3.7.47 \(\int \frac {1}{\sqrt {d+e x} (a+c x^2)^3} \, dx\) [647]

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

[Out]

1/4*(c*d*x+a*e)*(e*x+d)^(1/2)/a/(a*e^2+c*d^2)/(c*x^2+a)^2+1/16*(a*e*(7*a*e^2+c*d^2)+6*c*d*(2*a*e^2+c*d^2)*x)*(
e*x+d)^(1/2)/a^2/(a*e^2+c*d^2)^2/(c*x^2+a)+3/64*e*arctanh((-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*
d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(2*c^2*d^4+5*a*c*d^2*e^2+7*a^2*e^4+2*d*(2*a*e^2+c*d^
2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(1/4)/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-
3/64*e*arctanh((c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^
(1/2))^(1/2))*(2*c^2*d^4+5*a*c*d^2*e^2+7*a^2*e^4+2*d*(2*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(1/4)/
(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-3/128*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/
2)-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(2*c^2*d^4+5*a*c*d^2*e^2+7*a^2*e^4-2*d
*(2*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/c^(1/4)/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)
^(1/2))^(1/2)+3/128*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)+c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a*e^2+c
*d^2)^(1/2))^(1/2))*(2*c^2*d^4+5*a*c*d^2*e^2+7*a^2*e^4-2*d*(2*a*e^2+c*d^2)*c^(1/2)*(a*e^2+c*d^2)^(1/2))/a^2/c^
(1/4)/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 4.03, antiderivative size = 920, normalized size of antiderivative = 1.00, 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 = {755, 837, 841, 1183, 648, 632, 212, 642} \begin {gather*} \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 )} \end {gather*}

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 212

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

Rule 632

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 642

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]

Rule 648

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 755

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 837

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] +
Dist[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 841

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 1183

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]

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 {\text {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 {\text {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 {\text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.31, size = 382, normalized size = 0.42 \begin {gather*} \frac {\frac {2 \sqrt {a} \sqrt {d+e x} \left (11 a^3 e^3+6 c^3 d^3 x^3+a^2 c e \left (5 d^2+16 d e x+7 e^2 x^2\right )+a c^2 d x \left (10 d^2+d e x+12 e^2 x^2\right )\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )^2}+\frac {3 i \left (4 c d^2+10 i \sqrt {a} \sqrt {c} d e-7 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\left (\sqrt {c} d+i \sqrt {a} e\right )^2 \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}-\frac {3 i \left (4 c d^2-10 i \sqrt {a} \sqrt {c} d e-7 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right )^2 \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}}{32 a^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*Sqrt[a]*Sqrt[d + e*x]*(11*a^3*e^3 + 6*c^3*d^3*x^3 + a^2*c*e*(5*d^2 + 16*d*e*x + 7*e^2*x^2) + a*c^2*d*x*(10
*d^2 + d*e*x + 12*e^2*x^2)))/((c*d^2 + a*e^2)^2*(a + c*x^2)^2) + ((3*I)*(4*c*d^2 + (10*I)*Sqrt[a]*Sqrt[c]*d*e
- 7*a*e^2)*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/((Sqrt[c]*d +
I*Sqrt[a]*e)^2*Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]) - ((3*I)*(4*c*d^2 - (10*I)*Sqrt[a]*Sqrt[c]*d*e - 7*a*e^2)*
ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/((Sqrt[c]*d - I*Sqrt[a]*
e)^2*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]))/(32*a^(5/2))

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c \,x^{2}+a \right )^{3} \sqrt {e x +d}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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(x*e + d)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5408 vs. \(2 (752) = 1504\).
time = 8.44, size = 5408, normalized size = 5.88 \begin {gather*} \text {Too large to display} \end {gather*}

