3.623 \(\int \frac{1}{(d+e x)^{5/2} (a+c x^2)} \, dx\)

Optimal. Leaf size=736 \[ -\frac{c^{3/4} e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{c^{3/4} e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{c^{3/4} e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{c^{3/4} e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{4 c d e}{\sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 1.73181, antiderivative size = 736, 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 = {710, 829, 827, 1169, 634, 618, 206, 628} \[ -\frac{c^{3/4} e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{c^{3/4} e \left (2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{c^{3/4} e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{c^{3/4} e \left (-2 \sqrt{c} d \sqrt{a e^2+c d^2}-a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \left (a e^2+c d^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{4 c d e}{\sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(a + c*x^2)),x]

[Out]

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

Rule 710

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 +
a*e^2)), x] + Dist[c/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*(d - e*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d,
e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 829

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((e*f - d*g)*(d
+ e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f + a*
e*g - c*(e*f - d*g)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&
FractionQ[m] && LtQ[m, -1]

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

Mathematica [C]  time = 0.0645314, size = 135, normalized size = 0.18 \[ \frac{\frac{\, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}\right )}{\sqrt{-a} \sqrt{c} d-a e}-\frac{\, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-\sqrt{-a} e}\right )}{\sqrt{-a} \sqrt{c} d+a e}}{3 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)),x]

[Out]

(-(Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d + a*e))
+ Hypergeometric2F1[-3/2, 1, -1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]/(Sqrt[-a]*Sqrt[c]*d - a*e))/(
3*(d + e*x)^(3/2))

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Maple [B]  time = 0.254, size = 7264, normalized size = 9.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(c*x^2+a),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c x^{2} + a\right )}{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)*(e*x + d)^(5/2)), x)

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Fricas [B]  time = 3.11072, size = 10683, normalized size = 14.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+a),x, algorithm="fricas")

[Out]

1/6*(3*(c^2*d^6 + 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e + 2
*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 + 5*a^2*c^4*d^8
*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6
*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^
3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 +
120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2
+ 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))*log((5*c^4*d^4*e - 10*a*c^3*d^2*e^3
+ a^2*c^2*e^5)*sqrt(e*x + d) + (15*a*c^4*d^6*e^2 - 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 - a^4*c*e^8 + (a*c^
6*d^13 + 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 - 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 - 14*a^6*c*d^3*e^10
- 3*a^7*d*e^12)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3
*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8
+ 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^
18 + a^11*e^20)))*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a
^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 +
110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*
e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3
*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c
^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))) - 3*(c^2*d^6 + 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (
c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e + 2*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt(-(c^4*d^5 - 10*
a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 +
5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e
^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c
^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a
^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^
5*c*d^2*e^8 + a^6*e^10))*log((5*c^4*d^4*e - 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) - (15*a*c^4*d^6*e^2
- 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 - a^4*c*e^8 + (a*c^6*d^13 + 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 -
20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 - 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6
*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^
3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 +
120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e
^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2
*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^
3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8
+ 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e
^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8
+ a^6*e^10))) + 3*(c^2*d^6 + 2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c
^2*d^5*e + 2*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 + 5
*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2
- 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18
*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4
*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2
*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))*log((5*c^4*d^4*e - 10*a*
c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) + (15*a*c^4*d^6*e^2 - 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 - a^4*c
*e^8 - (a*c^6*d^13 + 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 - 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 - 14*a^6
*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e
^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c
^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a
^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 + 5*a^2*c^4*d^
8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^
6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a
^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 +
120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^
2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))) - 3*(c^2*d^6 + 2*a*c*d^4*e^2 + a^2
*d^2*e^4 + (c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e + 2*a*c*d^3*e^3 + a^2*d*e^5)*x)*sqrt(-(c
^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^
2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a
^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6
+ 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4
*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^
4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))*log((5*c^4*d^4*e - 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(e*x + d) - (15*a*
c^4*d^6*e^2 - 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 - a^4*c*e^8 - (a*c^6*d^13 + 2*a^2*c^5*d^11*e^2 - 5*a^3*c
^4*d^9*e^4 - 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 - 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt(-(25*c^7*d^8*e^2
- 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18
*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4
*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(c^4*d^5 - 10
*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 - (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 +
5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*
e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*
c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*
a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a
^5*c*d^2*e^8 + a^6*e^10))) - 4*(6*c*d*e^2*x + 7*c*d^2*e + a*e^3)*sqrt(e*x + d))/(c^2*d^6 + 2*a*c*d^4*e^2 + a^2
*d^2*e^4 + (c^2*d^4*e^2 + 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c^2*d^5*e + 2*a*c*d^3*e^3 + a^2*d*e^5)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(5/2)/(c*x**2+a),x)

[Out]

Integral(1/((a + c*x**2)*(d + e*x)**(5/2)), x)

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

[Out]

Timed out