3.872 \(\int \frac{\sqrt{c-d x^2}}{(e x)^{7/2} (a-b x^2)} \, dx\)

Optimal. Leaf size=457 \[ \frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-2 a d) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2} (5 b c-2 a d)}{5 a^2 c e^3 \sqrt{e x}}-\frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-2 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt{c-d x^2}}-\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{5 a e (e x)^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.983984, antiderivative size = 457, 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, 475, 583, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ -\frac{2 \sqrt{c-d x^2} (5 b c-2 a d)}{5 a^2 c e^3 \sqrt{e x}}+\frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-2 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt{c-d x^2}}-\frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} (5 b c-2 a d) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt{c-d x^2}}-\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{5 a e (e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - d*x^2]/((e*x)^(7/2)*(a - b*x^2)),x]

[Out]

(-2*Sqrt[c - d*x^2])/(5*a*e*(e*x)^(5/2)) - (2*(5*b*c - 2*a*d)*Sqrt[c - d*x^2])/(5*a^2*c*e^3*Sqrt[e*x]) - (2*d^
(1/4)*(5*b*c - 2*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(5*a^2
*c^(1/4)*e^(7/2)*Sqrt[c - d*x^2]) + (2*d^(1/4)*(5*b*c - 2*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticF[ArcSin[(d^(1/4)*S
qrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(5*a^2*c^(1/4)*e^(7/2)*Sqrt[c - d*x^2]) - (Sqrt[b]*c^(1/4)*(b*c - 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])/(a^(5/2)*d^(1/4)*e^(7/2)*Sqrt[c - d*x^2]) + (Sqrt[b]*c^(1/4)*(b*c - a*d)*Sqrt[1 - (d*x^2)/c]*Elliptic
Pi[(Sqrt[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a^(5/2)*d^(1/4)*e
^(7/2)*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 475

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*(m + 1)), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*
x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*b*(m + 1) + n*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + b*n*(p + q + 1))*x^n, x
], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[0, q, 1] && LtQ[m, -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)^{7/2} \left (a-b x^2\right )} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{d x^4}{e^2}}}{x^6 \left (a-\frac{b x^4}{e^2}\right )} \, dx,x,\sqrt{e x}\right )}{e}\\ &=-\frac{2 \sqrt{c-d x^2}}{5 a e (e x)^{5/2}}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{5 b c-2 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 )}{5 a e}\\ &=-\frac{2 \sqrt{c-d x^2}}{5 a e (e x)^{5/2}}-\frac{2 (5 b c-2 a d) \sqrt{c-d x^2}}{5 a^2 c e^3 \sqrt{e x}}-\frac{2 \operatorname{Subst}\left (\int \frac{x^2 \left (-\frac{5 b^2 c^2-10 a b c d+2 a^2 d^2}{e^4}-\frac{b d (5 b c-2 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 )}{5 a^2 c e}\\ &=-\frac{2 \sqrt{c-d x^2}}{5 a e (e x)^{5/2}}-\frac{2 (5 b c-2 a d) \sqrt{c-d x^2}}{5 a^2 c e^3 \sqrt{e x}}-\frac{2 \operatorname{Subst}\left (\int \left (\frac{d (5 b c-2 a d) x^2}{e^4 \sqrt{c-\frac{d x^4}{e^2}}}-\frac{5 \left (b^2 c^2-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 )}{5 a^2 c e}\\ &=-\frac{2 \sqrt{c-d x^2}}{5 a e (e x)^{5/2}}-\frac{2 (5 b c-2 a d) \sqrt{c-d x^2}}{5 a^2 c e^3 \sqrt{e x}}-\frac{(2 d (5 b c-2 a d)) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 a^2 c e^5}+\frac{(2 b (b c-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 )}{a^2 e^5}\\ &=-\frac{2 \sqrt{c-d x^2}}{5 a e (e x)^{5/2}}-\frac{2 (5 b c-2 a d) \sqrt{c-d x^2}}{5 a^2 c e^3 \sqrt{e x}}+\frac{\left (2 \sqrt{d} (5 b c-2 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{5 a^2 \sqrt{c} e^4}-\frac{\left (2 \sqrt{d} (5 b c-2 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 )}{5 a^2 \sqrt{c} e^4}+\frac{\left (\sqrt{b} (b c-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 )}{a^2 e^3}-\frac{\left (\sqrt{b} (b c-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 )}{a^2 e^3}\\ &=-\frac{2 \sqrt{c-d x^2}}{5 a e (e x)^{5/2}}-\frac{2 (5 b c-2 a d) \sqrt{c-d x^2}}{5 a^2 c e^3 \sqrt{e x}}+\frac{\left (2 \sqrt{d} (5 b c-2 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 )}{5 a^2 \sqrt{c} e^4 \sqrt{c-d x^2}}-\frac{\left (2 \sqrt{d} (5 b c-2 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 )}{5 a^2 \sqrt{c} e^4 \sqrt{c-d x^2}}+\frac{\left (\sqrt{b} (b c-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 )}{a^2 e^3 \sqrt{c-d x^2}}-\frac{\left (\sqrt{b} (b c-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 )}{a^2 e^3 \sqrt{c-d x^2}}\\ &=-\frac{2 \sqrt{c-d x^2}}{5 a e (e x)^{5/2}}-\frac{2 (5 b c-2 a d) \sqrt{c-d x^2}}{5 a^2 c e^3 \sqrt{e x}}+\frac{2 \sqrt [4]{d} (5 b c-2 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 )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt{c-d x^2}}-\frac{\sqrt{b} \sqrt [4]{c} (b c-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 )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} (b c-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 )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt{c-d x^2}}-\frac{\left (2 \sqrt{d} (5 b c-2 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 )}{5 a^2 \sqrt{c} e^4 \sqrt{c-d x^2}}\\ &=-\frac{2 \sqrt{c-d x^2}}{5 a e (e x)^{5/2}}-\frac{2 (5 b c-2 a d) \sqrt{c-d x^2}}{5 a^2 c e^3 \sqrt{e x}}-\frac{2 \sqrt [4]{d} (5 b c-2 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 )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt{c-d x^2}}+\frac{2 \sqrt [4]{d} (5 b c-2 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 )}{5 a^2 \sqrt [4]{c} e^{7/2} \sqrt{c-d x^2}}-\frac{\sqrt{b} \sqrt [4]{c} (b c-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 )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} (b c-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 )}{a^{5/2} \sqrt [4]{d} e^{7/2} \sqrt{c-d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.218595, size = 190, normalized size = 0.42 \[ \frac{x \left (14 x^4 \sqrt{1-\frac{d x^2}{c}} \left (2 a^2 d^2-10 a b c d+5 b^2 c^2\right ) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-6 \left (b d x^6 \sqrt{1-\frac{d x^2}{c}} (2 a d-5 b c) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+7 a \left (c-d x^2\right ) \left (a c-2 a d x^2+5 b c x^2\right )\right )\right )}{105 a^3 c (e x)^{7/2} \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[c - d*x^2]/((e*x)^(7/2)*(a - b*x^2)),x]

