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

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]*S
qrt[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 = 0.810686, 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 \[ \frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{d} \sqrt{c-d x^2}} \]

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]*S
qrt[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 in Sympy [A]  time = 101.271, size = 187, normalized size = 0.92 \[ - \frac{\sqrt [4]{c} \sqrt{e} \sqrt{1 - \frac{d x^{2}}{c}} \Pi \left (- \frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}}; \operatorname{asin}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | -1\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{d} \sqrt{c - d x^{2}}} + \frac{\sqrt [4]{c} \sqrt{e} \sqrt{1 - \frac{d x^{2}}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}}; \operatorname{asin}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | -1\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{d} \sqrt{c - d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(1/2)/(-b*x**2+a)/(-d*x**2+c)**(1/2),x)

[Out]

-c**(1/4)*sqrt(e)*sqrt(1 - d*x**2/c)*elliptic_pi(-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(
d)), asin(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)*elliptic_pi(sqrt(b)*sqrt
(c)/(sqrt(a)*sqrt(d)), asin(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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Mathematica [C]  time = 0.223889, size = 165, normalized size = 0.81 \[ -\frac{14 a c x \sqrt{e x} F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{3 \left (b x^2-a\right ) \sqrt{c-d x^2} \left (2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [B]  time = 0.032, size = 337, normalized size = 1.7 \[ -{\frac{d\sqrt{2}}{2\,x \left ( d{x}^{2}-c \right ) } \left ({\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) bc-\sqrt{cd}\sqrt{ab}{\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) +{\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) bc+\sqrt{cd}\sqrt{ab}{\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) \right ) \sqrt{-{dx{\frac{1}{\sqrt{cd}}}}}\sqrt{{1 \left ( -dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}}\sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}}\sqrt{-d{x}^{2}+c}\sqrt{ex} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}} \]

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-(c*d)^(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))+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+(c*d)^(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)))*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. \[ -\int \frac{\sqrt{e x}}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(e*x)/((b*x^2 - a)*sqrt(-d*x^2 + c)),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. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{e x}}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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