3.933 \(\int \frac{1}{(e x)^{5/2} (a-b x^2)^2 (c-d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=606 \[ \frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} \left (35 a^2 b c d^2-15 a^3 d^3-12 a b^2 c^2 d+7 b^3 c^3\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{6 a^2 c^{11/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{c-d x^2} \left (35 a^2 b c d^2-15 a^3 d^3-12 a b^2 c^2 d+7 b^3 c^3\right )}{6 a^2 c^3 e (e x)^{3/2} (b c-a d)^3}+\frac{d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{2 a c^2 e (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-17 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-17 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b}{2 a e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c e (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*e*(e*x)^(3/2)*(c - d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*e*(e*x)^(3/2)*(
a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(b^2*c^2 + 7*a*b*c*d - 3*a^2*d^2))/(2*a*c^2*(b*c - a*d)^3*e*(e*x)^(3/2)*Sqr
t[c - d*x^2]) - ((7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[c - d*x^2])/(6*a^2*c^3*(b*c -
a*d)^3*e*(e*x)^(3/2)) + (d^(3/4)*(7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[1 - (d*x^2)/
c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(6*a^2*c^(11/4)*(b*c - a*d)^3*e^(5/2)*Sqrt[c
- d*x^2]) + (b^3*c^(1/4)*(7*b*c - 17*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^3*d^(1/4)*(b*c - a*d)^3*e^(5/2)*Sqrt[c - d*x^2]) +
(b^3*c^(1/4)*(7*b*c - 17*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^3*d^(1/4)*(b*c - a*d)^3*e^(5/2)*Sqrt[c - d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.46119, antiderivative size = 606, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {466, 472, 579, 583, 523, 224, 221, 409, 1219, 1218} \[ \frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} \left (35 a^2 b c d^2-15 a^3 d^3-12 a b^2 c^2 d+7 b^3 c^3\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 a^2 c^{11/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{c-d x^2} \left (35 a^2 b c d^2-15 a^3 d^3-12 a b^2 c^2 d+7 b^3 c^3\right )}{6 a^2 c^3 e (e x)^{3/2} (b c-a d)^3}+\frac{d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{2 a c^2 e (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-17 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^3 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-17 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b}{2 a e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d (2 a d+3 b c)}{6 a c e (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

(d*(3*b*c + 2*a*d))/(6*a*c*(b*c - a*d)^2*e*(e*x)^(3/2)*(c - d*x^2)^(3/2)) + b/(2*a*(b*c - a*d)*e*(e*x)^(3/2)*(
a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(b^2*c^2 + 7*a*b*c*d - 3*a^2*d^2))/(2*a*c^2*(b*c - a*d)^3*e*(e*x)^(3/2)*Sqr
t[c - d*x^2]) - ((7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[c - d*x^2])/(6*a^2*c^3*(b*c -
a*d)^3*e*(e*x)^(3/2)) + (d^(3/4)*(7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*Sqrt[1 - (d*x^2)/
c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(6*a^2*c^(11/4)*(b*c - a*d)^3*e^(5/2)*Sqrt[c
- d*x^2]) + (b^3*c^(1/4)*(7*b*c - 17*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^3*d^(1/4)*(b*c - a*d)^3*e^(5/2)*Sqrt[c - d*x^2]) +
(b^3*c^(1/4)*(7*b*c - 17*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^3*d^(1/4)*(b*c - a*d)^3*e^(5/2)*Sqrt[c - d*x^2])

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 472

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 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 583

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 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

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 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 409

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

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]

