3.644 \(\int \frac{(d+e x)^{5/2}}{(a+c x^2)^3} \, dx\)

Optimal. Leaf size=846 \[ -\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left (c x^2+a\right )^2}-\frac{3 \left (a d e-\left (2 c d^2+a e^2\right ) x\right ) \sqrt{d+e x}}{16 a^2 c \left (c x^2+a\right )}+\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

[Out]

-((a*e - c*d*x)*(d + e*x)^(3/2))/(4*a*c*(a + c*x^2)^2) - (3*Sqrt[d + e*x]*(a*d*e - (2*c*d^2 + a*e^2)*x))/(16*a
^2*c*(a + c*x^2)) + (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(2*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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^(3/2)*d^3 + 2*
a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(2*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^(7/4)*Sqrt[c*d^2 + a*e^2]
*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(2*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] + S
qrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (3*e*(
2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(2*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^(7/4)*Sqrt[
c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

________________________________________________________________________________________

Rubi [A]  time = 3.39583, antiderivative size = 846, 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, 821, 827, 1169, 634, 618, 206, 628} \[ -\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left (c x^2+a\right )^2}-\frac{3 \left (a d e-\left (2 c d^2+a e^2\right ) x\right ) \sqrt{d+e x}}{16 a^2 c \left (c x^2+a\right )}+\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} e^2 d-\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)/(a + c*x^2)^3,x]

[Out]

-((a*e - c*d*x)*(d + e*x)^(3/2))/(4*a*c*(a + c*x^2)^2) - (3*Sqrt[d + e*x]*(a*d*e - (2*c*d^2 + a*e^2)*x))/(16*a
^2*c*(a + c*x^2)) + (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(2*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^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^(3/2)*d^3 + 2*
a*Sqrt[c]*d*e^2 + Sqrt[c*d^2 + a*e^2]*(2*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^(7/4)*Sqrt[c*d^2 + a*e^2]
*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(2*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] + S
qrt[c]*(d + e*x)])/(64*Sqrt[2]*a^2*c^(7/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (3*e*(
2*c^(3/2)*d^3 + 2*a*Sqrt[c]*d*e^2 - Sqrt[c*d^2 + a*e^2]*(2*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^(7/4)*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 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*
(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*
x^2)^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x
] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (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)^{5/2}}{\left (a+c x^2\right )^3} \, dx &=-\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}+\frac{\int \frac{\sqrt{d+e x} \left (\frac{3}{2} \left (2 c d^2+a e^2\right )+\frac{3}{2} c d e x\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac{3 \sqrt{d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}+\frac{\int \frac{\frac{3}{4} c d \left (4 c d^2+3 a e^2\right )+\frac{3}{4} c e \left (2 c d^2+a e^2\right ) x}{\sqrt{d+e x} \left (a+c x^2\right )} \, dx}{8 a^2 c^2}\\ &=-\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac{3 \sqrt{d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{4} c d e \left (2 c d^2+a e^2\right )+\frac{3}{4} c d e \left (4 c d^2+3 a e^2\right )+\frac{3}{4} c e \left (2 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}\\ &=-\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac{3 \sqrt{d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{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}{4} c d e \left (2 c d^2+a e^2\right )+\frac{3}{4} c d e \left (4 c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}-\left (-\frac{3}{4} c d e \left (2 c d^2+a e^2\right )-\frac{3}{4} \sqrt{c} e \sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )+\frac{3}{4} c d e \left (4 c d^2+3 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} \sqrt{c d^2+a e^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}{4} c d e \left (2 c d^2+a e^2\right )+\frac{3}{4} c d e \left (4 c d^2+3 a e^2\right )\right )}{\sqrt [4]{c}}+\left (-\frac{3}{4} c d e \left (2 c d^2+a e^2\right )-\frac{3}{4} \sqrt{c} e \sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )+\frac{3}{4} c d e \left (4 c d^2+3 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} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=-\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac{3 \sqrt{d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}-\frac{\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\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^{7/4} \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^{3/2} d^3+2 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\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^{7/4} \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^{3/2} d^3+2 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\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^2 \sqrt{c d^2+a e^2}}+\frac{\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\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^2 \sqrt{c d^2+a e^2}}\\ &=-\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac{3 \sqrt{d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}-\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt{2} a^2 c^{7/4} \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^{3/2} d^3+2 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\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^2 \sqrt{c d^2+a e^2}}-\frac{\left (3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (2 c d^2+a e^2\right )\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^2 \sqrt{c d^2+a e^2}}\\ &=-\frac{(a e-c d x) (d+e x)^{3/2}}{4 a c \left (a+c x^2\right )^2}-\frac{3 \sqrt{d+e x} \left (a d e-\left (2 c d^2+a e^2\right ) x\right )}{16 a^2 c \left (a+c x^2\right )}+\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (2 c d^2+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 )}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} d e^2+\sqrt{c d^2+a e^2} \left (2 c d^2+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 )}{32 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^{3/2} d^3+2 a \sqrt{c} d e^2-\sqrt{c d^2+a e^2} \left (2 c d^2+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 )}{64 \sqrt{2} a^2 c^{7/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ \end{align*}

