3.918 \(\int \frac {(e x)^{9/2}}{(a-b x^2)^2 (c-d x^2)^{3/2}} \, dx\)

Optimal. Leaf size=529 \[ \frac {\sqrt {a} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (7 b c-a d) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^2}-\frac {\sqrt {a} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (7 b c-a d) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^2}+\frac {c^{3/4} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (a d+2 b c) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 b d^{3/4} \sqrt {c-d x^2} (b c-a d)^2}-\frac {c^{3/4} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (a d+2 b c) E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 b d^{3/4} \sqrt {c-d x^2} (b c-a d)^2}+\frac {e^3 (e x)^{3/2} (a d+2 b c)}{2 b \sqrt {c-d x^2} (b c-a d)^2}+\frac {a e^3 (e x)^{3/2}}{2 b \left (a-b x^2\right ) \sqrt {c-d x^2} (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.08, antiderivative size = 529, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {466, 470, 579, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ \frac {\sqrt {a} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (7 b c-a d) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^2}-\frac {\sqrt {a} \sqrt [4]{c} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (7 b c-a d) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2} (b c-a d)^2}+\frac {c^{3/4} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (a d+2 b c) F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 b d^{3/4} \sqrt {c-d x^2} (b c-a d)^2}-\frac {c^{3/4} e^{9/2} \sqrt {1-\frac {d x^2}{c}} (a d+2 b c) E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 b d^{3/4} \sqrt {c-d x^2} (b c-a d)^2}+\frac {e^3 (e x)^{3/2} (a d+2 b c)}{2 b \sqrt {c-d x^2} (b c-a d)^2}+\frac {a e^3 (e x)^{3/2}}{2 b \left (a-b x^2\right ) \sqrt {c-d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]

[Out]

((2*b*c + a*d)*e^3*(e*x)^(3/2))/(2*b*(b*c - a*d)^2*Sqrt[c - d*x^2]) + (a*e^3*(e*x)^(3/2))/(2*b*(b*c - a*d)*(a
- b*x^2)*Sqrt[c - d*x^2]) - (c^(3/4)*(2*b*c + 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*b*d^(3/4)*(b*c - a*d)^2*Sqrt[c - d*x^2]) + (c^(3/4)*(2*b*c + a*d)*e^(9/2)*Sq
rt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*b*d^(3/4)*(b*c - a*d)^2*Sqr
t[c - d*x^2]) + (Sqrt[a]*c^(1/4)*(7*b*c - a*d)*e^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sqr
t[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*b^(3/2)*d^(1/4)*(b*c - a*d)^2*Sqrt[c -
d*x^2]) - (Sqrt[a]*c^(1/4)*(7*b*c - a*d)*e^(9/2)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqr
t[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*b^(3/2)*d^(1/4)*(b*c - a*d)^2*Sqrt[c - d*x^2])

Rule 221

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 224

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 307

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 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 466

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 470

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 490

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), In
t[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 579

