3.7.98 \(\int \frac {1}{(d+e x)^{3/2} (a+c x^2)^{5/2}} \, dx\) [698]

Optimal. Leaf size=532 \[ \frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {\sqrt {c} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \left (c d^2+a e^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {2 \sqrt {c} d \left (c d^2+3 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

1/3*(c*d*x+a*e)/a/(a*e^2+c*d^2)/(c*x^2+a)^(3/2)/(e*x+d)^(1/2)+1/6*(-a*e*(-7*a*e^2+c*d^2)+4*c*d*(3*a*e^2+c*d^2)
*x)/a^2/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)+1/6*e*(-21*a^2*e^4+15*a*c*d^2*e^2+4*c^2*d^4)*(c*x^2+a)^(
1/2)/a^2/(a*e^2+c*d^2)^3/(e*x+d)^(1/2)+1/6*(-21*a^2*e^4+15*a*c*d^2*e^2+4*c^2*d^4)*EllipticE(1/2*(1-x*c^(1/2)/(
-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*c^(1/2)*(e*x+d)^(1/2)*(c*x^2/a+1)^(1/2)/(
-a)^(3/2)/(a*e^2+c*d^2)^3/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-2/3*d*(3*a*e^2+c*d^
2)*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*c^(1/2)*(c
*x^2/a+1)^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)/(-a)^(3/2)/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)/(c*x
^2+a)^(1/2)

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Rubi [A]
time = 0.37, antiderivative size = 532, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {755, 837, 849, 858, 733, 435, 430} \begin {gather*} \frac {\sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \sqrt {a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {e \sqrt {a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 \sqrt {d+e x} \left (a e^2+c d^2\right )^3}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{6 a^2 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2}-\frac {2 \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \left (3 a e^2+c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*(a + c*x^2)^(5/2)),x]

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]*(a + c*x^2)^(3/2)) - (a*e*(c*d^2 - 7*a*e^2) - 4*c*d*(c*d^2 +
3*a*e^2)*x)/(6*a^2*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + c*x^2]) + (e*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*
e^4)*Sqrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]) + (Sqrt[c]*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*
e^4)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqr
t[-a]*Sqrt[c]*d - a*e)])/(6*(-a)^(3/2)*(c*d^2 + a*e^2)^3*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sq
rt[a + c*x^2]) - (2*Sqrt[c]*d*(c*d^2 + 3*a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c
*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(-
a)^(3/2)*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]*Sqrt[a + c*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 733

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*a*Rt[-c/a, 2]*(d + e*x)^m*(Sqrt[1
+ c*(x^2/a)]/(c*Sqrt[a + c*x^2]*(c*((d + e*x)/(c*d - a*e*Rt[-c/a, 2])))^m)), Subst[Int[(1 + 2*a*e*Rt[-c/a, 2]*
(x^2/(c*d - a*e*Rt[-c/a, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-c/a, 2]*x)/2]], x] /; FreeQ[{a, c, d, e},
x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

