\(\int \frac {1}{(e x)^{5/2} (a-b x^2)^2 (c-d x^2)^{5/2}} \, dx\) [933]

Optimal result

Integrand size = 30, antiderivative size = 606 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {d (3 b c+2 a d)}{6 a c (b c-a d)^2 e (e x)^{3/2} \left (c-d x^2\right )^{3/2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac {d \left (b^2 c^2+7 a b c d-3 a^2 d^2\right )}{2 a c^2 (b c-a d)^3 e (e x)^{3/2} \sqrt {c-d x^2}}-\frac {\left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {c-d x^2}}{6 a^2 c^3 (b c-a d)^3 e (e x)^{3/2}}+\frac {d^{3/4} \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 a^2 c^{11/4} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}}+\frac {b^3 \sqrt [4]{c} (7 b c-17 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}}+\frac {b^3 \sqrt [4]{c} (7 b c-17 a d) \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d)^3 e^{5/2} \sqrt {c-d x^2}} \]

[Out]

1/6*d*(2*a*d+3*b*c)/a/c/(-a*d+b*c)^2/e/(e*x)^(3/2)/(-d*x^2+c)^(3/2)+1/2*b/a/(-a*d+b*c)/e/(e*x)^(3/2)/(-b*x^2+a
)/(-d*x^2+c)^(3/2)+1/2*d*(-3*a^2*d^2+7*a*b*c*d+b^2*c^2)/a/c^2/(-a*d+b*c)^3/e/(e*x)^(3/2)/(-d*x^2+c)^(1/2)-1/6*
(-15*a^3*d^3+35*a^2*b*c*d^2-12*a*b^2*c^2*d+7*b^3*c^3)*(-d*x^2+c)^(1/2)/a^2/c^3/(-a*d+b*c)^3/e/(e*x)^(3/2)+1/6*
d^(3/4)*(-15*a^3*d^3+35*a^2*b*c*d^2-12*a*b^2*c^2*d+7*b^3*c^3)*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^(11/4)/(-a*d+b*c)^3/e^(5/2)/(-d*x^2+c)^(1/2)+1/4*b^3*c^(1/4)*(-17*a*d+7*b*c)*Elliptic
Pi(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/d^(1/4)/(-a*d
+b*c)^3/e^(5/2)/(-d*x^2+c)^(1/2)+1/4*b^3*c^(1/4)*(-17*a*d+7*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^3/d^(1/4)/(-a*d+b*c)^3/e^(5/2)/(-d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 606, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {477, 483, 593, 597, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {b^3 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (7 b c-17 a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2} (b c-a d)^3}+\frac {b^3 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (7 b c-17 a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2} (b c-a d)^3}+\frac {d \left (-3 a^2 d^2+7 a b c d+b^2 c^2\right )}{2 a c^2 e (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)^3}+\frac {d^{3/4} \sqrt {1-\frac {d x^2}{c}} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{6 a^2 c^{11/4} e^{5/2} \sqrt {c-d x^2} (b c-a d)^3}-\frac {\sqrt {c-d x^2} \left (-15 a^3 d^3+35 a^2 b c d^2-12 a b^2 c^2 d+7 b^3 c^3\right )}{6 a^2 c^3 e (e x)^{3/2} (b c-a d)^3}+\frac {b}{2 a e (e x)^{3/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac {d (2 a d+3 b c)}{6 a c e (e x)^{3/2} \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

[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 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 418

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 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 537