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^2*c^4*d^4*x^4 + 2*a^3*c^3*d^4*x^2 + a^4*c^2*d^4 + (a^4*c^2*x^4 + 2*a^5*c*x^2 + a^6)*e^4 + 2*(a^3*c^
3*d^2*x^4 + 2*a^4*c^2*d^2*x^2 + a^5*c*d^2)*e^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(x*e + 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^2*c^4*d^4*x^4 + 2*a^3*c^3*d^4*x^2 + a^4*c^2*d^4 + (a^4*c^2*x^4 + 2
*a^5*c*x^2 + a^6)*e^4 + 2*(a^3*c^3*d^2*x^4 + 2*a^4*c^2*d^2*x^2 + a^5*c*d^2)*e^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(x*e + 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^2*c^4*d^4*x^4 + 2*a^3*c^3*d^4*x^2
+ a^4*c^2*d^4 + (a^4*c^2*x^4 + 2*a^5*c*x^2 + a^6)*e^4 + 2*(a^3*c^3*d^2*x^4 + 2*a^4*c^2*d^2*x^2 + a^5*c*d^2)*e
^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 ...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 [A]
time = 3.06, size = 832, normalized size = 0.90 \begin {gather*} -\frac {3 \, {\left (4 \, c d^{2} + 10 \, \sqrt {-a c} d e - 7 \, a e^{2}\right )} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4} + \sqrt {{\left (a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4}\right )}^{2} - {\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} {\left (a^{2} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\right )}}}{a^{2} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}}}}\right )}{32 \, {\left (2 \, a^{3} c d e - \sqrt {-a c} a^{2} c d^{2} + \sqrt {-a c} a^{3} e^{2}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e}} - \frac {3 \, {\left (4 \, c d^{2} - 10 \, \sqrt {-a c} d e - 7 \, a e^{2}\right )} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4} - \sqrt {{\left (a^{2} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{3} e^{2} + a^{4} c d e^{4}\right )}^{2} - {\left (a^{2} c^{3} d^{6} + 3 \, a^{3} c^{2} d^{4} e^{2} + 3 \, a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} {\left (a^{2} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}\right )}}}{a^{2} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{2} e^{2} + a^{4} c e^{4}}}}\right )}{32 \, {\left (2 \, a^{3} c d e + \sqrt {-a c} a^{2} c d^{2} - \sqrt {-a c} a^{3} e^{2}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e}} + \frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{3} d^{3} e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{3} d^{4} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{3} d^{5} e - 6 \, \sqrt {x e + d} c^{3} d^{6} e + 12 \, {\left (x e + d\right )}^{\frac {7}{2}} a c^{2} d e^{3} - 35 \, {\left (x e + d\right )}^{\frac {5}{2}} a c^{2} d^{2} e^{3} + 44 \, {\left (x e + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{3} - 21 \, \sqrt {x e + d} a c^{2} d^{4} e^{3} + 7 \, {\left (x e + d\right )}^{\frac {5}{2}} a^{2} c e^{5} + 2 \, {\left (x e + d\right )}^{\frac {3}{2}} a^{2} c d e^{5} - 4 \, \sqrt {x e + d} a^{2} c d^{2} e^{5} + 11 \, \sqrt {x e + d} a^{3} e^{7}}{16 \, {\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )}^{2}} \end {gather*}

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]

-3/32*(4*c*d^2 + 10*sqrt(-a*c)*d*e - 7*a*e^2)*abs(c)*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d^5 + 2*a^3*c^2*d^3*e
^2 + a^4*c*d*e^4 + sqrt((a^2*c^3*d^5 + 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4)^2 - (a^2*c^3*d^6 + 3*a^3*c^2*d^4*e^2 +
3*a^4*c*d^2*e^4 + a^5*e^6)*(a^2*c^3*d^4 + 2*a^3*c^2*d^2*e^2 + a^4*c*e^4)))/(a^2*c^3*d^4 + 2*a^3*c^2*d^2*e^2 +
a^4*c*e^4)))/((2*a^3*c*d*e - sqrt(-a*c)*a^2*c*d^2 + sqrt(-a*c)*a^3*e^2)*sqrt(-c^2*d - sqrt(-a*c)*c*e)) - 3/32
*(4*c*d^2 - 10*sqrt(-a*c)*d*e - 7*a*e^2)*abs(c)*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d^5 + 2*a^3*c^2*d^3*e^2 +
a^4*c*d*e^4 - sqrt((a^2*c^3*d^5 + 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4)^2 - (a^2*c^3*d^6 + 3*a^3*c^2*d^4*e^2 + 3*a^
4*c*d^2*e^4 + a^5*e^6)*(a^2*c^3*d^4 + 2*a^3*c^2*d^2*e^2 + a^4*c*e^4)))/(a^2*c^3*d^4 + 2*a^3*c^2*d^2*e^2 + a^4*
c*e^4)))/((2*a^3*c*d*e + sqrt(-a*c)*a^2*c*d^2 - sqrt(-a*c)*a^3*e^2)*sqrt(-c^2*d + sqrt(-a*c)*c*e)) + 1/16*(6*(
x*e + d)^(7/2)*c^3*d^3*e - 18*(x*e + d)^(5/2)*c^3*d^4*e + 18*(x*e + d)^(3/2)*c^3*d^5*e - 6*sqrt(x*e + d)*c^3*d
^6*e + 12*(x*e + d)^(7/2)*a*c^2*d*e^3 - 35*(x*e + d)^(5/2)*a*c^2*d^2*e^3 + 44*(x*e + d)^(3/2)*a*c^2*d^3*e^3 -
21*sqrt(x*e + d)*a*c^2*d^4*e^3 + 7*(x*e + d)^(5/2)*a^2*c*e^5 + 2*(x*e + d)^(3/2)*a^2*c*d*e^5 - 4*sqrt(x*e + d)
*a^2*c*d^2*e^5 + 11*sqrt(x*e + d)*a^3*e^7)/((a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4)*((x*e + d)^2*c - 2*(x*e
+ d)*c*d + c*d^2 + a*e^2)^2)

