3.916 \(\int \frac{1}{(e x)^{3/2} (a-b x^2)^2 \sqrt{c-d x^2}} \, dx\)

Optimal. Leaf size=535 \[ \frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-4 a d) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{2 a^2 \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-4 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-7 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-7 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (5 b c-4 a d)}{2 a^2 c e \sqrt{e x} (b c-a d)}+\frac{b \sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right ) (b c-a d)} \]

[Out]

-((5*b*c - 4*a*d)*Sqrt[c - d*x^2])/(2*a^2*c*(b*c - a*d)*e*Sqrt[e*x]) + (b*Sqrt[c - d*x^2])/(2*a*(b*c - a*d)*e*
Sqrt[e*x]*(a - b*x^2)) - (d^(1/4)*(5*b*c - 4*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^
(1/4)*Sqrt[e])], -1])/(2*a^2*c^(1/4)*(b*c - a*d)*e^(3/2)*Sqrt[c - d*x^2]) + (d^(1/4)*(5*b*c - 4*a*d)*Sqrt[1 -
(d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a^2*c^(1/4)*(b*c - a*d)*e^(3/2)*Sq
rt[c - d*x^2]) - (Sqrt[b]*c^(1/4)*(5*b*c - 7*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)*e^(3/2)*Sqrt[c -
d*x^2]) + (Sqrt[b]*c^(1/4)*(5*b*c - 7*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)*e^(3/2)*Sqrt[c - d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.07725, antiderivative size = 535, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433, Rules used = {466, 472, 583, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ \frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-4 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-4 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-7 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-7 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (5 b c-4 a d)}{2 a^2 c e \sqrt{e x} (b c-a d)}+\frac{b \sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((5*b*c - 4*a*d)*Sqrt[c - d*x^2])/(2*a^2*c*(b*c - a*d)*e*Sqrt[e*x]) + (b*Sqrt[c - d*x^2])/(2*a*(b*c - a*d)*e*
Sqrt[e*x]*(a - b*x^2)) - (d^(1/4)*(5*b*c - 4*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^
(1/4)*Sqrt[e])], -1])/(2*a^2*c^(1/4)*(b*c - a*d)*e^(3/2)*Sqrt[c - d*x^2]) + (d^(1/4)*(5*b*c - 4*a*d)*Sqrt[1 -
(d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a^2*c^(1/4)*(b*c - a*d)*e^(3/2)*Sq
rt[c - d*x^2]) - (Sqrt[b]*c^(1/4)*(5*b*c - 7*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)*e^(3/2)*Sqrt[c -
d*x^2]) + (Sqrt[b]*c^(1/4)*(5*b*c - 7*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)*e^(3/2)*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 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 584

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 307

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 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 1200

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 1199

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 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 490

