\(\int \frac {1}{\sqrt {d+e x} (a+c x^2)^{5/2}} \, dx\) [697]

Optimal result

Integrand size = 21, antiderivative size = 450 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\frac {(a e+c d x) \sqrt {d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+5 a e^2\right )+4 c d \left (c d^2+2 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}}+\frac {2 \sqrt {c} d \left (c d^2+2 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {\left (4 c d^2+5 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \sqrt {c} \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \]

[Out]

1/3*(c*d*x+a*e)*(e*x+d)^(1/2)/a/(a*e^2+c*d^2)/(c*x^2+a)^(3/2)+1/6*(a*e*(5*a*e^2+c*d^2)+4*c*d*(2*a*e^2+c*d^2)*x
)*(e*x+d)^(1/2)/a^2/(a*e^2+c*d^2)^2/(c*x^2+a)^(1/2)+2/3*d*(2*a*e^2+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))*c^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/(-a)^(3/2
)/(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)-1/6*(5*a*e^2+4*c*d^2)*Ellip
ticF(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))*(1+c*x^2/a)^(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)/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2
)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {755, 837, 858, 733, 435, 430} \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (4 c d x \left (2 a e^2+c d^2\right )+a e \left (5 a e^2+c d^2\right )\right )}{6 a^2 \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2}-\frac {\sqrt {\frac {c x^2}{a}+1} \left (5 a e^2+4 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \sqrt {c} \sqrt {a+c x^2} \sqrt {d+e x} \left (a e^2+c d^2\right )}+\frac {2 \sqrt {c} d \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (2 a e^2+c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \sqrt {a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}+\frac {\sqrt {d+e x} (a e+c d x)}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )} \]

[In]

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

[Out]

((a*e + c*d*x)*Sqrt[d + e*x])/(3*a*(c*d^2 + a*e^2)*(a + c*x^2)^(3/2)) + (Sqrt[d + e*x]*(a*e*(c*d^2 + 5*a*e^2)
+ 4*c*d*(c*d^2 + 2*a*e^2)*x))/(6*a^2*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^2]) + (2*Sqrt[c]*d*(c*d^2 + 2*a*e^2)*Sqrt[
d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[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[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x
^2]) - ((4*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[A
rcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(6*(-a)^(3/2)*Sqrt[c]*(c*
d^2 + a*e^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 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*} \text {integral}& = \frac {(a e+c d x) \sqrt {d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {\frac {1}{2} \left (-4 c d^2-5 a e^2\right )-\frac {3}{2} c d e x}{\sqrt {d+e x} \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) \sqrt {d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+5 a e^2\right )+4 c d \left (c d^2+2 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}}+\frac {\int \frac {\frac {1}{4} a c e^2 \left (c d^2+5 a e^2\right )-c^2 d e \left (c d^2+2 a e^2\right ) x}{\sqrt {d+e x} \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) \sqrt {d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+5 a e^2\right )+4 c d \left (c d^2+2 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}}-\frac {\left (c d \left (c d^2+2 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{3 a^2 \left (c d^2+a e^2\right )^2}+\frac {\left (4 c d^2+5 a e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{12 a^2 \left (c d^2+a e^2\right )} \\ & = \frac {(a e+c d x) \sqrt {d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+5 a e^2\right )+4 c d \left (c d^2+2 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}}-\frac {\left (2 \sqrt {c} d \left (c d^2+2 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} 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 (\left (4 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 )}{6 \sqrt {-a} a \sqrt {c} \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \\ & = \frac {(a e+c d x) \sqrt {d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+5 a e^2\right )+4 c d \left (c d^2+2 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}}+\frac {2 \sqrt {c} d \left (c d^2+2 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 (-a)^{3/2} \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 {\left (4 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 )}{6 (-a)^{3/2} \sqrt {c} \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.27 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (a c d^2 e+5 a^2 e^3+4 c^2 d^3 x+8 a c d e^2 x+\frac {2 a \left (c d^2+a e^2\right ) (a e+c d x)}{a+c x^2}-\frac {4 d e \left (c d^2+2 a e^2\right ) \left (a+c x^2\right )}{d+e x}-\frac {4 i c d \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2+2 a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{e}+\frac {\sqrt {a} \left (4 c^{3/2} d^3+i \sqrt {a} c d^2 e+8 a \sqrt {c} d e^2+5 i a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}} \]

[In]

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

[Out]

(Sqrt[d + e*x]*(a*c*d^2*e + 5*a^2*e^3 + 4*c^2*d^3*x + 8*a*c*d*e^2*x + (2*a*(c*d^2 + a*e^2)*(a*e + c*d*x))/(a +
c*x^2) - (4*d*e*(c*d^2 + 2*a*e^2)*(a + c*x^2))/(d + e*x) - ((4*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2
+ 2*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))]*Sqr
t[d + e*x]*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqr
t[c]*d + I*Sqrt[a]*e)])/e + (Sqrt[a]*(4*c^(3/2)*d^3 + I*Sqrt[a]*c*d^2*e + 8*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))]*Sqrt[d + e*x
]*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)^2*Sqrt[a + c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(857\) vs. \(2(378)=756\).