Mathematica [A]  time = 0.725081, size = 277, normalized size = 0.33 \[ \frac{\frac{2 c^{3/4} \sqrt{d+e x} \left (-a^2 e (7 d+e x)+a c x \left (10 d^2+d e x+3 e^2 x^2\right )+6 c^2 d^2 x^3\right )}{a^2 \left (a+c x^2\right )^2}+\frac{3 \sqrt{\sqrt{c} d-\sqrt{-a} e} \left (2 \sqrt{-a} \sqrt{c} d e+a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{(-a)^{5/2}}-\frac{3 \sqrt{\sqrt{-a} e+\sqrt{c} d} \left (-2 \sqrt{-a} \sqrt{c} d e+a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{(-a)^{5/2}}}{32 c^{7/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(5/2)/(a + c*x^2)^3,x]

[Out]

((2*c^(3/4)*Sqrt[d + e*x]*(6*c^2*d^2*x^3 - a^2*e*(7*d + e*x) + a*c*x*(10*d^2 + d*e*x + 3*e^2*x^2)))/(a^2*(a +
c*x^2)^2) + (3*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(4*c*d^2 + 2*Sqrt[-a]*Sqrt[c]*d*e + a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d
+ e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(-a)^(5/2) - (3*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(4*c*d^2 - 2*Sqrt[-a]*Sqr
t[c]*d*e + a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(-a)^(5/2))/(32*c^(7/4))

________________________________________________________________________________________

Maple [B]  time = 0.245, size = 4264, normalized size = 5. \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)^(5/2)/(c*x^2+a)^3,x)

[Out]

