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

Optimal. Leaf size=444 \[ \frac{5 c^{3/4} \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{2 a^2 e^{3/2} \sqrt{c-d x^2}}-\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}-\frac{5 c^{3/4} \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 e^{3/2} \sqrt{c-d x^2}}-\frac{5 \sqrt{c-d x^2}}{2 a^2 e \sqrt{e x}}+\frac{\sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.911519, antiderivative size = 444, 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, 469, 583, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ -\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^{5/2} \sqrt{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{5 c^{3/4} \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 e^{3/2} \sqrt{c-d x^2}}-\frac{5 c^{3/4} \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a^2 e^{3/2} \sqrt{c-d x^2}}-\frac{5 \sqrt{c-d x^2}}{2 a^2 e \sqrt{e x}}+\frac{\sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[c - d*x^2])/(2*a^2*e*Sqrt[e*x]) + Sqrt[c - d*x^2]/(2*a*e*Sqrt[e*x]*(a - b*x^2)) - (5*c^(3/4)*d^(1/4)*
Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*a^2*e^(3/2)*Sqrt[c - d*x^
2]) + (5*c^(3/4)*d^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(2*
a^2*e^(3/2)*Sqrt[c - d*x^2]) - (c^(1/4)*(5*b*c - 3*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticPi[-((Sqrt[b]*Sqrt[c])/(Sq
rt[a]*Sqrt[d])), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^(5/2)*Sqrt[b]*d^(1/4)*e^(3/2)*Sqrt[c
- d*x^2]) + (c^(1/4)*(5*b*c - 3*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcS
in[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(4*a^(5/2)*Sqrt[b]*d^(1/4)*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 469

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[((e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(a*e*n*(p + 1)), x] + Dist[1/(a*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)
^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{
a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[0, q, 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{\sqrt{c-d x^2}}{(e x)^{3/2} \left (a-b x^2\right )^2} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{d x^4}{e^2}}}{x^2 \left (a-\frac{b x^4}{e^2}\right )^2} \, dx,x,\sqrt{e x}\right )}{e}\\ &=\frac{\sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{-5 c+\frac{3 d x^4}{e^2}}{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 e}\\ &=-\frac{5 \sqrt{c-d x^2}}{2 a^2 e \sqrt{e x}}+\frac{\sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (\frac{c (5 b c-8 a d)}{e^2}+\frac{5 b c d x^4}{e^4}\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 e}\\ &=-\frac{5 \sqrt{c-d x^2}}{2 a^2 e \sqrt{e x}}+\frac{\sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right )}+\frac{\operatorname{Subst}\left (\int \left (-\frac{5 c d x^2}{e^2 \sqrt{c-\frac{d x^4}{e^2}}}+\frac{\left (5 b c^2-3 a c d\right ) x^2}{e^2 \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 e}\\ &=-\frac{5 \sqrt{c-d x^2}}{2 a^2 e \sqrt{e x}}+\frac{\sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right )}-\frac{(5 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 e^3}+\frac{(5 b c-3 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 e^3}\\ &=-\frac{5 \sqrt{c-d x^2}}{2 a^2 e \sqrt{e x}}+\frac{\sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right )}+\frac{\left (5 \sqrt{c} \sqrt{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 e^2}-\frac{\left (5 \sqrt{c} \sqrt{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 e^2}+\frac{(5 b c-3 a d) \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 \sqrt{b} e}-\frac{(5 b c-3 a d) \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 \sqrt{b} e}\\ &=-\frac{5 \sqrt{c-d x^2}}{2 a^2 e \sqrt{e x}}+\frac{\sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right )}+\frac{\left (5 \sqrt{c} \sqrt{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 e^2 \sqrt{c-d x^2}}-\frac{\left (5 \sqrt{c} \sqrt{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 e^2 \sqrt{c-d x^2}}+\frac{\left ((5 b c-3 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 \sqrt{b} e \sqrt{c-d x^2}}-\frac{\left ((5 b c-3 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 \sqrt{b} e \sqrt{c-d x^2}}\\ &=-\frac{5 \sqrt{c-d x^2}}{2 a^2 e \sqrt{e x}}+\frac{\sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right )}+\frac{5 c^{3/4} \sqrt [4]{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 e^{3/2} \sqrt{c-d x^2}}-\frac{\sqrt [4]{c} (5 b c-3 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{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} (5 b c-3 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{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}-\frac{\left (5 \sqrt{c} \sqrt{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 e^2 \sqrt{c-d x^2}}\\ &=-\frac{5 \sqrt{c-d x^2}}{2 a^2 e \sqrt{e x}}+\frac{\sqrt{c-d x^2}}{2 a e \sqrt{e x} \left (a-b x^2\right )}-\frac{5 c^{3/4} \sqrt [4]{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 e^{3/2} \sqrt{c-d x^2}}+\frac{5 c^{3/4} \sqrt [4]{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 e^{3/2} \sqrt{c-d x^2}}-\frac{\sqrt [4]{c} (5 b c-3 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{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} (5 b c-3 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{b} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.177885, size = 182, normalized size = 0.41 \[ \frac{x \left (15 b d x^4 \left (b x^2-a\right ) \sqrt{1-\frac{d x^2}{c}} F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+7 x^2 \left (a-b x^2\right ) \sqrt{1-\frac{d x^2}{c}} (8 a d-5 b c) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+21 a \left (4 a-5 b x^2\right ) \left (c-d x^2\right )\right )}{42 a^3 (e x)^{3/2} \left (b x^2-a\right ) \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(21*a*(4*a - 5*b*x^2)*(c - d*x^2) + 7*(-5*b*c + 8*a*d)*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] + 15*b*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*(e*x)^(3/2)*(-a + b*x^2)*Sqrt[c - d*x^2])

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Maple [B]  time = 0.033, size = 2568, normalized size = 5.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(-d*x^2+c)^(1/2)*d*(-4*x^2*a*b^2*c*d-3*((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*d-3*((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*d-3*((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*d+20*x^4*a*b^2*d^2-20*x^4*b
^3*c*d+3*((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*d-3*((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*d-16*a*b^2*c^2-5*((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+3*((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*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+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*a*b^2*c*d-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*a*b^2*c*d+20*x^2*b^3*c^2-16*
x^2*a^2*b*d^2+16*a^2*b*c*d+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^2*b*c*d+3*((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*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))*a^2*b*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)*Ell
ipticPi(((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+3*((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*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-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^2+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^2-20*((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^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^2-5*((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
))*x^2*b^3*c^2+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^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^2+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^2)/e/(e*x)^(1/2)/(d*x^2-c)/a^2/(b*x^2-a)/((a*b)^(1/
2)*d+(c*d)^(1/2)*b)/((c*d)^(1/2)*b-(a*b)^(1/2)*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-d x^{2} + c}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-d x^{2} + c}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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