Rubi steps

\begin{align*} \int \frac{1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^4 \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{b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{e \operatorname{Subst}\left (\int \frac{\frac{7 b c-4 a d}{e^2}-\frac{13 b d x^4}{e^4}}{x^4 \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 a (b c-a d)}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}-\frac{e^3 \operatorname{Subst}\left (\int \frac{-\frac{6 \left (7 b^2 c^2-8 a b c d+6 a^2 d^2\right )}{e^4}+\frac{18 b d (3 b c+2 a d) x^4}{e^6}}{x^4 \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 a c (b c-a d)^2}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt{c-d x^2}}+\frac{e^5 \operatorname{Subst}\left (\int \frac{\frac{12 \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right )}{e^6}-\frac{60 b d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right ) x^4}{e^8}}{x^4 \left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{24 a c^2 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt{c-d x^2}}-\frac{\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}-\frac{e^5 \operatorname{Subst}\left (\int \frac{-\frac{12 \left (21 b^4 c^4-44 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+35 a^3 b c d^3-15 a^4 d^4\right )}{e^8}+\frac{12 b d \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) x^4}{e^{10}}}{\left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{72 a^2 c^3 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt{c-d x^2}}-\frac{\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}+\frac{\left (b^3 (7 b c-17 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\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)^3 e^3}+\frac{\left (d \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{6 a^2 c^3 (b c-a d)^3 e^3}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt{c-d x^2}}-\frac{\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}+\frac{\left (b^3 (7 b c-17 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{b} x^2}{\sqrt{a} e}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{4 a^3 (b c-a d)^3 e^3}+\frac{\left (b^3 (7 b c-17 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{b} x^2}{\sqrt{a} e}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{4 a^3 (b c-a d)^3 e^3}+\frac{\left (d \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \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 )}{6 a^2 c^3 (b c-a d)^3 e^3 \sqrt{c-d x^2}}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt{c-d x^2}}-\frac{\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}+\frac{d^{3/4} \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\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 )}{6 a^2 c^{11/4} (b c-a d)^3 e^{5/2} \sqrt{c-d x^2}}+\frac{\left (b^3 (7 b c-17 a d) \sqrt{1-\frac{d x^2}{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{\sqrt{b} x^2}{\sqrt{a} e}\right ) \sqrt{1-\frac{d x^4}{c e^2}}} \, dx,x,\sqrt{e x}\right )}{4 a^3 (b c-a d)^3 e^3 \sqrt{c-d x^2}}+\frac{\left (b^3 (7 b c-17 a d) \sqrt{1-\frac{d x^2}{c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{\sqrt{b} x^2}{\sqrt{a} e}\right ) \sqrt{1-\frac{d x^4}{c e^2}}} \, dx,x,\sqrt{e x}\right )}{4 a^3 (b c-a d)^3 e^3 \sqrt{c-d x^2}}\\ &=\frac{d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac{b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt{c-d x^2}}-\frac{\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt{c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}+\frac{d^{3/4} \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\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 )}{6 a^2 c^{11/4} (b c-a d)^3 e^{5/2} \sqrt{c-d x^2}}+\frac{b^3 \sqrt [4]{c} (7 b c-17 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^3 \sqrt [4]{d} (b c-a d)^3 e^{5/2} \sqrt{c-d x^2}}+\frac{b^3 \sqrt [4]{c} (7 b c-17 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^3 \sqrt [4]{d} (b c-a d)^3 e^{5/2} \sqrt{c-d x^2}}\\ \end{align*}

Mathematica [C]  time = 1.36853, size = 427, normalized size = 0.7 \[ \frac{x \left (-\frac{b d x^4 \sqrt{1-\frac{d x^2}{c}} \left (35 a^2 b c d^2-15 a^3 d^3-12 a b^2 c^2 d+7 b^3 c^3\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{(b c-a d)^3}+\frac{5 x^2 \sqrt{1-\frac{d x^2}{c}} \left (-12 a^2 b^2 c^2 d^2+35 a^3 b c d^3-15 a^4 d^4-44 a b^3 c^3 d+21 b^4 c^4\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{(b c-a d)^3}-\frac{5 a \left (a^2 b^2 c d \left (-12 c^2 d x^2+12 c^3-37 c d^2 x^4+35 d^3 x^6\right )-a^3 b d^2 \left (-45 c^2 d x^2+12 c^3+14 c d^2 x^4+15 d^3 x^6\right )+a^4 d^3 \left (4 c^2-21 c d x^2+15 d^2 x^4\right )-4 a b^3 c^2 \left (c-d x^2\right )^2 \left (c+3 d x^2\right )+7 b^4 c^3 x^2 \left (c-d x^2\right )^2\right )}{\left (a-b x^2\right ) \left (c-d x^2\right ) (a d-b c)^3}\right )}{30 a^3 c^3 (e x)^{5/2} \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(5/2)),x]

[Out]

(x*((-5*a*(7*b^4*c^3*x^2*(c - d*x^2)^2 - 4*a*b^3*c^2*(c - d*x^2)^2*(c + 3*d*x^2) + a^4*d^3*(4*c^2 - 21*c*d*x^2
+ 15*d^2*x^4) - a^3*b*d^2*(12*c^3 - 45*c^2*d*x^2 + 14*c*d^2*x^4 + 15*d^3*x^6) + a^2*b^2*c*d*(12*c^3 - 12*c^2*
d*x^2 - 37*c*d^2*x^4 + 35*d^3*x^6)))/((-(b*c) + a*d)^3*(a - b*x^2)*(c - d*x^2)) + (5*(21*b^4*c^4 - 44*a*b^3*c^
3*d - 12*a^2*b^2*c^2*d^2 + 35*a^3*b*c*d^3 - 15*a^4*d^4)*x^2*Sqrt[1 - (d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*
x^2)/c, (b*x^2)/a])/(b*c - a*d)^3 - (b*d*(7*b^3*c^3 - 12*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 15*a^3*d^3)*x^4*Sqrt[1
- (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(b*c - a*d)^3))/(30*a^3*c^3*(e*x)^(5/2)*Sqrt[c
- d*x^2])

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Maple [B]  time = 0.063, size = 5248, normalized size = 8.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)*(e*x)^(5/2)), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x)**(5/2)/(-b*x**2+a)**2/(-d*x**2+c)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)*(e*x)^(5/2)), x)