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 849

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

Rule 858

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

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {\frac {1}{2} \left (-4 c d^2-7 a e^2\right )-\frac {5}{2} c d e x}{(d+e x)^{3/2} \left (a+c x^2\right )^{3/2}} \, dx}{3 a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {\int \frac {-\frac {3}{4} a c e^2 \left (c d^2-7 a e^2\right )+c^2 d e \left (c d^2+3 a e^2\right ) x}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 a^2 c \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {2 \int \frac {-\frac {1}{8} a c^2 d e^2 \left (c d^2+33 a e^2\right )+\frac {1}{8} c^2 e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 a^2 c \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {\left (c d \left (c d^2+3 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 a^2 \left (c d^2+a e^2\right )^2}-\frac {\left (c \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{12 a^2 \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (\sqrt {c} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{6 \sqrt {-a} a \left (c d^2+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (2 \sqrt {c} d \left (c d^2+3 a e^2\right ) \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} e x^2}{\sqrt {-a} \left (c d-\frac {a \sqrt {c} e}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{3 \sqrt {-a} a \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {a e+c d x}{3 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}-\frac {a e \left (c d^2-7 a e^2\right )-4 c d \left (c d^2+3 a e^2\right ) x}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}+\frac {e \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {a+c x^2}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}+\frac {\sqrt {c} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \left (c d^2+a e^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {2 \sqrt {c} d \left (c d^2+3 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 22.72, size = 669, normalized size = 1.26 \begin {gather*} \frac {21 a^3 e^5-4 c^3 d^4 e x^2-12 a^2 e^5 \left (a+c x^2\right )+3 a^2 c e^3 \left (-5 d^2+7 e^2 x^2\right )-a c^2 d^2 e \left (4 d^2+15 e^2 x^2\right )+\frac {2 a c \left (c d^2+a e^2\right ) (d+e x) \left (c d^2 x+a e (2 d-e x)\right )}{a+c x^2}+c (d+e x) \left (4 c^2 d^4 x+3 a^2 e^3 (7 d-3 e x)+a c d^2 e (d+15 e x)\right )-\frac {i c \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (4 c^2 d^4+15 a c d^2 e^2-21 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{e}+\frac {\sqrt {a} \sqrt {c} \left (4 c^2 d^4+i \sqrt {a} c^{3/2} d^3 e+15 a c d^2 e^2+33 i a^{3/2} \sqrt {c} d e^3-21 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}}{6 a^2 \left (c d^2+a e^2\right )^3 \sqrt {d+e x} \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^(5/2)),x]

[Out]

(21*a^3*e^5 - 4*c^3*d^4*e*x^2 - 12*a^2*e^5*(a + c*x^2) + 3*a^2*c*e^3*(-5*d^2 + 7*e^2*x^2) - a*c^2*d^2*e*(4*d^2
+ 15*e^2*x^2) + (2*a*c*(c*d^2 + a*e^2)*(d + e*x)*(c*d^2*x + a*e*(2*d - e*x)))/(a + c*x^2) + c*(d + e*x)*(4*c^
2*d^4*x + 3*a^2*e^3*(7*d - 3*e*x) + a*c*d^2*e*(d + 15*e*x)) - (I*c*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(4*c^2*d^4
+ 15*a*c*d^2*e^2 - 21*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] -
e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]
*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/e + (Sqrt[a]*Sqrt[c]*(4*c^2*d^4 + I*Sqrt[a]*c^(3/2)*d^3*e + 15*a
*c*d^2*e^2 + (33*I)*a^(3/2)*Sqrt[c]*d*e^3 - 21*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-((
(I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/
Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]])/(6*a^2
*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3321\) vs. \(2(455)=910\).
time = 0.48, size = 3322, normalized size = 6.24

