3.10.24 \(\int \frac {1}{(e x)^{3/2} (a-b x^2)^2 (c-d x^2)^{3/2}} \, dx\) [924]

Optimal. Leaf size=628 \[ \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}-\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}} \]

[Out]

1/2*d*(2*a*d+b*c)/a/c/(-a*d+b*c)^2/e/(e*x)^(1/2)/(-d*x^2+c)^(1/2)+1/2*b/a/(-a*d+b*c)/e/(-b*x^2+a)/(e*x)^(1/2)/
(-d*x^2+c)^(1/2)-1/2*(6*a^2*d^2-8*a*b*c*d+5*b^2*c^2)*(-d*x^2+c)^(1/2)/a^2/c^2/(-a*d+b*c)^2/e/(e*x)^(1/2)-1/2*d
^(1/4)*(6*a^2*d^2-8*a*b*c*d+5*b^2*c^2)*EllipticE(d^(1/4)*(e*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/a^2/
c^(5/4)/(-a*d+b*c)^2/e^(3/2)/(-d*x^2+c)^(1/2)+1/2*d^(1/4)*(6*a^2*d^2-8*a*b*c*d+5*b^2*c^2)*EllipticF(d^(1/4)*(e
*x)^(1/2)/c^(1/4)/e^(1/2),I)*(1-d*x^2/c)^(1/2)/a^2/c^(5/4)/(-a*d+b*c)^2/e^(3/2)/(-d*x^2+c)^(1/2)-1/4*b^(3/2)*c
^(1/4)*(-11*a*d+5*b*c)*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^(5/2)/d^(1/4)/(-a*d+b*c)^2/e^(3/2)/(-d*x^2+c)^(1/2)+1/4*b^(3/2)*c^(1/4)*(-11*a*d+5*b*c)*Ellipt
icPi(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^(5/2)/d^(1/4)/
(-a*d+b*c)^2/e^(3/2)/(-d*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.94, antiderivative size = 628, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {477, 483, 593, 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}} (5 b c-11 a d) \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 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2} (b c-a d)^2}+\frac {b^{3/2} \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (5 b c-11 a d) \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 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt {c-d x^2} (b c-a d)^2}+\frac {\sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} \left (6 a^2 d^2-8 a b c d+5 b^2 c^2\right ) F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 c^{5/4} e^{3/2} \sqrt {c-d x^2} (b c-a d)^2}-\frac {\sqrt [4]{d} \sqrt {1-\frac {d x^2}{c}} \left (6 a^2 d^2-8 a b c d+5 b^2 c^2\right ) E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{2 a^2 c^{5/4} e^{3/2} \sqrt {c-d x^2} (b c-a d)^2}-\frac {\sqrt {c-d x^2} \left (6 a^2 d^2-8 a b c d+5 b^2 c^2\right )}{2 a^2 c^2 e \sqrt {e x} (b c-a d)^2}+\frac {b}{2 a e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2} (b c-a d)}+\frac {d (2 a d+b c)}{2 a c e \sqrt {e x} \sqrt {c-d x^2} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(b*c + 2*a*d))/(2*a*c*(b*c - a*d)^2*e*Sqrt[e*x]*Sqrt[c - d*x^2]) + b/(2*a*(b*c - a*d)*e*Sqrt[e*x]*(a - b*x^
2)*Sqrt[c - d*x^2]) - ((5*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*Sqrt[c - d*x^2])/(2*a^2*c^2*(b*c - a*d)^2*e*Sqrt[e*
x]) - (d^(1/4)*(5*b^2*c^2 - 8*a*b*c*d + 6*a^2*d^2)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c
^(1/4)*Sqrt[e])], -1])/(2*a^2*c^(5/4)*(b*c - a*d)^2*e^(3/2)*Sqrt[c - d*x^2]) + (d^(1/4)*(5*b^2*c^2 - 8*a*b*c*d
+ 6*a^2*d^2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a^2*c^(5/4)
*(b*c - a*d)^2*e^(3/2)*Sqrt[c - d*x^2]) - (b^(3/2)*c^(1/4)*(5*b*c - 11*a*d)*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*a^(5/2)*d^(1/4)*(b
*c - a*d)^2*e^(3/2)*Sqrt[c - d*x^2]) + (b^(3/2)*c^(1/4)*(5*b*c - 11*a*d)*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*a^(5/2)*d^(1/4)*(b*c - a
*d)^2*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 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 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 )^2 \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 )^2 \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {e \text {Subst}\left (\int \frac {\frac {5 b c-4 a d}{e^2}-\frac {7 b d x^4}{e^4}}{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 )}{2 a (b c-a d)}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {e^3 \text {Subst}\left (\int \frac {-\frac {2 \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right )}{e^4}+\frac {6 b d (b c+2 a d) x^4}{e^6}}{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 )}{4 a c (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {e^3 \text {Subst}\left (\int \frac {x^2 \left (\frac {2 \left (5 b^3 c^3-16 a b^2 c^2 d+8 a^2 b c d^2-6 a^3 d^3\right )}{e^6}+\frac {2 b d \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) x^4}{e^8}\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 a^2 c^2 (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {e^3 \text {Subst}\left (\int \left (-\frac {2 d \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) x^2}{e^6 \sqrt {c-\frac {d x^4}{e^2}}}+\frac {2 \left (5 b^3 c^3-11 a b^2 c^2 d\right ) x^2}{e^6 \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 a^2 c^2 (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {\left (b^2 (5 b c-11 a d)\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 a^2 (b c-a d)^2 e^3}-\frac {\left (d \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 c^2 (b c-a d)^2 e^3}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {\left (\sqrt {d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{2 a^2 c^{3/2} (b c-a d)^2 e^2}-\frac {\left (\sqrt {d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right )\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 a^2 c^{3/2} (b c-a d)^2 e^2}+\frac {\left (b^{3/2} (5 b c-11 a d)\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 a^2 (b c-a d)^2 e}-\frac {\left (b^{3/2} (5 b c-11 a d)\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 a^2 (b c-a d)^2 e}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {\left (\sqrt {d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \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 a^2 c^{3/2} (b c-a d)^2 e^2 \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \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 a^2 c^{3/2} (b c-a d)^2 e^2 \sqrt {c-d x^2}}+\frac {\left (b^{3/2} (5 b c-11 a d) \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 a^2 (b c-a d)^2 e \sqrt {c-d x^2}}-\frac {\left (b^{3/2} (5 b c-11 a d) \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 a^2 (b c-a d)^2 e \sqrt {c-d x^2}}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}+\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \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 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \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 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \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 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}-\frac {\left (\sqrt {d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \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 a^2 c^{3/2} (b c-a d)^2 e^2 \sqrt {c-d x^2}}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e \sqrt {e x} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e \sqrt {e x} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \sqrt {c-d x^2}}{2 a^2 c^2 (b c-a d)^2 e \sqrt {e x}}-\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \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 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {\sqrt [4]{d} \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) \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 a^2 c^{5/4} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}-\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \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 a^{5/2} \sqrt [4]{d} (b c-a d)^2 e^{3/2} \sqrt {c-d x^2}}+\frac {b^{3/2} \sqrt [4]{c} (5 b c-11 a d) \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 a^{5/2} \sqrt [4]{d} (b c-a d)^2 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 = 10.34, size = 319, normalized size = 0.51 \begin {gather*} \frac {x \left (21 a \left (2 a^3 d^2 \left (2 c-3 d x^2\right )-5 b^3 c^2 x^2 \left (c-d x^2\right )+4 a b^2 c \left (c^2+c d x^2-2 d^2 x^4\right )+2 a^2 b d \left (-4 c^2+2 c d x^2+3 d^2 x^4\right )\right )+7 \left (-5 b^3 c^3+16 a b^2 c^2 d-8 a^2 b c d^2+6 a^3 d^3\right ) x^2 \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 )+3 b d \left (5 b^2 c^2-8 a b c d+6 a^2 d^2\right ) x^4 \left (-a+b 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^3 c^2 (b c-a d)^2 (e x)^{3/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(21*a*(2*a^3*d^2*(2*c - 3*d*x^2) - 5*b^3*c^2*x^2*(c - d*x^2) + 4*a*b^2*c*(c^2 + c*d*x^2 - 2*d^2*x^4) + 2*a^
2*b*d*(-4*c^2 + 2*c*d*x^2 + 3*d^2*x^4)) + 7*(-5*b^3*c^3 + 16*a*b^2*c^2*d - 8*a^2*b*c*d^2 + 6*a^3*d^3)*x^2*(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] + 3*b*d*(5*b^2*c^2 - 8*a*b*c*d +
6*a^2*d^2)*x^4*(-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]))/(42*a^3*c^
2*(b*c - a*d)^2*(e*x)^(3/2)*(-a + b*x^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. \(3372\) vs. \(2(506)=1012\).
time = 0.14, size = 3373, normalized size = 5.37