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 584

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 1199

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 1200

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 1218

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

Rule 1219

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 )^{3/2}} \, dx &=\frac {2 \operatorname {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 )^{3/2}} \, 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 ) \sqrt {c-d x^2}}-\frac {e^3 \operatorname {Subst}\left (\int \frac {x^2 \left (3 a c+\frac {(4 b c-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 )^{3/2}} \, dx,x,\sqrt {e x}\right )}{2 b (b c-a d)}\\ &=\frac {(2 b c+a d) e^3 (e x)^{3/2}}{2 b (b c-a d)^2 \sqrt {c-d x^2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {e^5 \operatorname {Subst}\left (\int \frac {x^2 \left (-\frac {18 a b c^2}{e^2}+\frac {2 b c (2 b c+a d) x^4}{e^4}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{4 b c (b c-a d)^2}\\ &=\frac {(2 b c+a d) e^3 (e x)^{3/2}}{2 b (b c-a d)^2 \sqrt {c-d x^2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {e^5 \operatorname {Subst}\left (\int \left (-\frac {2 c (2 b c+a d) x^2}{e^2 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {2 \left (-7 a b c^2+a^2 c d\right ) x^2}{e^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 )}{4 b c (b c-a d)^2}\\ &=\frac {(2 b c+a d) e^3 (e x)^{3/2}}{2 b (b c-a d)^2 \sqrt {c-d x^2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (a (7 b c-a d) e^3\right ) \operatorname {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 (b c-a d)^2}-\frac {\left ((2 b c+a d) e^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b (b c-a d)^2}\\ &=\frac {(2 b c+a d) e^3 (e x)^{3/2}}{2 b (b c-a d)^2 \sqrt {c-d x^2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {\left (\sqrt {c} (2 b c+a d) e^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b \sqrt {d} (b c-a d)^2}-\frac {\left (\sqrt {c} (2 b c+a d) e^4\right ) \operatorname {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 b \sqrt {d} (b c-a d)^2}-\frac {\left (a (7 b c-a d) e^5\right ) \operatorname {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 b^{3/2} (b c-a d)^2}+\frac {\left (a (7 b c-a d) e^5\right ) \operatorname {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 b^{3/2} (b c-a d)^2}\\ &=\frac {(2 b c+a d) e^3 (e x)^{3/2}}{2 b (b c-a d)^2 \sqrt {c-d x^2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {\left (\sqrt {c} (2 b c+a d) e^4 \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{2 b \sqrt {d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} (2 b c+a d) e^4 \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {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 b \sqrt {d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\left (a (7 b c-a d) e^5 \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {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 b^{3/2} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {\left (a (7 b c-a d) e^5 \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {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 b^{3/2} (b c-a d)^2 \sqrt {c-d x^2}}\\ &=\frac {(2 b c+a d) e^3 (e x)^{3/2}}{2 b (b c-a d)^2 \sqrt {c-d x^2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {c^{3/4} (2 b c+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 b d^{3/4} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (7 b c-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 b^{3/2} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (7 b c-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 b^{3/2} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\left (\sqrt {c} (2 b c+a d) e^4 \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {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 b \sqrt {d} (b c-a d)^2 \sqrt {c-d x^2}}\\ &=\frac {(2 b c+a d) e^3 (e x)^{3/2}}{2 b (b c-a d)^2 \sqrt {c-d x^2}}+\frac {a e^3 (e x)^{3/2}}{2 b (b c-a d) \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {c^{3/4} (2 b c+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 b d^{3/4} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {c^{3/4} (2 b c+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 b d^{3/4} (b c-a d)^2 \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (7 b c-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 b^{3/2} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (7 b c-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 b^{3/2} \sqrt [4]{d} (b c-a d)^2 \sqrt {c-d x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.23, size = 189, normalized size = 0.36 \[ \frac {e^3 (e x)^{3/2} \left (x^2 \left (b x^2-a\right ) \sqrt {1-\frac {d x^2}{c}} (a d+2 b c) F_1\left (\frac {7}{4};\frac {1}{2},1;\frac {11}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+21 a c \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+7 a \left (-3 a c+a d x^2+2 b c x^2\right )\right )}{14 a \left (b x^2-a\right ) \sqrt {c-d x^2} (b c-a d)^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)),x]

[Out]

(e^3*(e*x)^(3/2)*(7*a*(-3*a*c + 2*b*c*x^2 + a*d*x^2) + 21*a*c*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/
2, 1, 7/4, (d*x^2)/c, (b*x^2)/a] + (2*b*c + a*d)*x^2*(-a + b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[7/4, 1/2, 1, 11
/4, (d*x^2)/c, (b*x^2)/a]))/(14*a*(b*c - a*d)^2*(-a + b*x^2)*Sqrt[c - d*x^2])

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fricas [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((e*x)^(9/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]

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)^(3/2),x, algorithm="giac")

[Out]

integrate((e*x)^(9/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)), x)

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maple [B]  time = 0.04, size = 2964, normalized size = 5.60 \[ \text {output too large to display} \]

