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

Optimal. Leaf size=382 \[ -\frac {8 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {8 c d e \sqrt {a+c x^2}}{3 \sqrt {d+e x} \left (a e^2+c d^2\right )^2}-\frac {2 e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}+\frac {2 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )} \]

[Out]

-2/3*e*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)/(e*x+d)^(3/2)-8/3*c*d*e*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)-8/3
*c^(3/2)*d*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))*(-
a)^(1/2)*(e*x+d)^(1/2)*(c*x^2/a+1)^(1/2)/(a*e^2+c*d^2)^2/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1
/2)))^(1/2)+2/3*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
))*(-a)^(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)/(e*x+d)
^(1/2)/(c*x^2+a)^(1/2)

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Rubi [A]  time = 0.27, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {745, 835, 844, 719, 424, 419} \[ -\frac {8 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {8 c d e \sqrt {a+c x^2}}{3 \sqrt {d+e x} \left (a e^2+c d^2\right )^2}-\frac {2 e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}+\frac {2 \sqrt {-a} \sqrt {c} \sqrt {\frac {c x^2}{a}+1} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e*Sqrt[a + c*x^2])/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) - (8*c*d*e*Sqrt[a + c*x^2])/(3*(c*d^2 + a*e^2)^2*Sq
rt[d + e*x]) - (8*Sqrt[-a]*c^(3/2)*d*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/S
qrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c
]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*S
qrt[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*(c*d^2 + a*e^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 745

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

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} \sqrt {a+c x^2}} \, dx &=-\frac {2 e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {(2 c) \int \frac {-\frac {3 d}{2}+\frac {e x}{2}}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx}{3 \left (c d^2+a e^2\right )}\\ &=-\frac {2 e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {8 c d e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {(4 c) \int \frac {\frac {1}{4} \left (3 c d^2-a e^2\right )+c d e x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 \left (c d^2+a e^2\right )^2}\\ &=-\frac {2 e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {8 c d e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {\left (4 c^2 d\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{3 \left (c d^2+a e^2\right )^2}-\frac {c \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{3 \left (c d^2+a e^2\right )}\\ &=-\frac {2 e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {8 c d e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {\left (8 a c^{3/2} d \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 )^2 \sqrt {\frac {c (d+e x)}{c d-\frac {a \sqrt {c} e}{\sqrt {-a}}}} \sqrt {a+c x^2}}-\frac {\left (2 a \sqrt {c} \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 ) \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=-\frac {2 e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {8 c d e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {8 \sqrt {-a} c^{3/2} d \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 \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} \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 \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 1.52, size = 494, normalized size = 1.29 \[ \frac {2 \left (-e^2 \left (a+c x^2\right ) \left (a e^2+c d (5 d+4 e x)\right )+\frac {c (d+e x) \left (i (d+e x)^{3/2} \left (4 i \sqrt {a} \sqrt {c} d e-a e^2+3 c d^2\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+4 d e^2 \left (a+c x^2\right ) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}+4 \sqrt {c} d (d+e x)^{3/2} \left (\sqrt {a} e-i \sqrt {c} d\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{3 e \sqrt {a+c x^2} (d+e x)^{3/2} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-(e^2*(a + c*x^2)*(a*e^2 + c*d*(5*d + 4*e*x))) + (c*(d + e*x)*(4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a
+ c*x^2) + 4*Sqrt[c]*d*((-I)*Sqrt[c]*d + Sqrt[a]*e)*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]]/Sqr
t[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + I*(3*c*d^2 + (4*I)*Sqrt[a]*Sqrt[c]*d*e - a
*e^2)*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]]))/(3*e*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)*Sqrt[a + c*x^2]
)

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fricas [F]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c x^{2} + a} \sqrt {e x + d}}{c e^{3} x^{5} + 3 \, c d e^{2} x^{4} + 3 \, a d^{2} e x + a d^{3} + {\left (3 \, c d^{2} e + a e^{3}\right )} x^{3} + {\left (c d^{3} + 3 \, a d e^{2}\right )} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.14, size = 1906, normalized size = 4.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(4*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*d*e^3*((-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)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)+4*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^2*d^3*e*((-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)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(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*a*c*d*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)-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*e^4*((-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)*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(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^3*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)-El
lipticF((-(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*d^2*
e^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)*(-a*
c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)+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)*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*d^2*e^2+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)*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^2*d^4-3*(-(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)*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^2*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)*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*d*e^3-3*(
-(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)*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^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)*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)*c*d^3*e+4*c^2*d*e^3*x^3+a*c*e^4*x^2+
5*c^2*d^2*e^2*x^2+4*a*c*d*e^3*x+a^2*e^4+5*a*c*d^2*e^2)/(c*x^2+a)^(1/2)/(a*e^2+c*d^2)^2/(e*x+d)^(3/2)/e

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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