3.10.26 \(\int \frac {(e x)^{9/2}}{(a-b x^2)^2 (c-d x^2)^{5/2}} \, dx\) [926]

Optimal. Leaf size=568 \[ \frac {(2 b c+3 a d) e^3 (e x)^{3/2}}{6 b (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {(b c+4 a d) e^3 (e x)^{3/2}}{2 (b c-a d)^3 \sqrt {c-d x^2}}-\frac {c^{3/4} (b c+4 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 d^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {c^{3/4} (b c+4 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 d^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (7 b c+3 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 \sqrt {b} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (7 b c+3 a d) e^{9/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 \sqrt {b} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}} \]

[Out]

1/6*(3*a*d+2*b*c)*e^3*(e*x)^(3/2)/b/(-a*d+b*c)^2/(-d*x^2+c)^(3/2)+1/2*a*e^3*(e*x)^(3/2)/b/(-a*d+b*c)/(-b*x^2+a
)/(-d*x^2+c)^(3/2)+1/2*(4*a*d+b*c)*e^3*(e*x)^(3/2)/(-a*d+b*c)^3/(-d*x^2+c)^(1/2)-1/2*c^(3/4)*(4*a*d+b*c)*e^(9/
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)^3/(-d*x^2+c)^(1/2)+1/
2*c^(3/4)*(4*a*d+b*c)*e^(9/2)*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)^3/(-d*x^2+c)^(1/2)+1/4*c^(1/4)*(3*a*d+7*b*c)*e^(9/2)*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)^3/b^(1/2)/(-d*x^2+c)^(1/2)-1/4*c^
(1/4)*(3*a*d+7*b*c)*e^(9/2)*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)^3/b^(1/2)/(-d*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.93, antiderivative size = 568, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {477, 481, 593, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \begin {gather*} \frac {c^{3/4} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (4 a d+b c) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 d^{3/4} \sqrt {c-d x^2} (b c-a d)^3}-\frac {c^{3/4} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (4 a d+b c) E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 d^{3/4} \sqrt {c-d x^2} (b c-a d)^3}+\frac {\sqrt {a} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (3 a d+7 b c) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^3}-\frac {\sqrt {a} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (3 a d+7 b c) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 \sqrt {b} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^3}+\frac {e^3 (e x)^{3/2} (4 a d+b c)}{2 \sqrt {c-d x^2} (b c-a d)^3}+\frac {e^3 (e x)^{3/2} (3 a d+2 b c)}{6 b \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac {a e^3 (e x)^{3/2}}{2 b \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

((2*b*c + 3*a*d)*e^3*(e*x)^(3/2))/(6*b*(b*c - a*d)^2*(c - d*x^2)^(3/2)) + (a*e^3*(e*x)^(3/2))/(2*b*(b*c - a*d)
*(a - b*x^2)*(c - d*x^2)^(3/2)) + ((b*c + 4*a*d)*e^3*(e*x)^(3/2))/(2*(b*c - a*d)^3*Sqrt[c - d*x^2]) - (c^(3/4)
*(b*c + 4*a*d)*e^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*d^
(3/4)*(b*c - a*d)^3*Sqrt[c - d*x^2]) + (c^(3/4)*(b*c + 4*a*d)*e^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^
(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*d^(3/4)*(b*c - a*d)^3*Sqrt[c - d*x^2]) + (Sqrt[a]*c^(1/4)*(7*b*c
+ 3*a*d)*e^(9/2)*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])/(4*Sqrt[b]*d^(1/4)*(b*c - a*d)^3*Sqrt[c - d*x^2]) - (Sqrt[a]*c^(1/4)*(7*b*c + 3*
a*d)*e^(9/2)*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])/(4*Sqrt[b]*d^(1/4)*(b*c - a*d)^3*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 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && 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 593

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p +
1))), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)
*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