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {\left (-\frac {\left (e^{2} a -c \,d^{2}\right ) x}{3 \left (e^{2} a +c \,d^{2}\right )^{2} a c}+\frac {2 d e}{3 \left (e^{2} a +c \,d^{2}\right )^{2} c}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 \left (c e x +c d \right ) \left (\frac {\left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right ) x}{12 \left (e^{2} a +c \,d^{2}\right )^{3} a^{2}}-\frac {d e \left (21 e^{2} a +c \,d^{2}\right )}{12 \left (e^{2} a +c \,d^{2}\right )^{3} a}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}-\frac {2 \left (c e \,x^{2}+a e \right ) e^{4}}{\left (e^{2} a +c \,d^{2}\right )^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 \left (\frac {2 c d \left (3 e^{2} a +c \,d^{2}\right )}{3 \left (e^{2} a +c \,d^{2}\right )^{2} a^{2}}-\frac {c \,e^{2} d \left (21 e^{2} a +c \,d^{2}\right )}{12 \left (e^{2} a +c \,d^{2}\right )^{3} a}+\frac {c d \left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right )}{6 \left (e^{2} a +c \,d^{2}\right )^{3} a^{2}}+\frac {e^{4} c d}{\left (e^{2} a +c \,d^{2}\right )^{3}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (\frac {c e \left (9 a^{2} e^{4}-15 a c \,d^{2} e^{2}-4 c^{2} d^{4}\right )}{12 \left (e^{2} a +c \,d^{2}\right )^{3} a^{2}}+\frac {e^{5} c}{\left (e^{2} a +c \,d^{2}\right )^{3}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, \EllipticF \left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(1000\)
default	\(\text {Expression too large to display}\)	\(3322\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/6*(21*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c
)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d
)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*c*e^6*x^2+19*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e
/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(
1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c^2*d^4*e^2-21*(-(e*x+d)*c/((-a*c)^
(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c
*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2
))*a^3*c*e^6*x^2-12*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*
e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1
/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*d*e^5+18*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*(
(-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF(
(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*c*d^2*e^4-3*(-
(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e
/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^
(1/2)*e+c*d))^(1/2))*a^2*c^2*d^4*e^2-6*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^
(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*
d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*c*d^2*e^4-12*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c
)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e
-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1
/2))*a^2*c*d*e^5*x^2-16*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1
/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d)
)^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^2*d^3*e^3*x^2+4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*
d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)
*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^4*d^6
*x^2+4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)
^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)
/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^3*d^6+12*a^2*c^2*d*e^5*x^3+16*a*c^3*d^3*e^3*x^3+14*a^3*c*d*e^5*x+20*a^2*c^2*
d^3*e^3*x+6*a*c^3*d^5*e*x+21*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*
d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),
(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^4*e^6-21*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+
(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x
+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^4*e^6-12*a^4*e^6+4*c^4
*d^5*e*x^3-4*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))
^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-(
(-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^3*d^5*e*x^2+18*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c
*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(
e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c^2*d^2*e^4*x^2-3
*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2)
)*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*
c)^(1/2)*e+c*d))^(1/2))*a*c^3*d^4*e^2*x^2-6*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-
a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)
*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c^2*d^2*e^4*x^2-16*(-a*c)^(1/2)*(-(e*x+
d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a
*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)
*e+c*d))^(1/2))*a^2*c*d^3*e^3-4*(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((
-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e...

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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/(e*x+d)^(3/2)/(c*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^(5/2)*(x*e + d)^(3/2)), x)

