\(\int \frac {(e x)^{5/2}}{(a-b x^2) (c-d x^2)^{3/2}} \, dx\) [890]

Optimal result

Integrand size = 30, antiderivative size = 414 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}+\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} (b c-a d) \sqrt {c-d x^2}}-\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{d^{3/4} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {477, 482, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{d^{3/4} \sqrt {c-d x^2} (b c-a d)}+\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{\sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)}-\frac {e (e x)^{3/2}}{\sqrt {c-d x^2} (b c-a d)} \]

[In]

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

[Out]

-((e*(e*x)^(3/2))/((b*c - a*d)*Sqrt[c - d*x^2])) + (c^(3/4)*e^(5/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1
/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(d^(3/4)*(b*c - a*d)*Sqrt[c - d*x^2]) - (c^(3/4)*e^(5/2)*Sqrt[1 - (d*x
^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(d^(3/4)*(b*c - a*d)*Sqrt[c - d*x^2]) - (
Sqrt[a]*c^(1/4)*e^(5/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d])), ArcSin[(d^(1/4)
*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(Sqrt[b]*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x^2]) + (Sqrt[a]*c^(1/4)*e^(5/2)*
Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e
])], -1])/(Sqrt[b]*d^(1/4)*(b*c - a*d)*Sqrt[c - d*x^2])

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + b*(x^4/a)]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + b*(x^4/
a)], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

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 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 482

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(n*(b*c - a*d)*(p + 1))), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 598

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 0]

Rule 1213

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

Rule 1214

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4], In
t[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] &&
!GtQ[a, 0]

Rule 1232

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[
a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rule 1233

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]
, Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a)]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^6}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = -\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}+\frac {e \text {Subst}\left (\int \frac {x^2 \left (3 a-\frac {b x^4}{e^2}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b c-a d} \\ & = -\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}+\frac {e \text {Subst}\left (\int \left (\frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}}+\frac {2 a x^2}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{b c-a d} \\ & = -\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}+\frac {e \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b c-a d}+\frac {(2 a e) \text {Subst}\left (\int \frac {x^2}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b c-a d} \\ & = -\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{\sqrt {d} (b c-a d)}+\frac {\left (\sqrt {c} e^2\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{\sqrt {d} (b c-a d)}+\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{\sqrt {b} (b c-a d)}-\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{\sqrt {b} (b c-a d)} \\ & = -\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} e^2 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{\sqrt {d} (b c-a d) \sqrt {c-d x^2}}+\frac {\left (\sqrt {c} e^2 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{\sqrt {d} (b c-a d) \sqrt {c-d x^2}}+\frac {\left (a e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{\sqrt {b} (b c-a d) \sqrt {c-d x^2}}-\frac {\left (a e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{\sqrt {b} (b c-a d) \sqrt {c-d x^2}} \\ & = -\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}-\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {\left (\sqrt {c} e^2 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}}{\sqrt {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}} \, dx,x,\sqrt {e x}\right )}{\sqrt {d} (b c-a d) \sqrt {c-d x^2}} \\ & = -\frac {e (e x)^{3/2}}{(b c-a d) \sqrt {c-d x^2}}+\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} (b c-a d) \sqrt {c-d x^2}}-\frac {c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{d^{3/4} (b c-a d) \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {b} \sqrt [4]{d} (b c-a d) \sqrt {c-d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.32 \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=-\frac {e (e x)^{3/2} \left (7 a-7 a \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+b x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{7 a (b c-a d) \sqrt {c-d x^2}} \]

[In]

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

[Out]

-1/7*(e*(e*x)^(3/2)*(7*a - 7*a*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + b*x^2*Sq
rt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a]))/(a*(b*c - a*d)*Sqrt[c - d*x^2])

Maple [A] (verified)

Time = 3.06 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.39

method	result	size
elliptic	\(\frac {\sqrt {e x}\, \sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {e^{3} x^{2}}{\left (a d -b c \right ) \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}+\frac {e^{3} c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{\left (a d -b c \right ) d \sqrt {-d e \,x^{3}+c e x}}-\frac {e^{3} c \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) d \sqrt {-d e \,x^{3}+c e x}}+\frac {e^{3} a \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {e^{3} a \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 \left (a d -b c \right ) b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{e x \sqrt {-d \,x^{2}+c}}\)	\(576\)
default	\(-\frac {\left (2 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d -2 \sqrt {2}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, E\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2}-\sqrt {2}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) a b c d +\sqrt {2}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, F\left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2} c^{2}-\sqrt {c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a d +\sqrt {c d}\, \sqrt {2}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a d +\sqrt {2}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a b c d +\sqrt {2}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right ) a b c d +2 x^{2} a b \,d^{2}-2 x^{2} b^{2} c d \right ) e^{2} \sqrt {e x}}{2 x \sqrt {-d \,x^{2}+c}\, \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {c d}\, b +\sqrt {a b}\, d \right ) \left (a d -b c \right )}\)	\(828\)

[In]

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

[Out]

1/e/x*(e*x)^(1/2)/(-d*x^2+c)^(1/2)*((-d*x^2+c)*e*x)^(1/2)*(e^3*x^2/(a*d-b*c)/(-(x^2-c/d)*d*e*x)^(1/2)+1/(a*d-b
*c)*e^3/d*c*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(
1/2)*EllipticE(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))-1/2/(a*d-b*c)*e^3/d*c*(d*x/(c*d)^(1/2)+1
)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(
1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))+1/2/(a*d-b*c)*e^3*a/b/d*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/
(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*Ellipt
icPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2)
)+1/2/(a*d-b*c)*e^3*a/b/d*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2)
)^(1/2)/(-d*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2
))^(1/2),-1/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2)),1/2*2^(1/2)))

Fricas [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=- \int \frac {\left (e x\right )^{\frac {5}{2}}}{- a c \sqrt {c - d x^{2}} + a d x^{2} \sqrt {c - d x^{2}} + b c x^{2} \sqrt {c - d x^{2}} - b d x^{4} \sqrt {c - d x^{2}}}\, dx \]

[In]

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

[Out]

-Integral((e*x)**(5/2)/(-a*c*sqrt(c - d*x**2) + a*d*x**2*sqrt(c - d*x**2) + b*c*x**2*sqrt(c - d*x**2) - b*d*x*
*4*sqrt(c - d*x**2)), x)

Maxima [F]

\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {\left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int { -\frac {\left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{5/2}}{\left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}}{\left (a-b\,x^2\right )\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]

[In]

int((e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(3/2)),x)

[Out]

int((e*x)^(5/2)/((a - b*x^2)*(c - d*x^2)^(3/2)), x)