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 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 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*} \text {integral}& = \frac {2 \text {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 \text {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 \text {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 \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 11.91 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\frac {x \left (-\frac {5 a \left (7 b^4 c^3 x^2 \left (c-d x^2\right )^2-4 a b^3 c^2 \left (c-d x^2\right )^2 \left (c+3 d x^2\right )+a^4 d^3 \left (4 c^2-21 c d x^2+15 d^2 x^4\right )-a^3 b d^2 \left (12 c^3-45 c^2 d x^2+14 c d^2 x^4+15 d^3 x^6\right )+a^2 b^2 c d \left (12 c^3-12 c^2 d x^2-37 c d^2 x^4+35 d^3 x^6\right )\right )}{(-b c+a d)^3 \left (a-b x^2\right ) \left (c-d x^2\right )}+\frac {5 \left (21 b^4 c^4-44 a b^3 c^3 d-12 a^2 b^2 c^2 d^2+35 a^3 b c d^3-15 a^4 d^4\right ) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )}{(b c-a d)^3}-\frac {b d \left (7 b^3 c^3-12 a b^2 c^2 d+35 a^2 b c d^2-15 a^3 d^3\right ) x^4 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )}{(b c-a d)^3}\right )}{30 a^3 c^3 (e x)^{5/2} \sqrt {c-d x^2}} \]

[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])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1463\) vs. \(2(506)=1012\).

Time = 3.16 (sec) , antiderivative size = 1464, normalized size of antiderivative = 2.42

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1464\)
default	\(\text {Expression too large to display}\)	\(5236\)

[In]

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

[Out]

((-d*x^2+c)*e*x)^(1/2)/(e*x)^(1/2)/(-d*x^2+c)^(1/2)*(1/2*d*b^4/e^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^2/(a*d-b*c)*(
-d*e*x^3+c*e*x)^(1/2)/(b*d*x^2-a*d)+1/3*d/e^3/c^2/(a*d-b*c)^2*(-d*e*x^3+c*e*x)^(1/2)/(x^2-c/d)^2+1/6*d^3/e^2*x
/c^3*(11*a*d-23*b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/(-(x^2-c/d)*d*e*x)^(1/2)-2/3/c^3/e^3/a^2*(-d*e*x^3+
c*e*x)^(1/2)/x^2-1/4*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/
2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^3/e^2/(a^2*d^2-2*
a*b*c*d+b^2*c^2)/a^2/(a*d-b*c)+11/12*d^3*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-
d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),1/2*2^(1/2))
/c^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/e^2*a-23/12*d^2*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)
^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^
(1/2),1/2*2^(1/2))/c^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/e^2*b+1/3*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(
-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/2)*EllipticF(((x+1/d*(c*d)^(1/2))*d/(
c*d)^(1/2))^(1/2),1/2*2^(1/2))/c^3/e^2/a^2-17/8*b^3/e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a/(a*d-b*c)/(a*b)^(1/2)*(c
*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*x)^(1/
2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1/2)/(-
1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2))+7/8*b^4/e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^2/(a*d-b*c)/(a*b)^(1/2
)/d*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3+c*e*
x)^(1/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*d)^(1
/2)/(-1/d*(c*d)^(1/2)-1/b*(a*b)^(1/2)),1/2*2^(1/2))*c+17/8*b^3/e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a/(a*d-b*c)/(a*
b)^(1/2)*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d*e*x^3
+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1/d*(c*
d)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2)),1/2*2^(1/2))-7/8*b^4/e^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^2/(a*d-b*c)
/(a*b)^(1/2)/d*(c*d)^(1/2)*(d*x/(c*d)^(1/2)+1)^(1/2)*(-2*d*x/(c*d)^(1/2)+2)^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)/(-d
*e*x^3+c*e*x)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2))*EllipticPi(((x+1/d*(c*d)^(1/2))*d/(c*d)^(1/2))^(1/2),-1
/d*(c*d)^(1/2)/(-1/d*(c*d)^(1/2)+1/b*(a*b)^(1/2)),1/2*2^(1/2))*c)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} - a\right )}^{2} {\left (-d x^{2} + c\right )}^{\frac {5}{2}} \left (e x\right )^{\frac {5}{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e x)^{5/2} \left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (e\,x\right )}^{5/2}\,{\left (a-b\,x^2\right )}^2\,{\left (c-d\,x^2\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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