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

Optimal. Leaf size=261 \[ \frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt [4]{d} \sqrt{c-d x^2}} \]

[Out]

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

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Rubi [A]  time = 1.00198, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233 \[ \frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt [4]{d} \sqrt{c-d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 156.888, size = 238, normalized size = 0.91 \[ - \frac{2 \sqrt [4]{c} e^{\frac{3}{2}} \sqrt{1 - \frac{d x^{2}}{c}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | -1\right )}{b \sqrt [4]{d} \sqrt{c - d x^{2}}} + \frac{\sqrt [4]{c} e^{\frac{3}{2}} \sqrt{1 - \frac{d x^{2}}{c}} \Pi \left (- \frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}}; \operatorname{asin}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | -1\right )}{b \sqrt [4]{d} \sqrt{c - d x^{2}}} + \frac{\sqrt [4]{c} e^{\frac{3}{2}} \sqrt{1 - \frac{d x^{2}}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}}; \operatorname{asin}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | -1\right )}{b \sqrt [4]{d} \sqrt{c - d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(3/2)/(-b*x**2+a)/(-d*x**2+c)**(1/2),x)

[Out]

-2*c**(1/4)*e**(3/2)*sqrt(1 - d*x**2/c)*elliptic_f(asin(d**(1/4)*sqrt(e*x)/(c**(
1/4)*sqrt(e))), -1)/(b*d**(1/4)*sqrt(c - d*x**2)) + c**(1/4)*e**(3/2)*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)/(b*d**(1/4)*sqrt(c - d*x**2)) + c**(1/4)*e**(3/2)*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)/(b*d**(1/4)*sqrt(c - d*x**2))

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Mathematica [C]  time = 0.229992, size = 165, normalized size = 0.63 \[ -\frac{18 a c x (e x)^{3/2} F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{5 \left (b x^2-a\right ) \sqrt{c-d x^2} \left (2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-18*a*c*x*(e*x)^(3/2)*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/(5*(-a
+ b*x^2)*Sqrt[c - d*x^2]*(9*a*c*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]
+ 2*x^2*(2*b*c*AppellF1[9/4, 1/2, 2, 13/4, (d*x^2)/c, (b*x^2)/a] + a*d*AppellF1
[9/4, 3/2, 1, 13/4, (d*x^2)/c, (b*x^2)/a])))

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Maple [B]  time = 0.034, size = 417, normalized size = 1.6 \[ -{\frac{\sqrt{2}e}{2\,x \left ( d{x}^{2}-c \right ) } \left ({\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) ab\sqrt{cd}-{\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) ad\sqrt{ab}-{\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) ab\sqrt{cd}-{\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) ad\sqrt{ab}+2\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},1/2\,\sqrt{2} \right ) ad\sqrt{ab}-2\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},1/2\,\sqrt{2} \right ) bc\sqrt{ab} \right ) \sqrt{-{dx{\frac{1}{\sqrt{cd}}}}}\sqrt{{1 \left ( -dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}}\sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}}\sqrt{cd}\sqrt{-d{x}^{2}+c}\sqrt{ex} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x)^(3/2)/((b*x^2 - a)*sqrt(-d*x^2 + c)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**(3/2)/(-b*x**2+a)/(-d*x**2+c)**(1/2),x)

[Out]

-Integral((e*x)**(3/2)/(-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. \[ \int -\frac{\left (e x\right )^{\frac{3}{2}}}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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