3.647 \(\int \frac{1}{\sqrt{d+e x} (a+c x^2)^3} \, dx\)

Optimal. Leaf size=920 \[ \frac{\sqrt{d+e x} (a e+c d x)}{4 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^2}+\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2+2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2+2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (c x^2+a\right )} \]

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(4*a*(c*d^2 + a*e^2)*(a + c*x^2)^2) + (Sqrt[d + e*x]*(a*e*(c*d^2 + 7*a*e^2) + 6*
c*d*(c*d^2 + 2*a*e^2)*x))/(16*a^2*(c*d^2 + a*e^2)^2*(a + c*x^2)) + (3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4
+ 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 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^(1/4)*(c*d^2 + a*e^2)^(5/2)*
Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 + 2*Sqrt[c]*d*Sqrt[c*d^2
+ a*e^2]*(c*d^2 + 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])/Sq
rt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(32*Sqrt[2]*a^2*c^(1/4)*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^
2 + a*e^2]]) - (3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 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^(1/4)*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (3*e*(2*c^2*d^4
+ 5*a*c*d^2*e^2 + 7*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 2*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] + Sq
rt[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^(1/4
)*(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 = 5.8513, antiderivative size = 920, 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 = {741, 823, 827, 1169, 634, 618, 206, 628} \[ \frac{\sqrt{d+e x} (a e+c d x)}{4 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^2}+\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2+2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2+2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}\right )}{32 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)-\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^2 d^4+5 a c e^2 d^2-2 \sqrt{c} \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right ) d+7 a^2 e^4\right ) \log \left (\sqrt{c} (d+e x)+\sqrt{2} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \sqrt{d+e x}+\sqrt{c d^2+a e^2}\right )}{64 \sqrt{2} a^2 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (c x^2+a\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(4*a*(c*d^2 + a*e^2)*(a + c*x^2)^2) + (Sqrt[d + e*x]*(a*e*(c*d^2 + 7*a*e^2) + 6*
c*d*(c*d^2 + 2*a*e^2)*x))/(16*a^2*(c*d^2 + a*e^2)^2*(a + c*x^2)) + (3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4
+ 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 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^(1/4)*(c*d^2 + a*e^2)^(5/2)*
Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 + 2*Sqrt[c]*d*Sqrt[c*d^2
+ a*e^2]*(c*d^2 + 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])/Sq
rt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(32*Sqrt[2]*a^2*c^(1/4)*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^
2 + a*e^2]]) - (3*e*(2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 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^(1/4)*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (3*e*(2*c^2*d^4
+ 5*a*c*d^2*e^2 + 7*a^2*e^4 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 2*a*e^2))*Log[Sqrt[c*d^2 + a*e^2] + Sq
rt[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^(1/4
)*(c*d^2 + a*e^2)^(5/2)*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 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (a+c x^2\right )^3} \, dx &=\frac{(a e+c d x) \sqrt{d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}-\frac{\int \frac{\frac{1}{2} \left (-6 c d^2-7 a e^2\right )-\frac{5}{2} c d e x}{\sqrt{d+e x} \left (a+c x^2\right )^2} \, dx}{4 a \left (c d^2+a e^2\right )}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{\int \frac{\frac{3}{4} c \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )+\frac{3}{2} c^2 d e \left (c d^2+2 a e^2\right ) x}{\sqrt{d+e x} \left (a+c x^2\right )} \, dx}{8 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )+\frac{3}{2} c^2 d e \left (c d^2+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 \left (c d^2+a e^2\right )^2}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}-\left (-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )-\frac{3}{2} c^{3/2} d e \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\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^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}+\left (-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )-\frac{3}{2} c^{3/2} d e \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\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^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{\left (\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac{3}{2} c^{3/2} d e \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right )-\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\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 )}{16 \sqrt{2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )-\frac{3}{2} c^{3/2} d e \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\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 )}{16 \sqrt{2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac{3}{2} c^{3/2} d e \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right )-\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}+\frac{2 \sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}\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 )}{16 \sqrt{2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{\left (\frac{2 \sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )-\frac{3}{2} c^{3/2} d e \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}\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 )}{16 \sqrt{2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}-\frac{3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-\sqrt{c} d \sqrt{c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-\sqrt{c} d \sqrt{c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{\left (\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac{3}{2} c^{3/2} d e \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right )-\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}+\frac{2 \sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}\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 )}{8 \sqrt{2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}-\frac{\left (\frac{2 \sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}-\frac{\sqrt{2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}} \left (-\frac{3}{2} c^2 d^2 e \left (c d^2+2 a e^2\right )-\frac{3}{2} c^{3/2} d e \sqrt{c d^2+a e^2} \left (c d^2+2 a e^2\right )+\frac{3}{4} c e \left (4 c^2 d^4+9 a c d^2 e^2+7 a^2 e^4\right )\right )}{\sqrt [4]{c}}\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 )}{8 \sqrt{2} a^2 c^{5/4} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ &=\frac{(a e+c d x) \sqrt{d+e x}}{4 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^2}+\frac{\sqrt{d+e x} \left (a e \left (c d^2+7 a e^2\right )+6 c d \left (c d^2+2 a e^2\right ) x\right )}{16 a^2 \left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}+\frac{3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4+\sqrt{c} d \sqrt{c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4+\sqrt{c} d \sqrt{c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d-\sqrt{c d^2+a e^2}}}-\frac{3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-\sqrt{c} d \sqrt{c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}+\frac{3 e \left (2 c^2 d^4+5 a c d^2 e^2+7 a^2 e^4-\sqrt{c} d \sqrt{c d^2+a e^2} \left (2 c d^2+4 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 \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt{\sqrt{c} d+\sqrt{c d^2+a e^2}}}\\ \end{align*}

