3.7.91 \(\int \frac {1}{(d+e x)^{5/2} (a+c x^2)^{3/2}} \, dx\) [691]

Optimal. Leaf size=485 \[ \frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (3 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {c^{3/2} d \left (3 c d^2-29 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} \left (c d^2+a e^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {\sqrt {c} \left (3 c d^2-5 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

(c*d*x+a*e)/a/(a*e^2+c*d^2)/(e*x+d)^(3/2)/(c*x^2+a)^(1/2)+1/3*e*(-5*a*e^2+3*c*d^2)*(c*x^2+a)^(1/2)/a/(a*e^2+c*
d^2)^2/(e*x+d)^(3/2)+1/3*c*d*e*(-29*a*e^2+3*c*d^2)*(c*x^2+a)^(1/2)/a/(a*e^2+c*d^2)^3/(e*x+d)^(1/2)-1/3*c^(3/2)
*d*(-29*a*e^2+3*c*d^2)*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))*(e*x+d)^(1/2)*(c*x^2/a+1)^(1/2)/(a*e^2+c*d^2)^3/(-a)^(1/2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^
(1/2)+d*c^(1/2)))^(1/2)+1/3*(-5*a*e^2+3*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*e^2+c*d^2)^2/(-a)^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]
time = 0.33, antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {755, 849, 858, 733, 435, 430} \begin {gather*} -\frac {c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 c d^2-29 a e^2\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 )}{3 \sqrt {-a} \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 {\sqrt {c} \sqrt {\frac {c x^2}{a}+1} \left (3 c d^2-5 a e^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 \sqrt {-a} \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2}+\frac {c d e \sqrt {a+c x^2} \left (3 c d^2-29 a e^2\right )}{3 a \sqrt {d+e x} \left (a e^2+c d^2\right )^3}+\frac {e \sqrt {a+c x^2} \left (3 c d^2-5 a e^2\right )}{3 a (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{a \sqrt {a+c x^2} (d+e x)^{3/2} \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*e + c*d*x)/(a*(c*d^2 + a*e^2)*(d + e*x)^(3/2)*Sqrt[a + c*x^2]) + (e*(3*c*d^2 - 5*a*e^2)*Sqrt[a + c*x^2])/(3
*a*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) + (c*d*e*(3*c*d^2 - 29*a*e^2)*Sqrt[a + c*x^2])/(3*a*(c*d^2 + a*e^2)^3*Sq
rt[d + e*x]) - (c^(3/2)*d*(3*c*d^2 - 29*a*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sq
rt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*Sqrt[-a]*(c*d^2 + a*e^2)^3*Sqrt[(Sqrt[c]
*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (Sqrt[c]*(3*c*d^2 - 5*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*Sqrt[-a]*(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 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)^{5/2} \left (a+c x^2\right )^{3/2}} \, dx &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}-\frac {\int \frac {-\frac {5 a e^2}{2}-\frac {3}{2} c d e x}{(d+e x)^{5/2} \sqrt {a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {2 \int \frac {6 a c d e^2+\frac {1}{4} c e \left (3 c d^2-5 a e^2\right ) x}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 a \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (3 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {4 \int \frac {-\frac {1}{8} a c e^2 \left (27 c d^2-5 a e^2\right )+\frac {1}{8} c^2 d e \left (3 c d^2-29 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 a \left (c d^2+a e^2\right )^3}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (3 