Time = 3.55 (sec) , antiderivative size = 858, normalized size of antiderivative = 1.91

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {\left (\frac {d x}{3 c a \left (e^{2} a +c \,d^{2}\right )}+\frac {e}{3 \left (e^{2} a +c \,d^{2}\right ) c^{2}}\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 {d \left (2 e^{2} a +c \,d^{2}\right ) x}{3 a^{2} \left (e^{2} a +c \,d^{2}\right )^{2}}-\frac {e \left (5 e^{2} a +c \,d^{2}\right )}{12 \left (e^{2} a +c \,d^{2}\right )^{2} a c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}+\frac {2 \left (\frac {5 e^{2} a +4 c \,d^{2}}{6 \left (e^{2} a +c \,d^{2}\right ) a^{2}}-\frac {e^{2} \left (5 e^{2} a +c \,d^{2}\right )}{12 \left (e^{2} a +c \,d^{2}\right )^{2} a}-\frac {2 c \,d^{2} \left (2 e^{2} a +c \,d^{2}\right )}{3 a^{2} \left (e^{2} a +c \,d^{2}\right )^{2}}\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}}}\, F\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 c d e \left (2 e^{2} a +c \,d^{2}\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 ) E\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}\, F\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 )}{3 \left (e^{2} a +c \,d^{2}\right )^{2} a^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(858\)
default	\(\text {Expression too large to display}\)	\(2673\)

[In]

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

[Out]

((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*((1/3/c*d/a/(a*e^2+c*d^2)*x+1/3*e/(a*e^2+c*d^2)/c^2)*(
c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x^2+1/c*a)^2-2*(c*e*x+c*d)*(-1/3*d*(2*a*e^2+c*d^2)/a^2/(a*e^2+c*d^2)^2*x-1/1
2*e*(5*a*e^2+c*d^2)/(a*e^2+c*d^2)^2/a/c)/((x^2+1/c*a)*(c*e*x+c*d))^(1/2)+2*(1/6/(a*e^2+c*d^2)*(5*a*e^2+4*c*d^2
)/a^2-1/12*e^2*(5*a*e^2+c*d^2)/(a*e^2+c*d^2)^2/a-2/3*c*d^2*(2*a*e^2+c*d^2)/a^2/(a*e^2+c*d^2)^2)*(d/e-(-a*c)^(1
/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/
c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/
2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))-2/3*c*d*e*(2*a*e^2+c*d^2)/(a*e^2+c*d^2)^2/a^2*(d/e-(-a
*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^
(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)*((-d/e-(-a*c)^(1/2)/c)*EllipticE(((x+d
/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/c*EllipticF(
((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\frac {{\left (4 \, a^{2} c^{2} d^{4} + 11 \, a^{3} c d^{2} e^{2} + 15 \, a^{4} e^{4} + {\left (4 \, c^{4} d^{4} + 11 \, a c^{3} d^{2} e^{2} + 15 \, a^{2} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (4 \, a c^{3} d^{4} + 11 \, a^{2} c^{2} d^{2} e^{2} + 15 \, a^{3} c e^{4}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (a^{2} c^{2} d^{3} e + 2 \, a^{3} c d e^{3} + {\left (c^{4} d^{3} e + 2 \, a c^{3} d e^{3}\right )} x^{4} + 2 \, {\left (a c^{3} d^{3} e + 2 \, a^{2} c^{2} d e^{3}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (3 \, a^{2} c^{2} d^{2} e^{2} + 7 \, a^{3} c e^{4} + 4 \, {\left (c^{4} d^{3} e + 2 \, a c^{3} d e^{3}\right )} x^{3} + {\left (a c^{3} d^{2} e^{2} + 5 \, a^{2} c^{2} e^{4}\right )} x^{2} + 2 \, {\left (3 \, a c^{3} d^{3} e + 5 \, a^{2} c^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{18 \, {\left (a^{4} c^{3} d^{4} e + 2 \, a^{5} c^{2} d^{2} e^{3} + a^{6} c e^{5} + {\left (a^{2} c^{5} d^{4} e + 2 \, a^{3} c^{4} d^{2} e^{3} + a^{4} c^{3} e^{5}\right )} x^{4} + 2 \, {\left (a^{3} c^{4} d^{4} e + 2 \, a^{4} c^{3} d^{2} e^{3} + a^{5} c^{2} e^{5}\right )} x^{2}\right )}} \]

[In]

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

[Out]

1/18*((4*a^2*c^2*d^4 + 11*a^3*c*d^2*e^2 + 15*a^4*e^4 + (4*c^4*d^4 + 11*a*c^3*d^2*e^2 + 15*a^2*c^2*e^4)*x^4 + 2
*(4*a*c^3*d^4 + 11*a^2*c^2*d^2*e^2 + 15*a^3*c*e^4)*x^2)*sqrt(c*e)*weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)/(c
*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 12*(a^2*c^2*d^3*e + 2*a^3*c*d*e^3 + (c^4*d^3*e
+ 2*a*c^3*d*e^3)*x^4 + 2*(a*c^3*d^3*e + 2*a^2*c^2*d*e^3)*x^2)*sqrt(c*e)*weierstrassZeta(4/3*(c*d^2 - 3*a*e^2)/
(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 +
9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e)) + 3*(3*a^2*c^2*d^2*e^2 + 7*a^3*c*e^4 + 4*(c^4*d^3*e + 2*a*c^3*d*e^3)*x
^3 + (a*c^3*d^2*e^2 + 5*a^2*c^2*e^4)*x^2 + 2*(3*a*c^3*d^3*e + 5*a^2*c^2*d*e^3)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d
))/(a^4*c^3*d^4*e + 2*a^5*c^2*d^2*e^3 + a^6*c*e^5 + (a^2*c^5*d^4*e + 2*a^3*c^4*d^2*e^3 + a^4*c^3*e^5)*x^4 + 2*
(a^3*c^4*d^4*e + 2*a^4*c^3*d^2*e^3 + a^5*c^2*e^5)*x^2)

Sympy [F]

\[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{\frac {5}{2}} \sqrt {d + e x}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,x^2+a\right )}^{5/2}\,\sqrt {d+e\,x}} \,d x \]

[In]

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

[Out]

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