3.10.25 \(\int \frac {1}{(e x)^{5/2} (a-b x^2)^2 (c-d x^2)^{3/2}} \, dx\) [925]

Optimal. Leaf size=512 \[ \frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e (e x)^{3/2} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) \sqrt {c-d x^2}}{6 a^2 c^2 (b c-a d)^2 e (e x)^{3/2}}+\frac {d^{3/4} \left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{6 a^2 c^{7/4} (b c-a d)^2 e^{5/2} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (7 b c-13 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d)^2 e^{5/2} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (7 b c-13 a d) \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} (b c-a d)^2 e^{5/2} \sqrt {c-d x^2}} \]

[Out]

1/2*d*(2*a*d+b*c)/a/c/(-a*d+b*c)^2/e/(e*x)^(3/2)/(-d*x^2+c)^(1/2)+1/2*b/a/(-a*d+b*c)/e/(e*x)^(3/2)/(-b*x^2+a)/
(-d*x^2+c)^(1/2)-1/6*(10*a^2*d^2-8*a*b*c*d+7*b^2*c^2)*(-d*x^2+c)^(1/2)/a^2/c^2/(-a*d+b*c)^2/e/(e*x)^(3/2)+1/6*
d^(3/4)*(10*a^2*d^2-8*a*b*c*d+7*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^(7/4)/(-a*d+b*c)^2/e^(5/2)/(-d*x^2+c)^(1/2)+1/4*b^2*c^(1/4)*(-13*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)^2/e^(5/2)/(-d*x^
2+c)^(1/2)+1/4*b^2*c^(1/4)*(-13*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)^2/e^(5/2)/(-d*x^2+c)^(1/2)