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1349\)
default	\(\text {Expression too large to display}\)	\(3373\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(11*((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)^(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))*(c*d)^(1/2)*a*b^2*c^2*d*x^2-11*((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)^(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))*(c*d)^(1/2)*a*b^2*c^2*d*x^2+48*a^3*b*c^2*d^2-48*a^2*b^
2*c^3*d+20*b^4*c^3*d*x^4-24*a^3*b*d^4*x^4-24*((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)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^3*
b*c*d^3*x^2+56*((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)*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*x^2-52*((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)*EllipticE((
(d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^3*c^3*d*x^2+12*((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)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))
^(1/2),1/2*2^(1/2))*a^3*b*c*d^3*x^2-28*((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)*EllipticF(((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*x^2+26*((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)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^3*c^3*d*x^2-5*((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)^(1/2)*Ellipt
icPi(((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))*(c*d)^(1/2
)*b^3*c^3*x^2+5*((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)^(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))*(c*d)^(1/2)*b^3*c^3*x^2-11*((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)*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))*a*b^3*c^3*d*x^2-11*((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)*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))*a*b^3*c^3*d*x^2+5*((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)^(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))*(c*d)^(1/2)*a*b^2*c^3-
5*((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)^(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))*(c*d)^(1/2)*a*b^2*c^3+24*((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)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a^4*c*d^3-20*(
(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)*E
llipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^3*c^4-12*((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)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1
/2))^(1/2),1/2*2^(1/2))*a^4*c*d^3+10*((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)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b^3*c^4-5*
((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)*
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))*a*b^
3*c^4+20*((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)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^4*c^4*x^2-10*((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)*EllipticF(((d*x+(c*d)^(1/
2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^4*c^4*x^2+5*((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)*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))*b^4*c^4*x^2+5*((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)*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*...

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

[Out]

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

[Out]

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

[Out]

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

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