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

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

[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 = 0.49, 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 = {741, 835, 844, 719, 424, 419} \[ -\frac {c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} \sqrt {a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {c d e \sqrt {a+c x^2} \left (3 c d^2-29 a e^2\right )}{3 a \sqrt {d+e x} \left (a e^2+c d^2\right )^3}+\frac {e \sqrt {a+c x^2} \left (3 c d^2-5 a e^2\right )}{3 a (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}+\frac {a e+c d x}{a \sqrt {a+c x^2} (d+e x)^{3/2} \left (a e^2+c d^2\right )}+\frac {\sqrt {c} \sqrt {\frac {c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {-a} \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )^2} \]

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 419

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

Rule 424

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

Rule 719

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 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 835

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 844

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 ) \operatorname {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 ) \operatorname {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]  time = 3.35, size = 634, normalized size = 1.31 \[ \frac {3 c (d+e x)^2 \left (-a^2 e^3+3 a c d e (d-e x)+c^2 d^3 x\right )+\frac {c (d+e x) \left (\sqrt {a} e (d+e x)^{3/2} \left (-5 i a^{3/2} e^3+27 i \sqrt {a} c d^2 e-29 a \sqrt {c} d e^2+3 c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {c} d (d+e x)^{3/2} \left (29 a^{3/2} e^3-3 \sqrt {a} c d^2 e-29 i a \sqrt {c} d e^2+3 i c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (-29 a^2 e^2+a c \left (3 d^2-29 e^2 x^2\right )+3 c^2 d^2 x^2\right )\right )}{e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}-2 a e^3 \left (a+c x^2\right ) \left (a e^2+c d^2\right )-20 a c d e^3 \left (a+c x^2\right ) (d+e x)}{3 a \sqrt {a+c x^2} (d+e x)^{3/2} \left (a e^2+c d^2\right )^3} \]

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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fricas [F]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} \sqrt {e x + d}}{c^{2} e^{3} x^{7} + 3 \, c^{2} d e^{2} x^{6} + 3 \, a^{2} d^{2} e x + {\left (3 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} x^{5} + a^{2} d^{3} + {\left (c^{2} d^{3} + 6 \, a c d e^{2}\right )} x^{4} + {\left (6 \, a c d^{2} e + a^{2} e^{3}\right )} x^{3} + {\left (2 \, a c d^{3} + 3 \, a^{2} d e^{2}\right )} x^{2}}, x\right ) \]

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]

integral(sqrt(c*x^2 + a)*sqrt(e*x + d)/(c^2*e^3*x^7 + 3*c^2*d*e^2*x^6 + 3*a^2*d^2*e*x + (3*c^2*d^2*e + 2*a*c*e
^3)*x^5 + a^2*d^3 + (c^2*d^3 + 6*a*c*d*e^2)*x^4 + (6*a*c*d^2*e + a^2*e^3)*x^3 + (2*a*c*d^3 + 3*a^2*d*e^2)*x^2)
, x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [B]  time = 0.16, size = 2623, normalized size = 5.41 \[ \text {result too large to display} \]

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)

[Out]

1/3*(5*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c
)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2
)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*(-a*c)^(1/2)*a^2*d*e^5-29*a*c^2*d*e^5*x^3-31*a*c^2*d^2*e^4*x^2+9*a*c^2*d^3*e
^3*x+3*c^3*d^5*e*x+3*c^3*d^3*e^3*x^3-5*a^2*c*e^6*x^2-25*a^2*c*d^2*e^4+9*a*c^2*d^4*e^2+3*EllipticE((-(e*x+d)/(-
c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*c^3*d^6*(-(e*x+d)/(-c*d+(-a*
c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)
*e)*e)^(1/2)-2*a^3*e^6+6*c^3*d^4*e^2*x^2+24*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^
(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x*a^2*c*d*e^5*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/
2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+24*EllipticF((-(e*x+d)/(-
c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x*a*c^2*d^3*e^3*(-(e*x+d)/(-
c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*
c)^(1/2)*e)*e)^(1/2)-3*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^
(1/2)*e))^(1/2))*x*c^2*d^4*e^2*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d
+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-29*EllipticE((-(e*x+d)/(-c*d+(-a*
c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x*a^2*c*d*e^5*(-(e*x+d)/(-c*d+(-a*c)
^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e
)*e)^(1/2)-26*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))
^(1/2))*x*a*c^2*d^3*e^3*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^
(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+2*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),
(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a*c*d^3*e^3*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*
c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)
+2*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x*a
*c*d^2*e^4*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^
(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-26*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)
,(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a*c^2*d^4*e^2*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*(
(-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+24*Ellipt
icF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^2*c*d^2*e^
4*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/
2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+24*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*
e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a*c^2*d^4*e^2*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c
*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+3*EllipticE((-(e*x+d)/(-c*d+(-a
*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x*c^3*d^5*e*(-(e*x+d)/(-c*d+(-a*c)^
(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)
*e)^(1/2)-3*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(
1/2))*c^2*d^5*e*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e
)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-29*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^
(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^2*c*d^2*e^4*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1
/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-26*a
^2*c*d*e^5*x+5*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e)
)^(1/2))*x*a^2*e^6*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2
)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2))/(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 \[ \int \frac {1}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{2}}}\,{d x} \]

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)*(e*x + d)^(5/2)), x)

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

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

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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