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Rubi [A]
time = 0.73, antiderivative size = 512, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {477, 483, 593, 597, 537, 230, 227, 418, 1233, 1232} \begin {gather*} \frac {b^2 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (7 b c-13 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^3 \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2} (b c-a d)^2}+\frac {b^2 \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} (7 b c-13 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^3 \sqrt [4]{d} e^{5/2} \sqrt {c-d x^2} (b c-a d)^2}+\frac {d^{3/4} \sqrt {1-\frac {d x^2}{c}} \left (10 a^2 d^2-8 a b c d+7 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 )}{6 a^2 c^{7/4} e^{5/2} \sqrt {c-d x^2} (b c-a d)^2}-\frac {\sqrt {c-d x^2} \left (10 a^2 d^2-8 a b c d+7 b^2 c^2\right )}{6 a^2 c^2 e (e x)^{3/2} (b c-a d)^2}+\frac {b}{2 a e (e x)^{3/2} \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 (e x)^{3/2} \sqrt {c-d x^2} (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((e*x)^(5/2)*(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*(e*x)^(3/2)*Sqrt[c - d*x^2]) + b/(2*a*(b*c - a*d)*e*(e*x)^(3/2)*(a -
b*x^2)*Sqrt[c - d*x^2]) - ((7*b^2*c^2 - 8*a*b*c*d + 10*a^2*d^2)*Sqrt[c - d*x^2])/(6*a^2*c^2*(b*c - a*d)^2*e*(e
*x)^(3/2)) + (d^(3/4)*(7*b^2*c^2 - 8*a*b*c*d + 10*a^2*d^2)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[
e*x])/(c^(1/4)*Sqrt[e])], -1])/(6*a^2*c^(7/4)*(b*c - a*d)^2*e^(5/2)*Sqrt[c - d*x^2]) + (b^2*c^(1/4)*(7*b*c - 1
3*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)^2*e^(5/2)*Sqrt[c - d*x^2]) + (b^2*c^(1/4)*(7*b*c - 13*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)^2*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*} \int \frac {1}{(e x)^{5/2} \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^4 \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 (e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}}+\frac {e \text {Subst}\left (\int \frac {\frac {7 b c-4 a d}{e^2}-\frac {9 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 )^{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 (e x)^{3/2} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {e^3 \text {Subst}\left (\int \frac {-\frac {2 \left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right )}{e^4}+\frac {10 b d (b c+2 a d) x^4}{e^6}}{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 )}{4 a c (b c-a d)^2}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e (e x)^{3/2} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) \sqrt {c-d x^2}}{6 a^2 c^2 (b c-a d)^2 e (e x)^{3/2}}+\frac {e^3 \text {Subst}\left (\int \frac {\frac {2 \left (21 b^3 c^3-32 a b^2 c^2 d-8 a^2 b c d^2+10 a^3 d^3\right )}{e^6}-\frac {2 b d \left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) x^4}{e^8}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{12 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 (e x)^{3/2} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) \sqrt {c-d x^2}}{6 a^2 c^2 (b c-a d)^2 e (e x)^{3/2}}+\frac {\left (b^2 (7 b c-13 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)^2 e^3}+\frac {\left (d \left (7 b^2 c^2-8 a b c d+10 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 )}{6 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 (e x)^{3/2} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) \sqrt {c-d x^2}}{6 a^2 c^2 (b c-a d)^2 e (e x)^{3/2}}+\frac {\left (b^2 (7 b c-13 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)^2 e^3}+\frac {\left (b^2 (7 b c-13 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)^2 e^3}+\frac {\left (d \left (7 b^2 c^2-8 a b c d+10 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 )}{6 a^2 c^2 (b c-a d)^2 e^3 \sqrt {c-d x^2}}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e (e x)^{3/2} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) \sqrt {c-d x^2}}{6 a^2 c^2 (b c-a d)^2 e (e x)^{3/2}}+\frac {d^{3/4} \left (7 b^2 c^2-8 a b c d+10 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 )}{6 a^2 c^{7/4} (b c-a d)^2 e^{5/2} \sqrt {c-d x^2}}+\frac {\left (b^2 (7 b c-13 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)^2 e^3 \sqrt {c-d x^2}}+\frac {\left (b^2 (7 b c-13 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)^2 e^3 \sqrt {c-d x^2}}\\ &=\frac {d (b c+2 a d)}{2 a c (b c-a d)^2 e (e x)^{3/2} \sqrt {c-d x^2}}+\frac {b}{2 a (b c-a d) e (e x)^{3/2} \left (a-b x^2\right ) \sqrt {c-d x^2}}-\frac {\left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) \sqrt {c-d x^2}}{6 a^2 c^2 (b c-a d)^2 e (e x)^{3/2}}+\frac {d^{3/4} \left (7 b^2 c^2-8 a b c d+10 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 )}{6 a^2 c^{7/4} (b c-a d)^2 e^{5/2} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (7 b c-13 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)^2 e^{5/2} \sqrt {c-d x^2}}+\frac {b^2 \sqrt [4]{c} (7 b c-13 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)^2 e^{5/2} \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 = 10.37, size = 318, normalized size = 0.62 \begin {gather*} \frac {x \left (5 a \left (2 a^3 d^2 \left (2 c-5 d x^2\right )-7 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+5 d^2 x^4\right )\right )+5 \left (21 b^3 c^3-32 a b^2 c^2 d-8 a^2 b c d^2+10 a^3 d^3\right ) x^2 \left (-a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} 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 d \left (7 b^2 c^2-8 a b c d+10 a^2 d^2\right ) x^4 \left (a-b x^2\right ) \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{30 a^3 c^2 (b c-a d)^2 (e x)^{5/2} \left (-a+b x^2\right ) \sqrt {c-d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x]

[Out]

(x*(5*a*(2*a^3*d^2*(2*c - 5*d*x^2) - 7*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 + 5*d^2*x^4)) + 5*(21*b^3*c^3 - 32*a*b^2*c^2*d - 8*a^2*b*c*d^2 + 10*a^3*d^3)*x^2*(-a
+ b*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*d*(7*b^2*c^2 - 8*a*b*c*d + 1
0*a^2*d^2)*x^4*(a - b*x^2)*Sqrt[1 - (d*x^2)/c]*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))/(30*a^3*c^2*
(b*c - a*d)^2*(e*x)^(5/2)*(-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. \(2858\) vs. \(2(418)=836\).
time = 0.13, size = 2859, normalized size = 5.58

