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

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

[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]])

________________________________________________________________________________________

Rubi [A]  time = 1.93555, antiderivative size = 769, normalized size of antiderivative = 1., 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} \[ -\frac{3 e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{3 e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{64 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{3 e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{32 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{3 e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{32 \sqrt{2} a^2 c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}+\frac{\sqrt{d+e x} (a e+6 c d x)}{16 a^2 c \left (a+c x^2\right )}-\frac{\sqrt{d+e x} (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]

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 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]

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 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 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 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]

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*}

Mathematica [A]  time = 1.22698, size = 430, normalized size = 0.56 \[ \frac{\frac{2 (d+e x)^{5/2} \left (3 a^2 e^3+a c d e (5 d+4 e x)+6 c^2 d^3 x\right )}{a+c x^2}+\frac{-2 \sqrt{-a} \sqrt [4]{c} e \sqrt{d+e x} \left (3 a^2 e^4+a c d e^2 (13 d+4 e x)+6 c^2 d^3 (2 d+e x)\right )+3 \sqrt{\sqrt{c} d-\sqrt{-a} e} \left (a e^2+c d^2\right ) \left (2 \sqrt{-a} c d^2 e+3 a \sqrt{c} d e^2+\sqrt{-a} a e^3+4 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )-3 \sqrt{\sqrt{-a} e+\sqrt{c} d} \left (a e^2+c d^2\right ) \left (-2 \sqrt{-a} c d^2 e+3 a \sqrt{c} d e^2+(-a)^{3/2} e^3+4 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} c^{5/4}}+\frac{8 a (d+e x)^{5/2} \left (a e^2+c d^2\right ) (a e+c d x)}{\left (a+c x^2\right )^2}}{32 a^2 \left (a e^2+c d^2\right )^2} \]

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)

________________________________________________________________________________________

Maple [B]  time = 0.254, size = 7538, normalized size = 9.8 \begin{align*} \text{output too large to display} \end{align*}

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Fricas [B]  time = 2.77599, size = 3384, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out