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), In
t[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 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)^{3/2} \left (a-b x^2\right )^2 \sqrt{c-d x^2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a-\frac{b x^4}{e^2}\right )^2 \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{e}\\ &=\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e \sqrt{e x} \left (a-b x^2\right )}+\frac{e \operatorname{Subst}\left (\int \frac{\frac{5 b c-4 a d}{e^2}-\frac{3 b d x^4}{e^4}}{x^2 \left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 a (b c-a d)}\\ &=-\frac{(5 b c-4 a d) \sqrt{c-d x^2}}{2 a^2 c (b c-a d) e \sqrt{e x}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e \sqrt{e x} \left (a-b x^2\right )}-\frac{e \operatorname{Subst}\left (\int \frac{x^2 \left (-\frac{(b c-2 a d) (5 b c-2 a d)}{e^4}-\frac{b d (5 b c-4 a d) x^4}{e^6}\right )}{\left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 a^2 c (b c-a d)}\\ &=-\frac{(5 b c-4 a d) \sqrt{c-d x^2}}{2 a^2 c (b c-a d) e \sqrt{e x}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e \sqrt{e x} \left (a-b x^2\right )}-\frac{e \operatorname{Subst}\left (\int \left (\frac{d (5 b c-4 a d) x^2}{e^4 \sqrt{c-\frac{d x^4}{e^2}}}-\frac{\left (5 b^2 c^2-7 a b c d\right ) x^2}{e^4 \left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}}\right ) \, dx,x,\sqrt{e x}\right )}{2 a^2 c (b c-a d)}\\ &=-\frac{(5 b c-4 a d) \sqrt{c-d x^2}}{2 a^2 c (b c-a d) e \sqrt{e x}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e \sqrt{e x} \left (a-b x^2\right )}+\frac{(b (5 b c-7 a d)) \operatorname{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) e^3}-\frac{(d (5 b c-4 a d)) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 a^2 c (b c-a d) e^3}\\ &=-\frac{(5 b c-4 a d) \sqrt{c-d x^2}}{2 a^2 c (b c-a d) e \sqrt{e x}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e \sqrt{e x} \left (a-b x^2\right )}+\frac{\left (\sqrt{d} (5 b c-4 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 a^2 \sqrt{c} (b c-a d) e^2}-\frac{\left (\sqrt{d} (5 b c-4 a d)\right ) \operatorname{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 \sqrt{c} (b c-a d) e^2}+\frac{\left (\sqrt{b} (5 b c-7 a d)\right ) \operatorname{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) e}-\frac{\left (\sqrt{b} (5 b c-7 a d)\right ) \operatorname{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) e}\\ &=-\frac{(5 b c-4 a d) \sqrt{c-d x^2}}{2 a^2 c (b c-a d) e \sqrt{e x}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e \sqrt{e x} \left (a-b x^2\right )}+\frac{\left (\sqrt{d} (5 b c-4 a d) \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 )}{2 a^2 \sqrt{c} (b c-a d) e^2 \sqrt{c-d x^2}}-\frac{\left (\sqrt{d} (5 b c-4 a d) \sqrt{1-\frac{d x^2}{c}}\right ) \operatorname{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 \sqrt{c} (b c-a d) e^2 \sqrt{c-d x^2}}+\frac{\left (\sqrt{b} (5 b c-7 a d) \sqrt{1-\frac{d x^2}{c}}\right ) \operatorname{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) e \sqrt{c-d x^2}}-\frac{\left (\sqrt{b} (5 b c-7 a d) \sqrt{1-\frac{d x^2}{c}}\right ) \operatorname{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) e \sqrt{c-d x^2}}\\ &=-\frac{(5 b c-4 a d) \sqrt{c-d x^2}}{2 a^2 c (b c-a d) e \sqrt{e x}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e \sqrt{e x} \left (a-b x^2\right )}+\frac{\sqrt [4]{d} (5 b c-4 a d) \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 \sqrt [4]{c} (b c-a d) e^{3/2} \sqrt{c-d x^2}}-\frac{\sqrt{b} \sqrt [4]{c} (5 b c-7 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) e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} (5 b c-7 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) e^{3/2} \sqrt{c-d x^2}}-\frac{\left (\sqrt{d} (5 b c-4 a d) \sqrt{1-\frac{d x^2}{c}}\right ) \operatorname{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 \sqrt{c} (b c-a d) e^2 \sqrt{c-d x^2}}\\ &=-\frac{(5 b c-4 a d) \sqrt{c-d x^2}}{2 a^2 c (b c-a d) e \sqrt{e x}}+\frac{b \sqrt{c-d x^2}}{2 a (b c-a d) e \sqrt{e x} \left (a-b x^2\right )}-\frac{\sqrt [4]{d} (5 b c-4 a d) \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 \sqrt [4]{c} (b c-a d) e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt [4]{d} (5 b c-4 a d) \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 \sqrt [4]{c} (b c-a d) e^{3/2} \sqrt{c-d x^2}}-\frac{\sqrt{b} \sqrt [4]{c} (5 b c-7 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) e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} (5 b c-7 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) e^{3/2} \sqrt{c-d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.276013, size = 235, normalized size = 0.44 \[ \frac{x \left (7 x^2 \left (b x^2-a\right ) \sqrt{1-\frac{d x^2}{c}} \left (4 a^2 d^2-12 a b c d+5 b^2 c^2\right ) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-21 a \left (c-d x^2\right ) \left (4 a^2 d-4 a b \left (c+d x^2\right )+5 b^2 c x^2\right )+3 b d x^4 \left (a-b x^2\right ) \sqrt{1-\frac{d x^2}{c}} (4 a d-5 b c) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}{42 a^3 c (e x)^{3/2} \left (b x^2-a\right ) \sqrt{c-d x^2} (b c-a d)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(-21*a*(c - d*x^2)*(4*a^2*d + 5*b^2*c*x^2 - 4*a*b*(c + d*x^2)) + 7*(5*b^2*c^2 - 12*a*b*c*d + 4*a^2*d^2)*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*c + 4*a*d)*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*(b*c - a*d)*(e
*x)^(3/2)*(-a + b*x^2)*Sqrt[c - d*x^2])

