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

Optimal. Leaf size=188 \[ \frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}} \]

[Out]

(c^(1/4)*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*d^(1/4)*Sqrt[e]*Sqrt[c - d*x^2]) + (c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*S
qrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a*d^(1/4)*Sqrt[e]*Sqrt[c - d*x
^2])

________________________________________________________________________________________

Rubi [A]  time = 0.251575, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {466, 409, 1219, 1218} \[ \frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^(1/4)*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*d^(1/4)*Sqrt[e]*Sqrt[c - d*x^2]) + (c^(1/4)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqrt[b]*S
qrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(a*d^(1/4)*Sqrt[e]*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 409

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 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}{\sqrt{e x} \left (a-b x^2\right ) \sqrt{c-d x^2}} \, dx &=\frac{2 \operatorname{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 )}{e}\\ &=\frac{\operatorname{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 )}{a e}+\frac{\operatorname{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 )}{a e}\\ &=\frac{\sqrt{1-\frac{d x^2}{c}} \operatorname{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 )}{a e \sqrt{c-d x^2}}+\frac{\sqrt{1-\frac{d x^2}{c}} \operatorname{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 )}{a e \sqrt{c-d x^2}}\\ &=\frac{\sqrt [4]{c} \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 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \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 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0443237, size = 68, normalized size = 0.36 \[ \frac{2 x \sqrt{\frac{c-d x^2}{c}} F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{a \sqrt{e x} \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*x*Sqrt[(c - d*x^2)/c]*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a])/(a*Sqrt[e*x]*Sqrt[c - d*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.025, size = 344, normalized size = 1.8 \begin{align*}{\frac{\sqrt{2}b}{2\,d{x}^{2}-2\,c} \left ( \sqrt{cd}{\it EllipticPi} \left ( \sqrt{{ \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) b+\sqrt{ab}{\it EllipticPi} \left ( \sqrt{{ \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) d-\sqrt{cd}{\it EllipticPi} \left ( \sqrt{{ \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) b+\sqrt{ab}{\it EllipticPi} \left ( \sqrt{{ \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) d \right ) \sqrt{-{dx{\frac{1}{\sqrt{cd}}}}}\sqrt{{ \left ( -dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}}\sqrt{{ \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}}\sqrt{cd}\sqrt{-d{x}^{2}+c} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}{\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ex}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c} \sqrt{e x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- a \sqrt{e x} \sqrt{c - d x^{2}} + b x^{2} \sqrt{e x} \sqrt{c - d x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c} \sqrt{e x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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