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Mupad [B]
time = 3.60, size = 2500, normalized size = 2.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 + 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 + 38912*
a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 + 4*a^9*c*d^2*e^6 + 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) -
((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(-a^15*c)^(1/2) + 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4
+ 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(-a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(-a^15*c)^(1/2)))/(4
096*(a^15*c*e^10 + a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 + 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 + 5*a^14*c^2
*d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 + 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6
+ 16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 + 4*a^7*c*d^2*e^6 + 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4
)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(-a^15*c)^(1/2) + 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 + 210*a^8*c^2
*d^3*e^6 - 21*c^2*d^4*e^5*(-a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(-a^15*c)^(1/2)))/(4096*(a^15*c*e
^10 + a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 + 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 + 5*a^14*c^2*d^2*e^8)))^(
1/2) - ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 + 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 + 990*a
^3*c^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 + 4*a^7*c*d^2*e^6 + 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*
(16*a^5*c^5*d^9 - 49*a^2*e^9*(-a^15*c)^(1/2) + 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 + 210*a^8*c^2*d^3*e^6
- 21*c^2*d^4*e^5*(-a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(-a^15*c)^(1/2)))/(4096*(a^15*c*e^10 + a^1
0*c^6*d^10 + 5*a^11*c^5*d^8*e^2 + 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 + 5*a^14*c^2*d^2*e^8)))^(1/2)*1i -
(((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3 + 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 + 38912*a^8*c
^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 + 4*a^9*c*d^2*e^6 + 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) + ((d +
e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(-a^15*c)^(1/2) + 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 + 210
*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(-a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(-a^15*c)^(1/2)))/(4096*(
a^15*c*e^10 + a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 + 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 + 5*a^14*c^2*d^2*
e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 + 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 + 163
84*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 + 4*a^7*c*d^2*e^6 + 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(
-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(-a^15*c)^(1/2) + 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 + 210*a^8*c^2*d^3*
e^6 - 21*c^2*d^4*e^5*(-a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(-a^15*c)^(1/2)))/(4096*(a^15*c*e^10 +
a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 + 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 + 5*a^14*c^2*d^2*e^8)))^(1/2)
+ ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d^8*e^2 + 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 + 990*a^3*c^
4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 + 4*a^7*c*d^2*e^6 + 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a
^5*c^5*d^9 - 49*a^2*e^9*(-a^15*c)^(1/2) + 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 + 210*a^8*c^2*d^3*e^6 - 21*
c^2*d^4*e^5*(-a^15*c)^(1/2) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(-a^15*c)^(1/2)))/(4096*(a^15*c*e^10 + a^10*c^6
*d^10 + 5*a^11*c^5*d^8*e^2 + 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 + 5*a^14*c^2*d^2*e^8)))^(1/2)*1i)/((3*(
144*c^6*d^7*e^3 + 684*a*c^5*d^5*e^5 + 882*a^3*c^3*d*e^9 + 1233*a^2*c^4*d^3*e^7))/(1024*(a^10*e^8 + a^6*c^4*d^8
+ 4*a^9*c*d^2*e^6 + 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) + (((3*(14336*a^9*c^3*e^11 + 4096*a^5*c^7*d^8*e^3
+ 18432*a^6*c^6*d^6*e^5 + 38912*a^7*c^5*d^4*e^7 + 38912*a^8*c^4*d^2*e^9))/(2048*(a^10*e^8 + a^6*c^4*d^8 + 4*a
^9*c*d^2*e^6 + 4*a^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) - ((d + e*x)^(1/2)*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(-
a^15*c)^(1/2) + 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 + 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(-a^15*c)^(1/2
) + 105*a^9*c*d*e^8 - 54*a*c*d^2*e^7*(-a^15*c)^(1/2)))/(4096*(a^15*c*e^10 + a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2
+ 10*a^12*c^4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 + 5*a^14*c^2*d^2*e^8)))^(1/2)*(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8
*d^9*e^2 + 16384*a^6*c^7*d^7*e^4 + 24576*a^7*c^6*d^5*e^6 + 16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8
+ 4*a^7*c*d^2*e^6 + 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(-a^15*c)^(1/2)
+ 84*a^6*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 + 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(-a^15*c)^(1/2) + 105*a^9*c
*d*e^8 - 54*a*c*d^2*e^7*(-a^15*c)^(1/2)))/(4096*(a^15*c*e^10 + a^10*c^6*d^10 + 5*a^11*c^5*d^8*e^2 + 10*a^12*c^
4*d^6*e^4 + 10*a^13*c^3*d^4*e^6 + 5*a^14*c^2*d^2*e^8)))^(1/2) - ((d + e*x)^(1/2)*(441*a^4*c^3*e^10 + 144*c^7*d
^8*e^2 + 612*a*c^6*d^6*e^4 + 1089*a^2*c^5*d^4*e^6 + 990*a^3*c^4*d^2*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 + 4*a^7*c
*d^2*e^6 + 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*(-(9*(16*a^5*c^5*d^9 - 49*a^2*e^9*(-a^15*c)^(1/2) + 84*a^6
*c^4*d^7*e^2 + 189*a^7*c^3*d^5*e^4 + 210*a^8*c^2*d^3*e^6 - 21*c^2*d^4*e^5*(-a^15*c)^(1/2) + 105*a^9*c*d*e^8 -
54*a*c*d^2*e^7*(-a^15*c)^(1/2)))/(4096*(a^15*c*...

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