqrt[c]*d + Sqrt[a]*e)])

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fricas [B]  time = 0.47, size = 1752, normalized size = 2.28

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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giac [A]  time = 0.61, size = 499, normalized size = 0.65 \begin {gather*} -\frac {3 \, {\left (4 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2} - {\left (2 \, \sqrt {-a c} c d^{2} e + \sqrt {-a c} a e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + 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}} - \frac {3 \, {\left (4 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2} + {\left (2 \, \sqrt {-a c} c d^{2} e + \sqrt {-a c} a e^{3}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + 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}} + \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{2} d e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{2} d^{2} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{2} d^{3} e - 6 \, \sqrt {x e + d} c^{2} d^{4} e + {\left (x e + d\right )}^{\frac {5}{2}} a c e^{3} + 8 \, {\left (x e + d\right )}^{\frac {3}{2}} a c d e^{3} - 9 \, \sqrt {x e + d} a c d^{2} e^{3} - 3 \, \sqrt {x e + d} a^{2} e^{5}}{16 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )}^{2} a^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/32*(4*c^3*d^3 + 3*a*c^2*d*e^2 - (2*sqrt(-a*c)*c*d^2*e + sqrt(-a*c)*a*e^3)*abs(c))*arctan(sqrt(x*e + 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)) - 3/32*(4*c^3*d^3 + 3*a*c^2*d*e^2 + (2*sqrt(-a*c)*c*d^2*e + sqrt(-a*c)*
a*e^3)*abs(c))*arctan(sqrt(x*e + 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)) + 1/16*(6*(x*e + d)^(7/2)*c^2*d
*e - 18*(x*e + d)^(5/2)*c^2*d^2*e + 18*(x*e + d)^(3/2)*c^2*d^3*e - 6*sqrt(x*e + d)*c^2*d^4*e + (x*e + d)^(5/2)
*a*c*e^3 + 8*(x*e + d)^(3/2)*a*c*d*e^3 - 9*sqrt(x*e + d)*a*c*d^2*e^3 - 3*sqrt(x*e + d)*a^2*e^5)/(((x*e + d)^2*
c - 2*(x*e + d)*c*d + c*d^2 + a*e^2)^2*a^2*c)

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maple [B]  time = 0.25, size = 7538, normalized size = 9.80 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.02, size = 3204, normalized size = 4.17

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(a + c*x^2)^3,x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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