[Out]

(x*(14*(5*b^2*c^2 - 10*a*b*c*d + 2*a^2*d^2)*x^4*Sqrt[1 - (d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x
^2)/a] - 6*(7*a*(c - d*x^2)*(a*c + 5*b*c*x^2 - 2*a*d*x^2) + b*d*(-5*b*c + 2*a*d)*x^6*Sqrt[1 - (d*x^2)/c]*Appel
lF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])))/(105*a^3*c*(e*x)^(7/2)*Sqrt[c - d*x^2])

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Maple [B]  time = 0.041, size = 1553, normalized size = 3.4 \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)^(7/2)/(-b*x^2+a),x)

[Out]

1/10*(-d*x^2+c)^(1/2)*d*b*(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*a*b*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*a*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)*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^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)*(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*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)*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^2*a^2*c*d^2+28*((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*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^2*c^3+4*((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*c*d^2-14*((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*c^2*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*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))*x^2*a*b*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*a*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)*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^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)*(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*c^2
-8*x^4*a^2*d^3+28*x^4*a*b*c*d^2-20*x^4*b^2*c^2*d+12*x^2*a^2*c*d^2-32*x^2*a*b*c^2*d+20*x^2*b^2*c^3-4*a^2*c^2*d+
4*a*b*c^3)/x^2/e^3/(e*x)^(1/2)/(d*x^2-c)/a^2/c/((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 )} \left (e x\right )^{\frac{7}{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)^(7/2)/(-b*x^2+a),x, algorithm="maxima")

[Out]

-integrate(sqrt(-d*x^2 + c)/((b*x^2 - a)*(e*x)^(7/2)), 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)^(7/2)/(-b*x^2+a),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)**(7/2)/(-b*x**2+a),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 )} \left (e x\right )^{\frac{7}{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)^(7/2)/(-b*x^2+a),x, algorithm="giac")

[Out]

integrate(-sqrt(-d*x^2 + c)/((b*x^2 - a)*(e*x)^(7/2)), x)