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

Optimal. Leaf size=625 \[ \frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{2 a c^{5/4} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d (e x)^{3/2} \left (-a^2 d^2+5 a b c d+b^2 c^2\right )}{2 a c^2 e \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a c^{5/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{b^{3/2} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-11 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^{3/2} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-11 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b (e x)^{3/2}}{2 a e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d (e x)^{3/2} (2 a d+3 b c)}{6 a c e \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.43087, antiderivative size = 625, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433, Rules used = {466, 472, 579, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ \frac{d (e x)^{3/2} \left (-a^2 d^2+5 a b c d+b^2 c^2\right )}{2 a c^2 e \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a c^{5/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a c^{5/4} \sqrt{c-d x^2} (b c-a d)^3}-\frac{b^{3/2} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-11 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b^{3/2} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-11 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{3/2} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{b (e x)^{3/2}}{2 a e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)}+\frac{d (e x)^{3/2} (2 a d+3 b c)}{6 a c e \left (c-d x^2\right )^{3/2} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(3*b*c + 2*a*d)*(e*x)^(3/2))/(6*a*c*(b*c - a*d)^2*e*(c - d*x^2)^(3/2)) + (b*(e*x)^(3/2))/(2*a*(b*c - a*d)*e
*(a - b*x^2)*(c - d*x^2)^(3/2)) + (d*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*(e*x)^(3/2))/(2*a*c^2*(b*c - a*d)^3*e*Sqr
t[c - d*x^2]) - (d^(1/4)*(b^2*c^2 + 5*a*b*c*d - a^2*d^2)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)
*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*c^(5/4)*(b*c - a*d)^3*Sqrt[c - d*x^2]) + (d^(1/4)*(b^2*c^2 + 5*a*b*c
*d - a^2*d^2)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a*c
^(5/4)*(b*c - a*d)^3*Sqrt[c - d*x^2]) - (b^(3/2)*c^(1/4)*(b*c - 11*a*d)*Sqrt[e]*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^(3/2)*d^(1/4
)*(b*c - a*d)^3*Sqrt[c - d*x^2]) + (b^(3/2)*c^(1/4)*(b*c - 11*a*d)*Sqrt[e]*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqr
t[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^(3/2)*d^(1/4)*(b*c -
a*d)^3*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 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{\sqrt{e x}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\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 (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{e \operatorname{Subst}\left (\int \frac{x^2 \left (\frac{b c-4 a d}{e^2}-\frac{7 b d x^4}{e^4}\right )}{\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) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac{b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}-\frac{e^3 \operatorname{Subst}\left (\int \frac{x^2 \left (-\frac{6 \left (b^2 c^2-8 a b c d+2 a^2 d^2\right )}{e^4}+\frac{6 b d (3 b c+2 a d) x^4}{e^6}\right )}{\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) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac{b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt{c-d x^2}}+\frac{e^5 \operatorname{Subst}\left (\int \frac{x^2 \left (\frac{12 \left (b^3 c^3-12 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right )}{e^6}+\frac{12 b d \left (b^2 c^2+5 a b c d-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 )}{24 a c^2 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac{b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt{c-d x^2}}+\frac{e^5 \operatorname{Subst}\left (\int \left (-\frac{12 d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) x^2}{e^6 \sqrt{c-\frac{d x^4}{e^2}}}+\frac{12 \left (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 )}{24 a c^2 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac{b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt{c-d x^2}}+\frac{\left (b^2 (b c-11 a d)\right ) \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 (b c-a d)^3 e}-\frac{\left (d \left (b^2 c^2+5 a b c d-a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 a c^2 (b c-a d)^3 e}\\ &=\frac{d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac{b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt{c-d x^2}}+\frac{\left (\sqrt{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 a c^{3/2} (b c-a d)^3}-\frac{\left (\sqrt{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right )\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 c^{3/2} (b c-a d)^3}+\frac{\left (b^{3/2} (b c-11 a d) e\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 (b c-a d)^3}-\frac{\left (b^{3/2} (b c-11 a d) e\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 (b c-a d)^3}\\ &=\frac{d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac{b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt{c-d x^2}}+\frac{\left (\sqrt{d} \left (b^2 c^2+5 a b c d-a^2 d^2\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 )}{2 a c^{3/2} (b c-a d)^3 \sqrt{c-d x^2}}-\frac{\left (\sqrt{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \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 c^{3/2} (b c-a d)^3 \sqrt{c-d x^2}}+\frac{\left (b^{3/2} (b c-11 a d) e \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 (b c-a d)^3 \sqrt{c-d x^2}}-\frac{\left (b^{3/2} (b c-11 a d) e \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 (b c-a d)^3 \sqrt{c-d x^2}}\\ &=\frac{d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac{b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt{c-d x^2}}+\frac{\sqrt [4]{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt{e} \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 c^{5/4} (b c-a d)^3 \sqrt{c-d x^2}}-\frac{b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt{e} \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^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt{c-d x^2}}+\frac{b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt{e} \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^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt{c-d x^2}}-\frac{\left (\sqrt{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \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 c^{3/2} (b c-a d)^3 \sqrt{c-d x^2}}\\ &=\frac{d (3 b c+2 a d) (e x)^{3/2}}{6 a c (b c-a d)^2 e \left (c-d x^2\right )^{3/2}}+\frac{b (e x)^{3/2}}{2 a (b c-a d) e \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d \left (b^2 c^2+5 a b c d-a^2 d^2\right ) (e x)^{3/2}}{2 a c^2 (b c-a d)^3 e \sqrt{c-d x^2}}-\frac{\sqrt [4]{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt{e} \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 c^{5/4} (b c-a d)^3 \sqrt{c-d x^2}}+\frac{\sqrt [4]{d} \left (b^2 c^2+5 a b c d-a^2 d^2\right ) \sqrt{e} \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 c^{5/4} (b c-a d)^3 \sqrt{c-d x^2}}-\frac{b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt{e} \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^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt{c-d x^2}}+\frac{b^{3/2} \sqrt [4]{c} (b c-11 a d) \sqrt{e} \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^{3/2} \sqrt [4]{d} (b c-a d)^3 \sqrt{c-d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.546268, size = 327, normalized size = 0.52 \[ \frac{\sqrt{e x} \left (3 b d x^3 \left (b x^2-a\right ) \left (c-d x^2\right ) \sqrt{1-\frac{d x^2}{c}} \left (-a^2 d^2+5 a b c d+b^2 c^2\right ) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+7 x \left (b x^2-a\right ) \left (c-d x^2\right ) \sqrt{1-\frac{d x^2}{c}} \left (-5 a^2 b c d^2+a^3 d^3-12 a b^2 c^2 d+b^3 c^3\right ) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+7 a x \left (a^2 b d^2 \left (-17 c^2+10 c d x^2+3 d^2 x^4\right )+a^3 d^3 \left (5 c-3 d x^2\right )+a b^2 c d^2 x^2 \left (17 c-15 d x^2\right )-3 b^3 c^2 \left (c-d x^2\right )^2\right )\right )}{42 a^2 c^2 \left (b x^2-a\right ) \left (c-d x^2\right )^{3/2} (b c-a d)^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[e*x]*(7*a*x*(a*b^2*c*d^2*x^2*(17*c - 15*d*x^2) + a^3*d^3*(5*c - 3*d*x^2) - 3*b^3*c^2*(c - d*x^2)^2 + a^2
*b*d^2*(-17*c^2 + 10*c*d*x^2 + 3*d^2*x^4)) + 7*(b^3*c^3 - 12*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*x*(-a + b*
x^2)*(c - d*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*(b^2*c^2 + 5*a*b
*c*d - a^2*d^2)*x^3*(-a + b*x^2)*(c - d*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^2*c^2*(b*c - a*d)^3*(-a + b*x^2)*(c - d*x^2)^(3/2))

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Maple [B]  time = 0.05, size = 5689, normalized size = 9.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(1/2)/(-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{\sqrt{e x}}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(e*x)/((b*x^2 - a)^2*(-d*x^2 + c)^(5/2)), 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((e*x)^(1/2)/(-b*x^2+a)^2/(-d*x^2+c)^(5/2),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((e*x)**(1/2)/(-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{\sqrt{e x}}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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