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

Optimal. Leaf size=726 \[ -\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{\sqrt{d+e x} (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(2*a*c*(a + c*x^2)) + (e*(c*d^2 + a*e^2 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTan
h[(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]
]])/(4*Sqrt[2]*a*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c*d^2 + a*e^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])/Sqr
t[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*
e^2]]) - (e*(c*d^2 + a*e^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[Sqr
t[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(8*Sqrt[2]*a*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqr
t[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c*d^2 + a*e^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)])/(8*Sqrt[2]*a*c^
(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

________________________________________________________________________________________

Rubi [A]  time = 1.29799, antiderivative size = 726, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {739, 827, 1169, 634, 618, 206, 628} \[ -\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{5/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{\sqrt{d+e x} (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(a + c*x^2)^2,x]

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(2*a*c*(a + c*x^2)) + (e*(c*d^2 + a*e^2 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTan
h[(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]
]])/(4*Sqrt[2]*a*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c*d^2 + a*e^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])/Sqr
t[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*
e^2]]) - (e*(c*d^2 + a*e^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[Sqr
t[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(8*Sqrt[2]*a*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqr
t[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c*d^2 + a*e^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)])/(8*Sqrt[2]*a*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 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 )^2} \, dx &=-\frac{(a e-c d x) \sqrt{d+e x}}{2 a c \left (a+c x^2\right )}+\frac{\int \frac{\frac{1}{2} \left (2 c d^2+a e^2\right )+\frac{1}{2} c d e x}{\sqrt{d+e x} \left (a+c x^2\right )} \, dx}{2 a c}\\ &=-\frac{(a e-c d x) \sqrt{d+e x}}{2 a c \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{2} c d^2 e+\frac{1}{2} e \left (2 c d^2+a e^2\right )+\frac{1}{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 )}{a c}\\ &=-\frac{(a e-c d x) \sqrt{d+e x}}{2 a 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{1}{2} c d^2 e+\frac{1}{2} e \left (2 c d^2+a e^2\right )\right )}{\sqrt [4]{c}}-\left (-\frac{1}{2} c d^2 e-\frac{1}{2} \sqrt{c} d e \sqrt{c d^2+a e^2}+\frac{1}{2} e \left (2 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 )}{2 \sqrt{2} a c^{5/4} \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{1}{2} c d^2 e+\frac{1}{2} e \left (2 c d^2+a e^2\right )\right )}{\sqrt [4]{c}}+\left (-\frac{1}{2} c d^2 e-\frac{1}{2} \sqrt{c} d e \sqrt{c d^2+a e^2}+\frac{1}{2} e \left (2 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 )}{2 \sqrt{2} a c^{5/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=-\frac{(a e-c d x) \sqrt{d+e x}}{2 a c \left (a+c x^2\right )}-\frac{\left (e \left (c d^2+a e^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 )}{8 \sqrt{2} a c^{5/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (e \left (c d^2+a e^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 )}{8 \sqrt{2} a c^{5/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (e \left (c d^2+a e^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 )}{8 a c^{3/2} \sqrt{c d^2+a e^2}}+\frac{\left (e \left (c d^2+a e^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 )}{8 a c^{3/2} \sqrt{c d^2+a e^2}}\\ &=-\frac{(a e-c d x) \sqrt{d+e x}}{2 a c \left (a+c x^2\right )}-\frac{e \left (c d^2+a e^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 )}{8 \sqrt{2} a c^{5/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c d^2+a e^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 )}{8 \sqrt{2} a c^{5/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{\left (e \left (c d^2+a e^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 )}{4 a c^{3/2} \sqrt{c d^2+a e^2}}-\frac{\left (e \left (c d^2+a e^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 )}{4 a c^{3/2} \sqrt{c d^2+a e^2}}\\ &=-\frac{(a e-c d x) \sqrt{d+e x}}{2 a c \left (a+c x^2\right )}+\frac{e \left (c d^2+a e^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 )}{4 \sqrt{2} a c^{5/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c d^2+a e^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 )}{4 \sqrt{2} a c^{5/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c d^2+a e^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 )}{8 \sqrt{2} a c^{5/4} \sqrt{c d^2+a e^2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c d^2+a e^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 )}{8 \sqrt{2} a 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 = 0.583789, size = 208, normalized size = 0.29 \[ \frac{\frac{2 a \sqrt [4]{c} \sqrt{d+e x} (c d x-a e)}{a+c x^2}-\sqrt{\sqrt{c} d-\sqrt{-a} e} \left (2 \sqrt{-a} \sqrt{c} d-a e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )+\sqrt{\sqrt{-a} e+\sqrt{c} d} \left (2 \sqrt{-a} \sqrt{c} d+a e\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{4 a^2 c^{5/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.23, size = 2851, normalized size = 3.9 \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)^2,x)

[Out]

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

________________________________________________________________________________________

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 )}^{2}}\,{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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 2.3905, size = 1432, normalized size = 1.97 \begin{align*} \frac{{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt{e x + d} +{\left (2 \, a^{3} c^{4} d \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) -{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt{e x + d} -{\left (2 \, a^{3} c^{4} d \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt{-\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) -{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt{e x + d} +{\left (2 \, a^{3} c^{4} d \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) +{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt{e x + d} -{\left (2 \, a^{3} c^{4} d \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt{\frac{a^{3} c^{2} \sqrt{-\frac{e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) + 4 \,{\left (c d x - a e\right )} \sqrt{e x + d}}{8 \,{\left (a c^{2} x^{2} + a^{2} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((a*c^2*x^2 + a^2*c)*sqrt(-(a^3*c^2*sqrt(-e^6/(a^3*c^5)) + 4*c*d^3 + 3*a*d*e^2)/(a^3*c^2))*log((4*c*d^2*e^
3 + a*e^5)*sqrt(e*x + d) + (2*a^3*c^4*d*sqrt(-e^6/(a^3*c^5)) + a^2*c*e^4)*sqrt(-(a^3*c^2*sqrt(-e^6/(a^3*c^5))
+ 4*c*d^3 + 3*a*d*e^2)/(a^3*c^2))) - (a*c^2*x^2 + a^2*c)*sqrt(-(a^3*c^2*sqrt(-e^6/(a^3*c^5)) + 4*c*d^3 + 3*a*d
*e^2)/(a^3*c^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) - (2*a^3*c^4*d*sqrt(-e^6/(a^3*c^5)) + a^2*c*e^4)*sqrt
(-(a^3*c^2*sqrt(-e^6/(a^3*c^5)) + 4*c*d^3 + 3*a*d*e^2)/(a^3*c^2))) - (a*c^2*x^2 + a^2*c)*sqrt((a^3*c^2*sqrt(-e
^6/(a^3*c^5)) - 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) + (2*a^3*c^4*d*sqrt(-e
^6/(a^3*c^5)) - a^2*c*e^4)*sqrt((a^3*c^2*sqrt(-e^6/(a^3*c^5)) - 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))) + (a*c^2*x^2
+ a^2*c)*sqrt((a^3*c^2*sqrt(-e^6/(a^3*c^5)) - 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e
*x + d) - (2*a^3*c^4*d*sqrt(-e^6/(a^3*c^5)) - a^2*c*e^4)*sqrt((a^3*c^2*sqrt(-e^6/(a^3*c^5)) - 4*c*d^3 - 3*a*d*
e^2)/(a^3*c^2))) + 4*(c*d*x - a*e)*sqrt(e*x + d))/(a*c^2*x^2 + a^2*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)**2,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)^2,x, algorithm="giac")

[Out]

Timed out