3.635 \(\int \frac{1}{\sqrt{d+e x} (a+c x^2)^2} \, dx\)

Optimal. Leaf size=739 \[ \frac{\sqrt{d+e x} (a e+c d x)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) + (e*(c*d^2 + 3*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])/Sqrt[Sqrt[c]*d - Sqrt
[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*c^(1/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c*d
^2 + 3*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])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*c^(1/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c
]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c*d^2 + 3*a*e^2 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - S
qrt[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^(1/4)*
(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c*d^2 + 3*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^(1/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 1.49835, antiderivative size = 739, 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 = {741, 827, 1169, 634, 618, 206, 628} \[ \frac{\sqrt{d+e x} (a e+c d x)}{2 a \left (a+c x^2\right ) \left (a e^2+c d^2\right )}-\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (-\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{c} d \sqrt{a e^2+c d^2}+3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a \sqrt [4]{c} \left (a e^2+c d^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(2*a*(c*d^2 + a*e^2)*(a + c*x^2)) + (e*(c*d^2 + 3*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])/Sqrt[Sqrt[c]*d - Sqrt
[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*c^(1/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c*d
^2 + 3*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])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*c^(1/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c
]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c*d^2 + 3*a*e^2 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - S
qrt[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^(1/4)*
(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c*d^2 + 3*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^(1/4)*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 741

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*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -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{1}{\sqrt{d+e x} \left (a+c x^2\right )^2} \, dx &=\frac{(a e+c d x) \sqrt{d+e x}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\int \frac{\frac{1}{2} \left (-2 c d^2-3 a e^2\right )-\frac{1}{2} c d e x}{\sqrt{d+e x} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{2 a \left (c d^2+a e^2\right ) \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^2-3 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 \left (c d^2+a e^2\right )}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \left (\frac{1}{2} c d^2 e+\frac{1}{2} e \left (-2 c d^2-3 a e^2\right )\right ) \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}-\left (\frac{1}{2} c d^2 e+\frac{1}{2} e \left (-2 c d^2-3 a e^2\right )+\frac{1}{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 )}{2 \sqrt{2} a \sqrt [4]{c} \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} \left (\frac{1}{2} c d^2 e+\frac{1}{2} e \left (-2 c d^2-3 a e^2\right )\right ) \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}{\sqrt [4]{c}}+\left (\frac{1}{2} c d^2 e+\frac{1}{2} e \left (-2 c d^2-3 a e^2\right )+\frac{1}{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 )}{2 \sqrt{2} a \sqrt [4]{c} \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}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{\left (e \left (c d^2+3 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (e \left (c d^2+3 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (e \left (c d^2+3 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 \sqrt{c} \left (c d^2+a e^2\right )^{3/2}}+\frac{\left (e \left (c d^2+3 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 \sqrt{c} \left (c d^2+a e^2\right )^{3/2}}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}-\frac{e \left (c d^2+3 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c d^2+3 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{\left (e \left (c d^2+3 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 \sqrt{c} \left (c d^2+a e^2\right )^{3/2}}-\frac{\left (e \left (c d^2+3 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 \sqrt{c} \left (c d^2+a e^2\right )^{3/2}}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{2 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )}+\frac{e \left (c d^2+3 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c d^2+3 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{e \left (c d^2+3 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{e \left (c d^2+3 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{3/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ \end{align*}

Mathematica [A]  time = 0.600634, size = 255, normalized size = 0.35 \[ \frac{\frac{\left (-\sqrt{-a} \sqrt{c} d e+3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt [4]{c} \sqrt{\sqrt{c} d-\sqrt{-a} e}}+\frac{\left (2 \sqrt{-a} c d^2-a \sqrt{c} d e+3 \sqrt{-a} a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{a \sqrt [4]{c} \sqrt{\sqrt{-a} e+\sqrt{c} d}}+\frac{2 \sqrt{d+e x} (a e+c d x)}{a+c x^2}}{4 a \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(a*e + c*d*x)*Sqrt[d + e*x])/(a + c*x^2) + ((2*c*d^2 - Sqrt[-a]*Sqrt[c]*d*e + 3*a*e^2)*ArcTanh[(c^(1/4)*Sq
rt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]])/(Sqrt[-a]*c^(1/4)*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]) + ((2*Sqrt[-a]*c*d
^2 - a*Sqrt[c]*d*e + 3*Sqrt[-a]*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(a*c^(1/
4)*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]))/(4*a*(c*d^2 + a*e^2))