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.92, size = 1103, normalized size = 2.07 \begin {gather*} \frac {2 \, {\left (2 \, c^{4} d^{6} x^{4} + 4 \, a c^{3} d^{6} x^{2} + 2 \, a^{2} c^{2} d^{6} + 39 \, {\left (a^{2} c^{2} d x^{5} + 2 \, a^{3} c d x^{3} + a^{4} d x\right )} e^{5} + 39 \, {\left (a^{2} c^{2} d^{2} x^{4} + 2 \, a^{3} c d^{2} x^{2} + a^{4} d^{2}\right )} e^{4} + 9 \, {\left (a c^{3} d^{3} x^{5} + 2 \, a^{2} c^{2} d^{3} x^{3} + a^{3} c d^{3} x\right )} e^{3} + 9 \, {\left (a c^{3} d^{4} x^{4} + 2 \, a^{2} c^{2} d^{4} x^{2} + a^{3} c d^{4}\right )} e^{2} + 2 \, {\left (c^{4} d^{5} x^{5} + 2 \, a c^{3} d^{5} x^{3} + a^{2} c^{2} d^{5} x\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right ) - 3 \, {\left (21 \, {\left (a^{2} c^{2} x^{5} + 2 \, a^{3} c x^{3} + a^{4} x\right )} e^{6} + 21 \, {\left (a^{2} c^{2} d x^{4} + 2 \, a^{3} c d x^{2} + a^{4} d\right )} e^{5} - 15 \, {\left (a c^{3} d^{2} x^{5} + 2 \, a^{2} c^{2} d^{2} x^{3} + a^{3} c d^{2} x\right )} e^{4} - 15 \, {\left (a c^{3} d^{3} x^{4} + 2 \, a^{2} c^{2} d^{3} x^{2} + a^{3} c d^{3}\right )} e^{3} - 4 \, {\left (c^{4} d^{4} x^{5} + 2 \, a c^{3} d^{4} x^{3} + a^{2} c^{2} d^{4} x\right )} e^{2} - 4 \, {\left (c^{4} d^{5} x^{4} + 2 \, a c^{3} d^{5} x^{2} + a^{2} c^{2} d^{5}\right )} e\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )} e^{\left (-2\right )}}{3 \, c}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )} e^{\left (-3\right )}}{27 \, c}, \frac {1}{3} \, {\left (3 \, x e + d\right )} e^{\left (-1\right )}\right )\right ) - 3 \, \sqrt {c x^{2} + a} {\left ({\left (21 \, a^{2} c^{2} x^{4} + 35 \, a^{3} c x^{2} + 12 \, a^{4}\right )} e^{6} - 2 \, {\left (6 \, a^{2} c^{2} d x^{3} + 7 \, a^{3} c d x\right )} e^{5} - {\left (15 \, a c^{3} d^{2} x^{4} + 36 \, a^{2} c^{2} d^{2} x^{2} + 25 \, a^{3} c d^{2}\right )} e^{4} - 4 \, {\left (4 \, a c^{3} d^{3} x^{3} + 5 \, a^{2} c^{2} d^{3} x\right )} e^{3} - {\left (4 \, c^{4} d^{4} x^{4} + 7 \, a c^{3} d^{4} x^{2} + 5 \, a^{2} c^{2} d^{4}\right )} e^{2} - 2 \, {\left (2 \, c^{4} d^{5} x^{3} + 3 \, a c^{3} d^{5} x\right )} e\right )} \sqrt {x e + d}}{18 \, {\left ({\left (a^{5} c^{2} x^{5} + 2 \, a^{6} c x^{3} + a^{7} x\right )} e^{8} + {\left (a^{5} c^{2} d x^{4} + 2 \, a^{6} c d x^{2} + a^{7} d\right )} e^{7} + 3 \, {\left (a^{4} c^{3} d^{2} x^{5} + 2 \, a^{5} c^{2} d^{2} x^{3} + a^{6} c d^{2} x\right )} e^{6} + 3 \, {\left (a^{4} c^{3} d^{3} x^{4} + 2 \, a^{5} c^{2} d^{3} x^{2} + a^{6} c d^{3}\right )} e^{5} + 3 \, {\left (a^{3} c^{4} d^{4} x^{5} + 2 \, a^{4} c^{3} d^{4} x^{3} + a^{5} c^{2} d^{4} x\right )} e^{4} + 3 \, {\left (a^{3} c^{4} d^{5} x^{4} + 2 \, a^{4} c^{3} d^{5} x^{2} + a^{5} c^{2} d^{5}\right )} e^{3} + {\left (a^{2} c^{5} d^{6} x^{5} + 2 \, a^{3} c^{4} d^{6} x^{3} + a^{4} c^{3} d^{6} x\right )} e^{2} + {\left (a^{2} c^{5} d^{7} x^{4} + 2 \, a^{3} c^{4} d^{7} x^{2} + a^{4} c^{3} d^{7}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

