3.9.94 \(\int \frac {1}{(e x)^{3/2} (a-b x^2) (c-d x^2)^{3/2}} \, dx\) [894]

Optimal. Leaf size=493 \[ -\frac {d}{c (b c-a d) e \sqrt {e x} \sqrt {c-d x^2}}-\frac {(2 b c-3 a d) \sqrt {c-d x^2}}{a c^2 (b c-a d) e \sqrt {e x}}-\frac {\sqrt [4]{d} (2 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a c^{5/4} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (2 b c-3 a d) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a c^{5/4} (b c-a d) e^{3/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} (b c-a d) e^{3/2} \sqrt {c-d x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.70, antiderivative size = 493, 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 = {477, 483, 597, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \begin {gather*} -\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{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 )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2} (b c-a d)}+\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{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 )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2} (b c-a d)}+\frac {\sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (2 b c-3 a d) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a c^{5/4} e^{3/2} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} (2 b c-3 a d) E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a c^{5/4} e^{3/2} \sqrt {c-d x^2} (b c-a d)}-\frac {\sqrt {c-d x^2} (2 b c-3 a d)}{a c^2 e \sqrt {e x} (b c-a d)}-\frac {d}{c e \sqrt {e x} \sqrt {c-d x^2} (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]

[Out]

-(d/(c*(b*c - a*d)*e*Sqrt[e*x]*Sqrt[c - d*x^2])) - ((2*b*c - 3*a*d)*Sqrt[c - d*x^2])/(a*c^2*(b*c - a*d)*e*Sqrt
[e*x]) - (d^(1/4)*(2*b*c - 3*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])],
-1])/(a*c^(5/4)*(b*c - a*d)*e^(3/2)*Sqrt[c - d*x^2]) + (d^(1/4)*(2*b*c - 3*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticF
[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a*c^(5/4)*(b*c - a*d)*e^(3/2)*Sqrt[c - d*x^2]) - (b^(3/2
)*c^(1/4)*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])/(a^(3/2)*d^(1/4)*(b*c - a*d)*e^(3/2)*Sqrt[c - d*x^2]) + (b^(3/2)*c^(1/4)*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])/(a^(
3/2)*d^(1/4)*(b*c - a*d)*e^(3/2)*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 483