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (c^2 d \left (3 c d^2-29 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{6 a \left (c d^2+a e^2\right )^3}+\frac {\left (c \left (3 c d^2-5 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{6 a \left (c d^2+a e^2\right )^2}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (3 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {\left (c^{3/2} d \left (3 c d^2-29 a e^2\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 )}{3 \sqrt {-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 (\sqrt {c} \left (3 c d^2-5 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} \left (c d^2+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=\frac {a e+c d x}{a \left (c d^2+a e^2\right ) (d+e x)^{3/2} \sqrt {a+c x^2}}+\frac {e \left (3 c d^2-5 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {c d e \left (3 c d^2-29 a e^2\right ) \sqrt {a+c x^2}}{3 a \left (c d^2+a e^2\right )^3 \sqrt {d+e x}}-\frac {c^{3/2} d \left (3 c d^2-29 a e^2\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 )}{3 \sqrt {-a} \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 {\sqrt {c} \left (3 c d^2-5 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 \sqrt {-a} \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 = 22.32, size = 634, normalized size = 1.31 \begin {gather*} \frac {-2 a e^3 \left (c d^2+a e^2\right ) \left (a+c x^2\right )-20 a c d e^3 (d+e x) \left (a+c x^2\right )+3 c (d+e x)^2 \left (-a^2 e^3+c^2 d^3 x+3 a c d e (d-e x)\right )+\frac {c (d+e x) \left (-d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (-29 a^2 e^2+3 c^2 d^2 x^2+a c \left (3 d^2-29 e^2 x^2\right )\right )+\sqrt {c} d \left (3 i c^{3/2} d^3-3 \sqrt {a} c d^2 e-29 i a \sqrt {c} d e^2+29 a^{3/2} e^3\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 )+\sqrt {a} e \left (3 c^{3/2} d^3+27 i \sqrt {a} c d^2 e-29 a \sqrt {c} d e^2-5 i a^{3/2} e^3\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 )\right )}{e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}}{3 a \left (c d^2+a e^2\right )^3 (d+e x)^{3/2} \sqrt {a+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*e^3*(c*d^2 + a*e^2)*(a + c*x^2) - 20*a*c*d*e^3*(d + e*x)*(a + c*x^2) + 3*c*(d + e*x)^2*(-(a^2*e^3) + c^2
*d^3*x + 3*a*c*d*e*(d - e*x)) + (c*(d + e*x)*(-(d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-29*a^2*e^2 + 3*c^2*d^
2*x^2 + a*c*(3*d^2 - 29*e^2*x^2))) + Sqrt[c]*d*((3*I)*c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e - (29*I)*a*Sqrt[c]*d*e^2
+ 29*a^(3/2)*e^3)*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)] + Sqrt[a]*e*(3*c^(3/2)*d^3 + (27*I)*Sqrt[a]*c*d^2*e - 29*a*Sqrt[c]*d*e^2 - (5*
I)*a^(3/2)*e^3)*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)]))/(e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(3*a*(c*d^2 + a*e^2)^3*(d + e*x)^(3/2)*Sq
rt[a + c*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2622\) vs. \(2(409)=818\).
time = 0.47, size = 2623, normalized size = 5.41