-3/32/e/a^3/c^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e
*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-
2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^3+3/64*e/a^2/c
^(5/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(
2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))
*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)-3/64*
e/a^2/c^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^
(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)
^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d-3/64/e/a^3/c^2*ln((e
*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1/2)*(
2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d-1/2*e^3/(c*e^2*x^2+a*e^2)^2*d/a*(e*x+d)^(5/2)
+17/16*e^3/(c*e^2*x^2+a*e^2)^2/a*(e*x+d)^(3/2)*d^2-3/4*e^3/(c*e^2*x^2+a*e^2)^2*d^3/a*(e*x+d)^(1/2)-3/8*e^5/(c*
e^2*x^2+a*e^2)^2*d/c*(e*x+d)^(1/2)-3/32/e/a^3/c^2/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c
*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1
/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(2*(a*
c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d-3/32/e/a^3/c^2/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c
*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4
*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(
a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d+3/64/e/a^3/c*ln((e*x+d)*c
^(1/2)-(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1/2)*(2*(a*c*
e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^2+3/64/e/a^3/c^(3/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(
1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^2-3/6
4/e/a^3/c^(3/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*
(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^2-3/64/e/a^3/c*ln((e*x+d)*c^(1/2)+(e*x+d)^(1
/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2
)+2*c*d)^(1/2)*d^2+3/8*e/a^2/c/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2
*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d
^2))^(1/2)-2*c*d)^(1/2))*(a*e^2+c*d^2)^(1/2)*d+3/16*e^3/(c*e^2*x^2+a*e^2)^2/a*(e*x+d)^(7/2)-1/16*e^5/(c*e^2*x^
2+a*e^2)^2/c*(e*x+d)^(3/2)+3/32/e/a^3/c^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^
(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-
2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c
*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^2+3/32/e/a^3/c^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1
/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/
2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2
)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^2+3/32/e/a^3/c/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2
))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2
)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^
(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^2+3/32/e/a^3/c/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^
2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^
2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)
^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^2+3/64/e/a^3/c^2*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*(c*(a*e^
2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(a*e^2+c*d^2)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*
(a*c*e^2+c^2*d^2)^(1/2)*d-3/64*e/a^2/c^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(
1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2
*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*
d)^(1/2)*d+3/64*e/a^2/c^(5/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*
c^(1/2)*(e*x+d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^
2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e
^2+c^2*d^2)^(1/2)-3/32/e/a^3/c^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arc
tan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e
^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)
*d^3+3/8*e/a^2/c/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+
d)^(1/2)-(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c
*d)^(1/2))*(a*e^2+c*d^2)^(1/2)*d+3/64/e/a^3/c^(1/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2
)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d^3-3/64/e/a^3/c^(1/2)*ln((e*x+d)*
c^(1/2)-(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+
2*c*d)^(1/2)*d^3+3/8*e/(c*e^2*x^2+a*e^2)^2/a^2*(e*x+d)^(7/2)*c*d^2-9/8*e/(c*e^2*x^2+a*e^2)^2*d^3/a^2*(e*x+d)^(
5/2)*c+9/8*e/(c*e^2*x^2+a*e^2)^2/a^2*c*(e*x+d)^(3/2)*d^4-3/8*e/(c*e^2*x^2+a*e^2)^2*d^5/a^2*c*(e*x+d)^(1/2)+3/1
28*e/a^2/c^(3/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))
*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d-3/128*e/a^2/c^(5/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*(c*(a*e^2+c
*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)
-3/128*e/a^2/c^(3/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1
/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d+3/128*e/a^2/c^(5/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*(c*(a*e
^2+c*d^2))^(1/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(
1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{5}{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)^(5/2)/(c*x^2+a)^3,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(5/2)/(c*x^2 + a)^3, x)

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Fricas [A]  time = 2.52199, size = 2194, normalized size = 2.59 \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)^(5/2)/(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(-(a^5*c^3*sqrt(-e^10/(a^5*c^7)) + 16*c^2*d^5 + 20*a*c*d^3*e
^2 + 5*a^2*d*e^4)/(a^5*c^3))*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*e^6 - (4*a^5*c^6*d^2 + a^6*c^5*e^2)*sqrt(-e^10/(a^5*c^7)))*sqrt(-(a^5*c^3*sqrt(-e^10/(a^5*c^7)) + 16*c^2*d^5
+ 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))) - 3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(a^5*c^3*sqrt(-e^
10/(a^5*c^7)) + 16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))*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*e^6 - (4*a^5*c^6*d^2 + a^6*c^5*e^2)*sqrt(-e^10/(a^5*c^7)))*sqrt(-(a
^5*c^3*sqrt(-e^10/(a^5*c^7)) + 16*c^2*d^5 + 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))) + 3*(a^2*c^3*x^4 + 2*a^3
*c^2*x^2 + a^4*c)*sqrt((a^5*c^3*sqrt(-e^10/(a^5*c^7)) - 16*c^2*d^5 - 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))*
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*e^6 + (4*a^5*c^6*d^2 + a^6*
c^5*e^2)*sqrt(-e^10/(a^5*c^7)))*sqrt((a^5*c^3*sqrt(-e^10/(a^5*c^7)) - 16*c^2*d^5 - 20*a*c*d^3*e^2 - 5*a^2*d*e^
4)/(a^5*c^3))) - 3*(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt((a^5*c^3*sqrt(-e^10/(a^5*c^7)) - 16*c^2*d^5 - 20
*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))*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*e^6 + (4*a^5*c^6*d^2 + a^6*c^5*e^2)*sqrt(-e^10/(a^5*c^7)))*sqrt((a^5*c^3*sqrt(-e^10/(a^5*c^7)) - 1
6*c^2*d^5 - 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))) + 4*(a*c*d*e*x^2 - 7*a^2*d*e + 3*(2*c^2*d^2 + a*c*e^2)*x
^3 + (10*a*c*d^2 - a^2*e^2)*x)*sqrt(e*x + d))/(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)

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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)**(5/2)/(c*x**2+a)**3,x)

[Out]

Timed out

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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)^(5/2)/(c*x^2+a)^3,x, algorithm="giac")

[Out]

Timed out