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Maple [B]  time = 0.038, size = 2982, normalized size = 5.6 \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)^(3/2)/(-b*x^2+a)^2/(-d*x^2+c)^(1/2),x)

[Out]

-1/8*(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)*(-x*d/(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^2*c^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)*(-x*d/(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^2*c^3-8*((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)*(-x*d/(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*c*d^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)*(-x*d/(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^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)*(-x*d/(c
*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),
1/2*2^(1/2))*a*b^2*c^3+7*((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)*
(-x*d/(c*d)^(1/2))^(1/2)*EllipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(
1/2)*b),1/2*2^(1/2))*x^2*a*b^2*c^2*d+36*((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)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b^2
*c^2*d-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)*(-x*d/(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)*x^2*b^2*c^2-16*((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)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a
^2*b*c*d^2+36*x^4*a*b^2*c*d^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)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*b^3*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)*(-x*d/(c*d)^(1/2))^(1/2)*El
lipticPi(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*x^2*b^
3*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)*(-x*d/(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))*x^2*b^3*c^3-18*((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)*(-x*d/
(c*d)^(1/2))^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a*b^2*c^2*d-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)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticE((
(d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*b^3*c^3+16*((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)*(-x*d/(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*c*d^2+7*((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)*(-x*d/(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))*x^2*a*b^2*c^2*d+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)*(-x*d/(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/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*(c*d)^(1/2)*x^2*b^2*c^2+8*((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)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticF(((d*x+(c*d)^(1/2))/(c*d)
^(1/2))^(1/2),1/2*2^(1/2))*x^2*a^2*b*c*d^2+7*((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)*(-x*d/(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/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*(c*d)^(1/2)*a^2*c*d-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)*(-x*d/(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/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*(c*d)^(1/2)*a*b*c^2+7*((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)*(-x*d/(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)*x^2*a*b*c*d-16*a*b^2*c^3-7*((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)*(-x*d/(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/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*(c*d)^(1/2)*x^2*a*b*c*d-16*x^2*a^2*b*c*d^2-20*x^2*a*b^2*c^2*
d+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)*(-x*d/(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*c^2+16*x^2*a^3*d^3+20*x^2*b^3*c^3-16*x^4*a^2*b*d^3-20*x^4*b^3*c^2*d-16*a^3*c*d^2+3
2*a^2*b*c^2*d-7*((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)*(-x*d/(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^2*c*d-7*((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)*(-x*d/(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^2*b*c^2*d-36*((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)*(-x*d/(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*c^2*d+18*((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)*(-
x*d/(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*c^2*d-7*((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)*(-x*d/(c*d)^(1/2))^(1/2)*EllipticPi((
(d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),(c*d)^(1/2)*b/((a*b)^(1/2)*d+(c*d)^(1/2)*b),1/2*2^(1/2))*a^2*b*c^2*d)*b*d
*(-d*x^2+c)^(1/2)/c/((c*d)^(1/2)*b-(a*b)^(1/2)*d)/((a*b)^(1/2)*d+(c*d)^(1/2)*b)/(b*x^2-a)/(a*d-b*c)/a^2/(d*x^2
-c)/e/(e*x)^(1/2)

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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} \sqrt{-d x^{2} + c} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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)^(1/2),x, algorithm="maxima")

[Out]

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

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

[Out]

Timed out

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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)**(3/2)/(-b*x**2+a)**2/(-d*x**2+c)**(1/2),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} \sqrt{-d x^{2} + c} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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)^(1/2),x, algorithm="giac")

[Out]

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