1/18*(2*(2*c^4*d^6*x^4 + 4*a*c^3*d^6*x^2 + 2*a^2*c^2*d^6 + 39*(a^2*c^2*d*x^5 + 2*a^3*c*d*x^3 + a^4*d*x)*e^5 +
39*(a^2*c^2*d^2*x^4 + 2*a^3*c*d^2*x^2 + a^4*d^2)*e^4 + 9*(a*c^3*d^3*x^5 + 2*a^2*c^2*d^3*x^3 + a^3*c*d^3*x)*e^3
+ 9*(a*c^3*d^4*x^4 + 2*a^2*c^2*d^4*x^2 + a^3*c*d^4)*e^2 + 2*(c^4*d^5*x^5 + 2*a*c^3*d^5*x^3 + a^2*c^2*d^5*x)*e
)*sqrt(c)*e^(1/2)*weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)*e^(-2)/c, -8/27*(c*d^3 + 9*a*d*e^2)*e^(-3)/c, 1/3*
(3*x*e + d)*e^(-1)) - 3*(21*(a^2*c^2*x^5 + 2*a^3*c*x^3 + a^4*x)*e^6 + 21*(a^2*c^2*d*x^4 + 2*a^3*c*d*x^2 + a^4*
d)*e^5 - 15*(a*c^3*d^2*x^5 + 2*a^2*c^2*d^2*x^3 + a^3*c*d^2*x)*e^4 - 15*(a*c^3*d^3*x^4 + 2*a^2*c^2*d^3*x^2 + a^
3*c*d^3)*e^3 - 4*(c^4*d^4*x^5 + 2*a*c^3*d^4*x^3 + a^2*c^2*d^4*x)*e^2 - 4*(c^4*d^5*x^4 + 2*a*c^3*d^5*x^2 + a^2*
c^2*d^5)*e)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c*d^2 - 3*a*e^2)*e^(-2)/c, -8/27*(c*d^3 + 9*a*d*e^2)*e^(-3)/c
, weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)*e^(-2)/c, -8/27*(c*d^3 + 9*a*d*e^2)*e^(-3)/c, 1/3*(3*x*e + d)*e^(-
1))) - 3*sqrt(c*x^2 + a)*((21*a^2*c^2*x^4 + 35*a^3*c*x^2 + 12*a^4)*e^6 - 2*(6*a^2*c^2*d*x^3 + 7*a^3*c*d*x)*e^5
- (15*a*c^3*d^2*x^4 + 36*a^2*c^2*d^2*x^2 + 25*a^3*c*d^2)*e^4 - 4*(4*a*c^3*d^3*x^3 + 5*a^2*c^2*d^3*x)*e^3 - (4
*c^4*d^4*x^4 + 7*a*c^3*d^4*x^2 + 5*a^2*c^2*d^4)*e^2 - 2*(2*c^4*d^5*x^3 + 3*a*c^3*d^5*x)*e)*sqrt(x*e + d))/((a^
5*c^2*x^5 + 2*a^6*c*x^3 + a^7*x)*e^8 + (a^5*c^2*d*x^4 + 2*a^6*c*d*x^2 + a^7*d)*e^7 + 3*(a^4*c^3*d^2*x^5 + 2*a^
5*c^2*d^2*x^3 + a^6*c*d^2*x)*e^6 + 3*(a^4*c^3*d^3*x^4 + 2*a^5*c^2*d^3*x^2 + a^6*c*d^3)*e^5 + 3*(a^3*c^4*d^4*x^
5 + 2*a^4*c^3*d^4*x^3 + a^5*c^2*d^4*x)*e^4 + 3*(a^3*c^4*d^5*x^4 + 2*a^4*c^3*d^5*x^2 + a^5*c^2*d^5)*e^3 + (a^2*
c^5*d^6*x^5 + 2*a^3*c^4*d^6*x^3 + a^4*c^3*d^6*x)*e^2 + (a^2*c^5*d^7*x^4 + 2*a^3*c^4*d^7*x^2 + a^4*c^3*d^7)*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + c x^{2}\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(c*x**2+a)**(5/2),x)

[Out]

Integral(1/((a + c*x**2)**(5/2)*(d + e*x)**(3/2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + a)^(5/2)*(x*e + d)^(3/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (c\,x^2+a\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^2)^(5/2)*(d + e*x)^(3/2)),x)

[Out]

int(1/((a + c*x^2)^(5/2)*(d + e*x)^(3/2)), x)

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