Mathematica [A]  time = 0.763325, size = 464, normalized size = 0.5 \[ \frac{\frac{\sqrt{d+e x} \left (7 a^2 e^3+a c d e (d+12 e x)+6 c^2 d^3 x\right )}{2 \left (a+c x^2\right )}+\frac{3 \left (\frac{\left (7 a^2 e^4+5 a c d^2 e^2+2 c^2 d^4\right ) \left (\sqrt{\sqrt{-a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )-\sqrt{\sqrt{c} d-\sqrt{-a} e} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )\right )}{\sqrt{\sqrt{c} d-\sqrt{-a} e} \sqrt{\sqrt{-a} e+\sqrt{c} d}}+2 \sqrt{c} d \left (2 a e^2+c d^2\right ) \left (\sqrt{\sqrt{c} d-\sqrt{-a} e} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )-\sqrt{\sqrt{-a} e+\sqrt{c} d} \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )\right )\right )}{4 \sqrt{-a} \sqrt [4]{c}}+\frac{2 a \sqrt{d+e x} \left (a e^2+c d^2\right ) (a e+c d x)}{\left (a+c x^2\right )^2}}{8 a^2 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a*(c*d^2 + a*e^2)*(a*e + c*d*x)*Sqrt[d + e*x])/(a + c*x^2)^2 + (Sqrt[d + e*x]*(7*a^2*e^3 + 6*c^2*d^3*x + a
*c*d*e*(d + 12*e*x)))/(2*(a + c*x^2)) + (3*(((2*c^2*d^4 + 5*a*c*d^2*e^2 + 7*a^2*e^4)*(Sqrt[Sqrt[c]*d + Sqrt[-a
]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] - Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*ArcTanh[(c^(
1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]]))/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]
) + 2*Sqrt[c]*d*(c*d^2 + 2*a*e^2)*(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d
- Sqrt[-a]*e]] - Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]]))
)/(4*Sqrt[-a]*c^(1/4)))/(8*a^2*(c*d^2 + a*e^2)^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 ) ^{3}}{\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)^3/(e*x+d)^(1/2),x)

[Out]

int(1/(c*x^2+a)^3/(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 )}^{3} \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)^3/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^3*sqrt(e*x + d)), x)

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Fricas [B]  time = 13.3048, size = 12473, normalized size = 13.56 \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)^3/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