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)^(3/2),x)

[Out]

-1/8*(7*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b+(a*b)^(1/2)*d)*b,1
/2*2^(1/2))*x^2*a*b^2*c*d*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*
d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)-7*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(
1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*x^2*a*b^2*c*d*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(
c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)-4*x^4*a^2*b^2*d^3+8*x^4*b^4*
c^2*d-4*x^4*a*b^3*c*d^2+2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b+(a
*b)^(1/2)*d)*b,1/2*2^(1/2))*x^2*a^2*b^2*c*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^
(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)-7*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)
/((c*d)^(1/2)*b+(a*b)^(1/2)*d)*b,1/2*2^(1/2))*x^2*a*b^3*c^2*d*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*
d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)+2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/
2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*x^2*a^2*b^2*c*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^
(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)-7*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2)
)/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*x^2*a*b^3*c^2*d*((d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)+8*((d*x+(c*d)^(1/2))/(
c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticE(((d*x+(c
*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*b^4*c^3-4*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+
(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2
*2^(1/2))*x^2*b^4*c^3-8*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(
-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^3*c^3+4*((d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticF
(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^3*c^3-4*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*
((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1
/2),1/2*2^(1/2))*x^2*a^2*b^2*c*d^2-4*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(
1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b^3*
c^2*d+2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d
*x)^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a^2*b^2*c*d^2+2*((d*x+(c*d)^(1/2))/
(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticF(((d*x+(
c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b^3*c^2*d-2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))
^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b+(a*b)^(1/2)*d)*b,1/2*2^(1/2))*x^2*a^2*b*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^
(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+2^(1/2)*Ellipt
icPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*x^2*a^2*b*
d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c
*d)^(1/2)*(a*b)^(1/2)-7*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b+(a
*b)^(1/2)*d)*b,1/2*2^(1/2))*a^2*b*c*d*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+7*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(
1/2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*a^2*b*c*d*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+4*((d*x+(c*d)^(1/2))
/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticE(((d*x+
(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^3*b*c*d^2+4*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*
x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1
/2*2^(1/2))*a^2*b^2*c^2*d-2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/
2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^3*b*c*d^2-2*((d*x
+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*Ell
ipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^2*b^2*c^2*d+12*x^2*a^2*b^2*c*d^2-12*x^2*a*b^3*c^2*
d-2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b+(a*b)^(1/2)*d)*b,1/2*2^(
1/2))*a^3*b*c*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)
*d*x)^(1/2)+2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b+(a*b)^(1/2)*d)
*b,1/2*2^(1/2))*a^3*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)
^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+7*2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/
2)/((c*d)^(1/2)*b+(a*b)^(1/2)*d)*b,1/2*2^(1/2))*a^2*b^2*c^2*d*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*
d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)-2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/
2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*a^3*b*c*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*
((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)-2^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^
(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*a^3*d^2*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^
(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2)*(c*d)^(1/2)*(a*b)^(1/2)+7*2^(1/2)*Elli
pticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)*b,1/2*2^(1/2))*a^2*b^2*
c^2*d*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-1/(c*d)^(1/2)*d*x)^(1/2))
*(-d*x^2+c)^(1/2)*e^4*(e*x)^(1/2)/x/((c*d)^(1/2)*b-(a*b)^(1/2)*d)/((c*d)^(1/2)*b+(a*b)^(1/2)*d)/(b*x^2-a)/(a*d
-b*c)^2/(d*x^2-c)/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{\frac {9}{2}}}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]

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)^(3/2),x, algorithm="maxima")

[Out]

integrate((e*x)^(9/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^{9/2}}{{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]

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)^(3/2)),x)

[Out]

int((e*x)^(9/2)/((a - b*x^2)^2*(c - d*x^2)^(3/2)), x)

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sympy [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((e*x)**(9/2)/(-b*x**2+a)**2/(-d*x**2+c)**(3/2),x)

[Out]

Timed out

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