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*} \int \frac {(e x)^{9/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {x^{10}}{\left (a-\frac {b x^4}{e^2}\right )^2 \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}-\frac {e^3 \text {Subst}\left (\int \frac {x^2 \left (3 a c+\frac {(4 b c+3 a d) x^4}{e^2}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt {e x}\right )}{2 b (b c-a d)}\\ &=\frac {(2 b c+3 a d) e^3 (e x)^{3/2}}{6 b (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {e^5 \text {Subst}\left (\int \frac {x^2 \left (-\frac {30 a b c^2}{e^2}-\frac {6 b c (2 b c+3 a d) x^4}{e^4}\right )}{\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 )}{12 b c (b c-a d)^2}\\ &=\frac {(2 b c+3 a d) e^3 (e x)^{3/2}}{6 b (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {(b c+4 a d) e^3 (e x)^{3/2}}{2 (b c-a d)^3 \sqrt {c-d x^2}}-\frac {e^7 \text {Subst}\left (\int \frac {x^2 \left (\frac {12 a b c^2 (8 b c+7 a d)}{e^4}-\frac {12 b^2 c^2 (b c+4 a d) x^4}{e^6}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{24 b c^2 (b c-a d)^3}\\ &=\frac {(2 b c+3 a d) e^3 (e x)^{3/2}}{6 b (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {(b c+4 a d) e^3 (e x)^{3/2}}{2 (b c-a d)^3 \sqrt {c-d x^2}}-\frac {e^7 \text {Subst}\left (\int \left (\frac {12 b c^2 (b c+4 a d) x^2}{e^4 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {12 \left (7 a b^2 c^3+3 a^2 b c^2 d\right ) x^2}{e^4 \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{24 b c^2 (b c-a d)^3}\\ &=\frac {(2 b c+3 a d) e^3 (e x)^{3/2}}{6 b (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {(b c+4 a d) e^3 (e x)^{3/2}}{2 (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (a (7 b c+3 a d) e^3\right ) \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 )}{2 (b c-a d)^3}-\frac {\left ((b c+4 a d) e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 (b c-a d)^3}\\ &=\frac {(2 b c+3 a d) e^3 (e x)^{3/2}}{6 b (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {(b c+4 a d) e^3 (e x)^{3/2}}{2 (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\left (\sqrt {c} (b c+4 a d) e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 \sqrt {d} (b c-a d)^3}-\frac {\left (\sqrt {c} (b c+4 a d) e^4\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 )}{2 \sqrt {d} (b c-a d)^3}-\frac {\left (a (7 b c+3 a d) e^5\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 )}{4 \sqrt {b} (b c-a d)^3}+\frac {\left (a (7 b c+3 a d) e^5\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 )}{4 \sqrt {b} (b c-a d)^3}\\ &=\frac {(2 b c+3 a d) e^3 (e x)^{3/2}}{6 b (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {(b c+4 a d) e^3 (e x)^{3/2}}{2 (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\left (\sqrt {c} (b c+4 a d) e^4 \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 )}{2 \sqrt {d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} (b c+4 a d) e^4 \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 )}{2 \sqrt {d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (a (7 b c+3 a d) e^5 \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 )}{4 \sqrt {b} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\left (a (7 b c+3 a d) e^5 \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 )}{4 \sqrt {b} (b c-a d)^3 \sqrt {c-d x^2}}\\ &=\frac {(2 b c+3 a d) e^3 (e x)^{3/2}}{6 b (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {(b c+4 a d) e^3 (e x)^{3/2}}{2 (b c-a d)^3 \sqrt {c-d x^2}}+\frac {c^{3/4} (b c+4 a d) e^{9/2} \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 )}{2 d^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (7 b c+3 a d) e^{9/2} \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 )}{4 \sqrt {b} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (7 b c+3 a d) e^{9/2} \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 )}{4 \sqrt {b} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} (b c+4 a d) e^4 \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 )}{2 \sqrt {d} (b c-a d)^3 \sqrt {c-d x^2}}\\ &=\frac {(2 b c+3 a d) e^3 (e x)^{3/2}}{6 b (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {(b c+4 a d) e^3 (e x)^{3/2}}{2 (b c-a d)^3 \sqrt {c-d x^2}}-\frac {c^{3/4} (b c+4 a d) e^{9/2} \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 )}{2 d^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {c^{3/4} (b c+4 a d) e^{9/2} \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 )}{2 d^{3/4} (b c-a d)^3 \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (7 b c+3 a d) e^{9/2} \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 )}{4 \sqrt {b} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (7 b c+3 a d) e^{9/2} \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 )}{4 \sqrt {b} \sqrt [4]{d} (b c-a d)^3 \sqrt {c-d x^2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 10.30, size = 256, normalized size = 0.45 \begin {gather*} -\frac {e^3 (e x)^{3/2} \left (7 a \left (a^2 d \left (7 c-9 d x^2\right )+b^2 c x^2 \left (-5 c+3 d x^2\right )+4 a b \left (2 c^2-4 c d x^2+3 d^2 x^4\right )\right )+7 a (8 b c+7 a d) \left (-a+b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+3 b (b c+4 a d) x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{42 a (b c-a d)^3 \left (-a+b x^2\right ) \left (c-d x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

-1/42*(e^3*(e*x)^(3/2)*(7*a*(a^2*d*(7*c - 9*d*x^2) + b^2*c*x^2*(-5*c + 3*d*x^2) + 4*a*b*(2*c^2 - 4*c*d*x^2 + 3
*d^2*x^4)) + 7*a*(8*b*c + 7*a*d)*(-a + b*x^2)*(c - d*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^
2)/c, (b*x^2)/a] + 3*b*(b*c + 4*a*d)*x^2*(a - b*x^2)*(c - d*x^2)*Sqrt[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)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5113\) vs. \(2(446)=892\).
time = 0.15, size = 5114, normalized size = 9.00

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1409\)
default	\(\text {Expression too large to display}\)	\(5114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

e^(9/2)*integrate(x^(9/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(9/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)

[Out]

Timed out

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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((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(x^(9/2)*e^(9/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x\right )}^{9/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)),x)

[Out]

int((e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(5/2)), x)

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