1/64*(3*(a^4*c^2*d^4 + 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 + 2*(a^
3*c^3*d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt(-(16*c^4*d^9 + 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 2
10*a^3*c*d^3*e^6 + 105*a^4*d*e^8 + (a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6
+ 5*a^9*c*d^2*e^8 + a^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a
^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*
e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*
c^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 +
10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10))*log(27*(336*c^4*d^8*e^5 + 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2
*d^4*e^9 + 4802*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x + d) + 27*(42*a^3*c^4*d^8*e^6 + 213*a^4*c^3*d^6*e^8 +
515*a^5*c^2*d^4*e^10 + 623*a^6*c*d^2*e^12 + 343*a^7*e^14 + (4*a^5*c^8*d^15 + 31*a^6*c^7*d^13*e^2 + 106*a^7*c^
6*d^11*e^4 + 205*a^8*c^5*d^9*e^6 + 240*a^9*c^4*d^7*e^8 + 169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 + 11*a^1
2*c*d*e^14)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401
*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^
12*e^8 + 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a^
14*c^2*d^2*e^18 + a^15*c*e^20)))*sqrt(-(16*c^4*d^9 + 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^
6 + 105*a^4*d*e^8 + (a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*
e^8 + a^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 +
2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c
^7*d^12*e^8 + 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 +
10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*
e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10))) - 3*(a^4*c^2*d^4 + 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d
^2*e^2 + a^4*c^2*e^4)*x^4 + 2*(a^3*c^3*d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt(-(16*c^4*d^9 + 84*a*c^3*
d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 + (a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7
*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12
+ 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7
*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 +
120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^
4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10))*log(27*(336*c^4*d^8*e^5 +
1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^9 + 4802*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x + d) - 27*(42*a^3*c^
4*d^8*e^6 + 213*a^4*c^3*d^6*e^8 + 515*a^5*c^2*d^4*e^10 + 623*a^6*c*d^2*e^12 + 343*a^7*e^14 + (4*a^5*c^8*d^15 +
31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 + 205*a^8*c^5*d^9*e^6 + 240*a^9*c^4*d^7*e^8 + 169*a^10*c^3*d^5*e^1
0 + 66*a^11*c^2*d^3*e^12 + 11*a^12*c*d*e^14)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*
e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*
a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^
14 + 45*a^13*c^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))*sqrt(-(16*c^4*d^9 + 84*a*c^3*d^7*e^2 + 189*a
^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 + (a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 +
10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2
*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 +
120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d
^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*
a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10))) + 3*(a^4*c^2*d^4 + 2*a^5*c*d^2*e^2 + a^6
*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 + 2*(a^3*c^3*d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x
^2)*sqrt(-(16*c^4*d^9 + 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 - (a^5*c^5*
d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10)*sqrt(-(441*c
^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^2
0 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^
10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c
*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*
e^10))*log(27*(336*c^4*d^8*e^5 + 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^9 + 4802*a^3*c*d^2*e^11 + 2401*a^4*e^
13)*sqrt(e*x + d) + 27*(42*a^3*c^4*d^8*e^6 + 213*a^4*c^3*d^6*e^8 + 515*a^5*c^2*d^4*e^10 + 623*a^6*c*d^2*e^12 +
343*a^7*e^14 - (4*a^5*c^8*d^15 + 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4 + 205*a^8*c^5*d^9*e^6 + 240*a^9*c
^4*d^7*e^8 + 169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 + 11*a^12*c*d*e^14)*sqrt(-(441*c^4*d^8*e^10 + 2268*a
*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18
*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^10*e^10 + 210*a^11*c
^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))*sqrt(-(16*c
^4*d^9 + 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4*d*e^8 - (a^5*c^5*d^10 + 5*a^6*c^
4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2
268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10
*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 252*a^10*c^6*d^10*e^10 + 210*a
^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c
^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10))) - 3*(a^
4*c^2*d^4 + 2*a^5*c*d^2*e^2 + a^6*e^4 + (a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 + 2*(a^3*c^3*d^4 +
2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*x^2)*sqrt(-(16*c^4*d^9 + 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^
3*e^6 + 105*a^4*d*e^8 - (a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*
d^2*e^8 + a^10*e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^
16 + 2401*a^4*e^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a
^9*c^7*d^12*e^8 + 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^1
6 + 10*a^14*c^2*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*
d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*e^10))*log(27*(336*c^4*d^8*e^5 + 1788*a*c^3*d^6*e^7 + 4189*a^2*c^2*d^4*e^9 +
4802*a^3*c*d^2*e^11 + 2401*a^4*e^13)*sqrt(e*x + d) - 27*(42*a^3*c^4*d^8*e^6 + 213*a^4*c^3*d^6*e^8 + 515*a^5*c^
2*d^4*e^10 + 623*a^6*c*d^2*e^12 + 343*a^7*e^14 - (4*a^5*c^8*d^15 + 31*a^6*c^7*d^13*e^2 + 106*a^7*c^6*d^11*e^4
+ 205*a^8*c^5*d^9*e^6 + 240*a^9*c^4*d^7*e^8 + 169*a^10*c^3*d^5*e^10 + 66*a^11*c^2*d^3*e^12 + 11*a^12*c*d*e^14)
*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e^18)/
(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8 + 25
2*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a^14*c^2*d^2*
e^18 + a^15*c*e^20)))*sqrt(-(16*c^4*d^9 + 84*a*c^3*d^7*e^2 + 189*a^2*c^2*d^5*e^4 + 210*a^3*c*d^3*e^6 + 105*a^4
*d*e^8 - (a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9*c*d^2*e^8 + a^10*
e^10)*sqrt(-(441*c^4*d^8*e^10 + 2268*a*c^3*d^6*e^12 + 4974*a^2*c^2*d^4*e^14 + 5292*a^3*c*d^2*e^16 + 2401*a^4*e
^18)/(a^5*c^11*d^20 + 10*a^6*c^10*d^18*e^2 + 45*a^7*c^9*d^16*e^4 + 120*a^8*c^8*d^14*e^6 + 210*a^9*c^7*d^12*e^8
+ 252*a^10*c^6*d^10*e^10 + 210*a^11*c^5*d^8*e^12 + 120*a^12*c^4*d^6*e^14 + 45*a^13*c^3*d^4*e^16 + 10*a^14*c^2
*d^2*e^18 + a^15*c*e^20)))/(a^5*c^5*d^10 + 5*a^6*c^4*d^8*e^2 + 10*a^7*c^3*d^6*e^4 + 10*a^8*c^2*d^4*e^6 + 5*a^9
*c*d^2*e^8 + a^10*e^10))) + 4*(5*a^2*c*d^2*e + 11*a^3*e^3 + 6*(c^3*d^3 + 2*a*c^2*d*e^2)*x^3 + (a*c^2*d^2*e + 7
*a^2*c*e^3)*x^2 + 2*(5*a*c^2*d^3 + 8*a^2*c*d*e^2)*x)*sqrt(e*x + d))/(a^4*c^2*d^4 + 2*a^5*c*d^2*e^2 + a^6*e^4 +
(a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4)*x^4 + 2*(a^3*c^3*d^4 + 2*a^4*c^2*d^2*e^2 + a^5*c*e^4)*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)**3/(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)^3/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

Timed out