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

Optimal. Leaf size=203 \[ \frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2}} \]

[Out]

-((c^(1/4)*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])/(Sqrt[a]*Sqrt[b]*d^(1/4)*Sqrt[c - d*x^2])) + (c^(1/4)*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])/(Sqrt[
a]*Sqrt[b]*d^(1/4)*Sqrt[c - d*x^2])

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Rubi [A]  time = 0.260219, antiderivative size = 203, 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, 490, 1219, 1218} \[ \frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [C]  time = 0.041193, size = 70, normalized size = 0.34 \[ \frac{2 x \sqrt{e x} \sqrt{\frac{c-d x^2}{c}} F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{3 a \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*x*Sqrt[e*x]*Sqrt[(c - d*x^2)/c]*AppellF1[3/4, 1/2, 1, 7/4, (d*x^2)/c, (b*x^2)/a])/(3*a*Sqrt[c - d*x^2])

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Maple [B]  time = 0.019, size = 337, normalized size = 1.7 \begin{align*} -{\frac{d\sqrt{2}}{2\,x \left ( d{x}^{2}-c \right ) } \left ({\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 ) bc-\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 ) \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 ) bc+\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 ) \sqrt{cd} \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{-d{x}^{2}+c}\sqrt{ex} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} \sqrt{-d x^{2} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} \sqrt{-d x^{2} + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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