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}}{\frac{1}{\sqrt{ex+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 3.67187, size = 6525, normalized size = 8.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((a^2*c*d^2 + a^3*e^2 + (a*c^2*d^2 + a^2*c*e^2)*x^2)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a
^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^
10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*
c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))*log((20*c^2*d^4*e^
3 + 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(e*x + d) + (5*a^2*c^2*d^4*e^4 + 24*a^3*c*d^2*e^6 + 27*a^4*e^8 + 2*(a^3*c
^5*d^9 + 5*a^4*c^4*d^7*e^2 + 9*a^5*c^3*d^5*e^4 + 7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt(-(25*c^2*d^4*e^6 + 90
*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*
a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c
^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/
(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*
d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))) - (a^2*c*d^2 + a^3*e^
2 + (a*c^2*d^2 + a^2*c*e^2)*x^2)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c^3*d^6 + 3*a^4*c^2*d
^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^
4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^1
2)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))*log((20*c^2*d^4*e^3 + 81*a*c*d^2*e^5 + 81*
a^2*e^7)*sqrt(e*x + d) - (5*a^2*c^2*d^4*e^4 + 24*a^3*c*d^2*e^6 + 27*a^4*e^8 + 2*(a^3*c^5*d^9 + 5*a^4*c^4*d^7*e
^2 + 9*a^5*c^3*d^5*e^4 + 7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^
10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*
c^2*d^2*e^10 + a^9*c*e^12)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 + (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e
^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^
6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))
/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))) + (a^2*c*d^2 + a^3*e^2 + (a*c^2*d^2 + a^2*c*e
^2)*x^2)*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4
+ a^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*
c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^
4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))*log((20*c^2*d^4*e^3 + 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(e*x + d) +
(5*a^2*c^2*d^4*e^4 + 24*a^3*c*d^2*e^6 + 27*a^4*e^8 - 2*(a^3*c^5*d^9 + 5*a^4*c^4*d^7*e^2 + 9*a^5*c^3*d^5*e^4 +
7*a^6*c^2*d^3*e^6 + 2*a^7*c*d*e^8)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^
4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^1
2)))*sqrt(-(4*c^2*d^5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a
^6*e^6)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*
d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^
2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6))) - (a^2*c*d^2 + a^3*e^2 + (a*c^2*d^2 + a^2*c*e^2)*x^2)*sqrt(-(4*c^2*d^
5 + 15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^
2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4
*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*
d^2*e^4 + a^6*e^6))*log((20*c^2*d^4*e^3 + 81*a*c*d^2*e^5 + 81*a^2*e^7)*sqrt(e*x + d) - (5*a^2*c^2*d^4*e^4 + 24
*a^3*c*d^2*e^6 + 27*a^4*e^8 - 2*(a^3*c^5*d^9 + 5*a^4*c^4*d^7*e^2 + 9*a^5*c^3*d^5*e^4 + 7*a^6*c^2*d^3*e^6 + 2*a
^7*c*d*e^8)*sqrt(-(25*c^2*d^4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*
c^5*d^8*e^4 + 20*a^6*c^4*d^6*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))*sqrt(-(4*c^2*d^5 +
15*a*c*d^3*e^2 + 15*a^2*d*e^4 - (a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*e^4 + a^6*e^6)*sqrt(-(25*c^2*d^
4*e^6 + 90*a*c*d^2*e^8 + 81*a^2*e^10)/(a^3*c^7*d^12 + 6*a^4*c^6*d^10*e^2 + 15*a^5*c^5*d^8*e^4 + 20*a^6*c^4*d^6
*e^6 + 15*a^7*c^3*d^4*e^8 + 6*a^8*c^2*d^2*e^10 + a^9*c*e^12)))/(a^3*c^3*d^6 + 3*a^4*c^2*d^4*e^2 + 3*a^5*c*d^2*
e^4 + a^6*e^6))) + 4*(c*d*x + a*e)*sqrt(e*x + d))/(a^2*c*d^2 + a^3*e^2 + (a*c^2*d^2 + a^2*c*e^2)*x^2)

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

[Out]

Timed out