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*(e*
x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a*d)
*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*(b*c - a*d)*(p + 1) + d*b*(m + n
*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ
[p, -1] && 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 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -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 {1}{(e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{x^2 \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 {d}{c (b c-a d) e \sqrt {e x} \sqrt {c-d x^2}}-\frac {e \text {Subst}\left (\int \frac {-\frac {2 b c-3 a d}{e^2}-\frac {3 b d x^4}{e^4}}{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 )}{c (b c-a d)}\\ &=-\frac {d}{c (b c-a d) e \sqrt {e x} \sqrt {c-d x^2}}-\frac {(2 b c-3 a d) \sqrt {c-d x^2}}{a c^2 (b c-a d) e \sqrt {e x}}+\frac {e \text {Subst}\left (\int \frac {x^2 \left (\frac {2 b^2 c^2-2 a b c d+3 a^2 d^2}{e^4}+\frac {b d (2 b c-3 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 )}{a c^2 (b c-a d)}\\ &=-\frac {d}{c (b c-a d) e \sqrt {e x} \sqrt {c-d x^2}}-\frac {(2 b c-3 a d) \sqrt {c-d x^2}}{a c^2 (b c-a d) e \sqrt {e x}}+\frac {e \text {Subst}\left (\int \left (-\frac {d (2 b c-3 a d) x^2}{e^4 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {2 b^2 c^2 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 )}{a c^2 (b c-a d)}\\ &=-\frac {d}{c (b c-a d) e \sqrt {e x} \sqrt {c-d x^2}}-\frac {(2 b c-3 a d) \sqrt {c-d x^2}}{a c^2 (b c-a d) e \sqrt {e x}}+\frac {\left (2 b^2\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 )}{a (b c-a d) e^3}-\frac {(d (2 b c-3 a d)) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a c^2 (b c-a d) e^3}\\ &=-\frac {d}{c (b c-a d) e \sqrt {e x} \sqrt {c-d x^2}}-\frac {(2 b c-3 a d) \sqrt {c-d x^2}}{a c^2 (b c-a d) e \sqrt {e x}}+\frac {\left (\sqrt {d} (2 b c-3 a d)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a c^{3/2} (b c-a d) e^2}-\frac {\left (\sqrt {d} (2 b c-3 a d)\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 )}{a c^{3/2} (b c-a d) e^2}+\frac {b^{3/2} \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 )}{a (b c-a d) e}-\frac {b^{3/2} \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 )}{a (b c-a d) e}\\ &=-\frac {d}{c (b c-a d) e \sqrt {e x} \sqrt {c-d x^2}}-\frac {(2 b c-3 a d) \sqrt {c-d x^2}}{a c^2 (b c-a d) e \sqrt {e x}}+\frac {\left (\sqrt {d} (2 b c-3 a d) \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 )}{a c^{3/2} (b c-a d) e^2 \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} (2 b c-3 a d) \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 )}{a c^{3/2} (b c-a d) e^2 \sqrt {c-d x^2}}+\frac {\left (b^{3/2} \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 )}{a (b c-a d) e \sqrt {c-d x^2}}-\frac {\left (b^{3/2} \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 )}{a (b c-a d) e \sqrt {c-d x^2}}\\ &=-\frac {d}{c (b c-a d) e \sqrt {e x} \sqrt {c-d x^2}}-\frac {(2 b c-3 a d) \sqrt {c-d x^2}}{a c^2 (b c-a d) e \sqrt {e x}}+\frac {\sqrt [4]{d} (2 b c-3 a d) \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 )}{a c^{5/4} (b c-a d) e^{3/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} \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 )}{a^{3/2} \sqrt [4]{d} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} \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 )}{a^{3/2} \sqrt [4]{d} (b c-a d) e^{3/2} \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} (2 b c-3 a d) \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 )}{a c^{3/2} (b c-a d) e^2 \sqrt {c-d x^2}}\\ &=-\frac {d}{c (b c-a d) e \sqrt {e x} \sqrt {c-d x^2}}-\frac {(2 b c-3 a d) \sqrt {c-d x^2}}{a c^2 (b c-a d) e \sqrt {e x}}-\frac {\sqrt [4]{d} (2 b c-3 a d) \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 )}{a c^{5/4} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} (2 b c-3 a d) \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 )}{a c^{5/4} (b c-a d) e^{3/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} \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 )}{a^{3/2} \sqrt [4]{d} (b c-a d) e^{3/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} \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 )}{a^{3/2} \sqrt [4]{d} (b c-a d) e^{3/2} \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 = 10.19, size = 198, normalized size = 0.40 \begin {gather*} \frac {x \left (21 a \left (a d \left (2 c-3 d x^2\right )-2 b c \left (c-d x^2\right )\right )+7 \left (2 b^2 c^2-2 a b c d+3 a^2 d^2\right ) x^2 \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 d (2 b c-3 a d) x^4 \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 )}{21 a^2 c^2 (b c-a d) (e x)^{3/2} \sqrt {c-d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1046\) vs. \(2(389)=778\).
time = 0.13, size = 1047, normalized size = 2.12 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x)^(3/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(6*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*((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)*(-d*x/(c*d)^(1/2))^(1/2)*a^2*c*d^2-10*EllipticE(((d*x+(c*d)^(1/2))
/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*((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)*(-d*x/(c*d)^(1/2))^(1/2)*a*b*c^2*d+4*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*((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)*(-d*x/(c*d)^(1/2))^(1/2)*b^2*
c^3-3*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*((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)*(-d*x/(c*d)^(1/2))^(1/2)*a^2*c*d^2+5*EllipticF(((d*x+(c*d)^(1/2))/(c
*d)^(1/2))^(1/2),1/2*2^(1/2))*((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)*(-d*x/(c*d)^(1/2))^(1/2)*a*b*c^2*d-2*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*((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)*(-d*x/(c*d)^(1/2))^(1/2)*b^2*c^3
-(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(1/2)*b+(a*b)^(1/2)*d),1/2*
2^(1/2))*((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)*(-d*x/(c*d)^(1/2
))^(1/2)*(c*d)^(1/2)*b*c^2+(a*b)^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((c*d)^(
1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*((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)*(-d*x/(c*d)^(1/2))^(1/2)*(c*d)^(1/2)*b*c^2+EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1
/2)*b/((c*d)^(1/2)*b+(a*b)^(1/2)*d),1/2*2^(1/2))*((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)*(-d*x/(c*d)^(1/2))^(1/2)*b^2*c^3+EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d
)^(1/2)*b/((c*d)^(1/2)*b-(a*b)^(1/2)*d),1/2*2^(1/2))*((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)*(-d*x/(c*d)^(1/2))^(1/2)*b^2*c^3+6*a^2*d^3*x^2-10*a*b*c*d^2*x^2+4*b^2*c^2*d*x^2-4*
a^2*c*d^2+8*a*b*c^2*d-4*b^2*c^3)*b*d/(-d*x^2+c)^(1/2)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)/((c*d)^(1/2)*b+(a*b)^(1/2)
*d)/a/(a*d-b*c)/c^2/e/(e*x)^(1/2)

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

[Out]

-e^(-3/2)*integrate(1/((b*x^2 - a)*(-d*x^2 + c)^(3/2)*x^(3/2)), 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(1/(e*x)^(3/2)/(-b*x^2+a)/(-d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {1}{- a c \left (e x\right )^{\frac {3}{2}} \sqrt {c - d x^{2}} + a d x^{2} \left (e x\right )^{\frac {3}{2}} \sqrt {c - d x^{2}} + b c x^{2} \left (e x\right )^{\frac {3}{2}} \sqrt {c - d x^{2}} - b d x^{4} \left (e x\right )^{\frac {3}{2}} \sqrt {c - d x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x)**(3/2)/(-b*x**2+a)/(-d*x**2+c)**(3/2),x)

[Out]

-Integral(1/(-a*c*(e*x)**(3/2)*sqrt(c - d*x**2) + a*d*x**2*(e*x)**(3/2)*sqrt(c - d*x**2) + b*c*x**2*(e*x)**(3/
2)*sqrt(c - d*x**2) - b*d*x**4*(e*x)**(3/2)*sqrt(c - d*x**2)), x)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)),x)

[Out]

int(1/((e*x)^(3/2)*(a - b*x^2)*(c - d*x^2)^(3/2)), x)

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