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 e \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 \left (e^{2} a +c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{2}}-\frac {20 \left (c e \,x^{2}+a e \right ) e^{2} c d}{3 \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 (c e x +c d \right ) \left (\frac {\left (3 e^{2} a -c \,d^{2}\right ) c d x}{2 \left (e^{2} a +c \,d^{2}\right )^{3} a}+\frac {e \left (e^{2} a -3 c \,d^{2}\right )}{2 \left (e^{2} a +c \,d^{2}\right )^{3}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}+\frac {2 \left (-\frac {e^{2} c}{3 \left (e^{2} a +c \,d^{2}\right )^{2}}+\frac {10 d^{2} e^{2} c^{2}}{3 \left (e^{2} a +c \,d^{2}\right )^{3}}-\frac {\left (e^{2} a -c \,d^{2}\right ) c}{\left (e^{2} a +c \,d^{2}\right )^{2} a}+\frac {c \,e^{2} \left (e^{2} a -3 c \,d^{2}\right )}{2 \left (e^{2} a +c \,d^{2}\right )^{3}}+\frac {c^{2} d^{2} \left (3 e^{2} a -c \,d^{2}\right )}{\left (e^{2} a +c \,d^{2}\right )^{3} a}\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 {10 d \,e^{3} c^{2}}{3 \left (e^{2} a +c \,d^{2}\right )^{3}}+\frac {c^{2} d e \left (3 e^{2} a -c \,d^{2}\right )}{2 \left (e^{2} a +c \,d^{2}\right )^{3} a}\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}}\)	\(945\)
default	\(\text {Expression too large to display}\)	\(2623\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(-29*a*c^2*d*e^5*x^3+9*d^4*e^2*c^2*a-25*e^4*d^2*a^2*c+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)*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^3*d^5*e*x+5*((-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)^(1/2)*(-(e*x+d)*c/((-a*
c)^(1/2)*e-c*d))^(1/2)*a^2*d*e^5-2*e^6*a^3+5*(-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*e^6*x-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)^(1/2)*c^2*d^5*e+24*(-(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^2*e^4+24*(-(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)*Ellipti
cF((-(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^4*e^2-2
9*(-(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*d^2*e^4-26*(-(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^2*d^4*e^2+3*c^3*d^3*e^3*x^3+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)^(1/2)*a*c*d^2*e^4*x+6*c^3*d^4*e^2*x^2-29*(-(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*d*e^5*x-26*(-(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^2*d^3*e^3*x+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)^(1/2)*a*c*d^3*e^3-3*(-(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)
^(1/2)*c^2*d^4*e^2*x+24*(-(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+24*(-(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+3*c^3*d
^5*e*x-5*a^2*c*e^6*x^2-31*a*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)*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^3*d^6-26*a^2*c*d*e^5*x+9*a*c^2*d^3*e^3*
x)/(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^3/a/(e*x+d)^(3/2)/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)^(5/2)/(c*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.38, size = 883, normalized size = 1.82 \begin {gather*} \frac {{\left (3 \, c^{3} d^{6} x^{2} + 3 \, a c^{2} d^{6} - 15 \, {\left (a^{2} c x^{4} + a^{3} x^{2}\right )} e^{6} - 30 \, {\left (a^{2} c d x^{3} + a^{3} d x\right )} e^{5} + {\left (52 \, a c^{2} d^{2} x^{4} + 37 \, a^{2} c d^{2} x^{2} - 15 \, a^{3} d^{2}\right )} e^{4} + 104 \, {\left (a c^{2} d^{3} x^{3} + a^{2} c d^{3} x\right )} e^{3} + {\left (3 \, c^{3} d^{4} x^{4} + 