method	result	size
elliptic	\(\frac {\sqrt {\left (-d \,x^{2}+c \right ) e x}\, \left (\frac {b^{3} \sqrt {-d e \,x^{3}+c e x}}{2 \left (a d -b c \right )^{2} a^{2} e^{3} \left (-b \,x^{2}+a \right )}+\frac {d^{3} x}{e^{2} c^{2} \left (a d -b c \right )^{2} \sqrt {-\left (x^{2}-\frac {c}{d}\right ) d e x}}-\frac {2 \sqrt {-d e \,x^{3}+c e x}}{3 c^{2} e^{3} a^{2} x^{2}}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right ) b^{2}}{4 \sqrt {-d e \,x^{3}+c e x}\, e^{2} a^{2} \left (a d -b c \right )^{2}}+\frac {d^{2} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {-d e \,x^{3}+c e x}\, c^{2} e^{2} \left (a d -b c \right )^{2}}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{3 \sqrt {-d e \,x^{3}+c e x}\, c^{2} e^{2} a^{2}}+\frac {13 b^{2} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{8 e^{2} a \left (a d -b c \right )^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {7 b^{3} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 e^{2} a^{2} \left (a d -b c \right )^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}-\frac {13 b^{2} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{8 e^{2} a \left (a d -b c \right )^{2} \sqrt {a b}\, \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}+\frac {7 b^{3} \sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right ) c}{8 e^{2} a^{2} \left (a d -b c \right )^{2} \sqrt {a b}\, d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\)	\(1094\)
default	\(\text {Expression too large to display}\)	\(2859\)

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)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/24*b*d*(-39*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))*2^(1/2)*a*b^4*c^3*d*x^3*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2
)*(-d*x/(c*d)^(1/2))^(1/2)+48*a^3*b*c^2*d^2*(a*b)^(1/2)+40*a^4*d^4*x^2*(a*b)^(1/2)+30*EllipticF(((d*x+(c*d)^(1
/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^2*b^2*c^2*d*x*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)
*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-48*a^2*b^2*c^3*d*(a*b)^(1/2)-40*a
^3*b*d^4*x^4*(a*b)^(1/2)+28*b^4*c^3*d*x^4*(a*b)^(1/2)+21*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))*2^(1/2)*b^5*c^4*x^3*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)
*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)-21*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))*2^(1/2)*b^5*c^4*x^3*((d*x+(c*d)^(1/2))/(c*d)^(
1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)+72*a^2*b^2*c*d^3*x^4*(a*b)^(1/2)-6
0*a*b^3*c^2*d^2*x^4*(a*b)^(1/2)-56*a^3*b*c*d^3*x^2*(a*b)^(1/2)+44*a*b^3*c^3*d*x^2*(a*b)^(1/2)-21*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))*2^(1/2)*b^4*c^3*x
^3*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))
^(1/2)*(a*b)^(1/2)+39*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))*2^(1/2)*a*b^4*c^3*d*x^3*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2
))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)-21*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))*2^(1/2)*b^4*c^3*x^3*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+
(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)+14*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/
2))^(1/2),1/2*2^(1/2))*2^(1/2)*b^4*c^3*x^3*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)+39*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))*2^(1/2)*a^2*b^3*c^3*d*x*((d*x+(c*d)^(1/2))/(c*d)^(1/
2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)-39*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))*2^(1/2)*a^2*b^3*c^3*d*x*((d*x+(c*d)^(
1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)+20*EllipticF(((d*x+(c
*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^4*d^3*x*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)
*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-28*b^4*c^4*x^2*(a*b)^(1/2)-16*a^4
*c*d^3*(a*b)^(1/2)+16*a*b^3*c^4*(a*b)^(1/2)-20*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^
(1/2)*a^3*b*d^3*x^3*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(
-d*x/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)+21*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))*2^(1/2)*a*b^3*c^3*x*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*
x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)+21*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))*2^(1/2)*a*b^3*c^3*x*(c*d)^(1/2)*((d*x+(c
*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-14*E
llipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a*b^3*c^3*x*(c*d)^(1/2)*((d*x+(c*d)^(1/2))
/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-39*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))*2^(1/2)*a^2*b^2
*c^2*d*x*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^
(1/2))^(1/2)*(a*b)^(1/2)+39*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))*2^(1/2)*a*b^3*c^2*d*x^3*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)-21*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))*2^(1/2)*a*b^4*c^4*x*((d*x+(c*d)^(1/2))/(c*d)^(1
/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)+21*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))*2^(1/2)*a*b^4*c^4*x*((d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-d*x/(c*d)^(1/2))^(1/2)+39*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))*2^(1/2)*a*b^3*c^2*d*x^3*(
c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(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)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

e^(-5/2)*integrate(1/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)*x^(5/2)), 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)^(5/2)/(-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)**(5/2)/(-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)^(5/2)/(-b*x^2+a)^2/(-d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate(e^(-5/2)/((b*x^2 - a)^2*(-d*x^2 + c)^(3/2)*x^(5/2)), 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 )}^{5/2}\,{\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)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)),x)

[Out]

int(1/((e*x)^(5/2)*(a - b*x^2)^2*(c - d*x^2)^(3/2)), x)

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