55 \, a c^{2} d^{4} x^{2} + 52 \, a^{2} c d^{4}\right )} e^{2} + 6 \, {\left (c^{3} d^{5} x^{3} + a 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 (29 \, {\left (a c^{2} d x^{4} + a^{2} c d x^{2}\right )} e^{5} + 58 \, {\left (a c^{2} d^{2} x^{3} + a^{2} c d^{2} x\right )} e^{4} - {\left (3 \, c^{3} d^{3} x^{4} - 26 \, a c^{2} d^{3} x^{2} - 29 \, a^{2} c d^{3}\right )} e^{3} - 6 \, {\left (c^{3} d^{4} x^{3} + a c^{2} d^{4} x\right )} e^{2} - 3 \, {\left (c^{3} d^{5} x^{2} + a 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 \, {\left (3 \, c^{3} d^{5} x e - {\left (5 \, a^{2} c x^{2} + 2 \, a^{3}\right )} e^{6} - {\left (29 \, a c^{2} d x^{3} + 26 \, a^{2} c d x\right )} e^{5} - {\left (31 \, a c^{2} d^{2} x^{2} + 25 \, a^{2} c d^{2}\right )} e^{4} + 3 \, {\left (c^{3} d^{3} x^{3} + 3 \, a c^{2} d^{3} x\right )} e^{3} + 3 \, {\left (2 \, c^{3} d^{4} x^{2} + 3 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c x^{2} + a} \sqrt {x e + d}}{9 \, {\left ({\left (a^{4} c x^{4} + a^{5} x^{2}\right )} e^{9} + 2 \, {\left (a^{4} c d x^{3} + a^{5} d x\right )} e^{8} + {\left (3 \, a^{3} c^{2} d^{2} x^{4} + 4 \, a^{4} c d^{2} x^{2} + a^{5} d^{2}\right )} e^{7} + 6 \, {\left (a^{3} c^{2} d^{3} x^{3} + a^{4} c d^{3} x\right )} e^{6} + 3 \, {\left (a^{2} c^{3} d^{4} x^{4} + 2 \, a^{3} c^{2} d^{4} x^{2} + a^{4} c d^{4}\right )} e^{5} + 6 \, {\left (a^{2} c^{3} d^{5} x^{3} + a^{3} c^{2} d^{5} x\right )} e^{4} + {\left (a c^{4} d^{6} x^{4} + 4 \, a^{2} c^{3} d^{6} x^{2} + 3 \, a^{3} c^{2} d^{6}\right )} e^{3} + 2 \, {\left (a c^{4} d^{7} x^{3} + a^{2} c^{3} d^{7} x\right )} e^{2} + {\left (a c^{4} d^{8} x^{2} + a^{2} c^{3} d^{8}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*((3*c^3*d^6*x^2 + 3*a*c^2*d^6 - 15*(a^2*c*x^4 + a^3*x^2)*e^6 - 30*(a^2*c*d*x^3 + a^3*d*x)*e^5 + (52*a*c^2*
d^2*x^4 + 37*a^2*c*d^2*x^2 - 15*a^3*d^2)*e^4 + 104*(a*c^2*d^3*x^3 + a^2*c*d^3*x)*e^3 + (3*c^3*d^4*x^4 + 55*a*c
^2*d^4*x^2 + 52*a^2*c*d^4)*e^2 + 6*(c^3*d^5*x^3 + a*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*(29*(a*c^2*d*x^4 + a^2
*c*d*x^2)*e^5 + 58*(a*c^2*d^2*x^3 + a^2*c*d^2*x)*e^4 - (3*c^3*d^3*x^4 - 26*a*c^2*d^3*x^2 - 29*a^2*c*d^3)*e^3 -
6*(c^3*d^4*x^3 + a*c^2*d^4*x)*e^2 - 3*(c^3*d^5*x^2 + a*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*(3*c^3*d^5*x*e - (5*a^2*c*x^2 + 2*a^3)*e^6 -
(29*a*c^2*d*x^3 + 26*a^2*c*d*x)*e^5 - (31*a*c^2*d^2*x^2 + 25*a^2*c*d^2)*e^4 + 3*(c^3*d^3*x^3 + 3*a*c^2*d^3*x)*
e^3 + 3*(2*c^3*d^4*x^2 + 3*a*c^2*d^4)*e^2)*sqrt(c*x^2 + a)*sqrt(x*e + d))/((a^4*c*x^4 + a^5*x^2)*e^9 + 2*(a^4*
c*d*x^3 + a^5*d*x)*e^8 + (3*a^3*c^2*d^2*x^4 + 4*a^4*c*d^2*x^2 + a^5*d^2)*e^7 + 6*(a^3*c^2*d^3*x^3 + a^4*c*d^3*
x)*e^6 + 3*(a^2*c^3*d^4*x^4 + 2*a^3*c^2*d^4*x^2 + a^4*c*d^4)*e^5 + 6*(a^2*c^3*d^5*x^3 + a^3*c^2*d^5*x)*e^4 + (
a*c^4*d^6*x^4 + 4*a^2*c^3*d^6*x^2 + 3*a^3*c^2*d^6)*e^3 + 2*(a*c^4*d^7*x^3 + a^2*c^3*d^7*x)*e^2 + (a*c^4*d^8*x^
2 + a^2*c^3*d^8)*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 {3}{2}} \left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/((c*x^2 + a)^(3/2)*(x*e + d)^(5/2)), 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 )}^{3/2}\,{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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