3.881 \(\int \frac{(e x)^{5/2}}{\left (a-b x^2\right ) \sqrt{c-d x^2}} \, dx\)

Optimal. Leaf size=349 \[ -\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b d^{3/4} \sqrt{c-d x^2}}-\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b d^{3/4} \sqrt{c-d x^2}} \]

[Out]

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

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Rubi [A]  time = 1.47099, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.367 \[ -\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt{a} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b d^{3/4} \sqrt{c-d x^2}}-\frac{2 c^{3/4} e^{5/2} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b d^{3/4} \sqrt{c-d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^(5/2)/((a - b*x^2)*Sqrt[c - d*x^2]),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(5/2)/(-b*x**2+a)/(-d*x**2+c)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.223809, size = 165, normalized size = 0.47 \[ -\frac{22 a c x (e x)^{5/2} F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{7 \left (b x^2-a\right ) \sqrt{c-d x^2} \left (2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(e*x)^(5/2)/((a - b*x^2)*Sqrt[c - d*x^2]),x]

[Out]

(-22*a*c*x*(e*x)^(5/2)*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])/(7*(-a
+ b*x^2)*Sqrt[c - d*x^2]*(11*a*c*AppellF1[7/4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)
/a] + 2*x^2*(2*b*c*AppellF1[11/4, 1/2, 2, 15/4, (d*x^2)/c, (b*x^2)/a] + a*d*Appe
llF1[11/4, 3/2, 1, 15/4, (d*x^2)/c, (b*x^2)/a])))

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Maple [A]  time = 0.049, size = 472, normalized size = 1.4 \[{\frac{\sqrt{2}{e}^{2}}{2\,x \left ( d{x}^{2}-c \right ) b} \left ( 4\,{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},1/2\,\sqrt{2} \right ) abcd-4\,{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{2}-2\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},1/2\,\sqrt{2} \right ) abcd+2\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},1/2\,\sqrt{2} \right ){b}^{2}{c}^{2}+\sqrt{cd}\sqrt{ab}{\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) ad-\sqrt{cd}\sqrt{ab}{\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) ad-{\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) abcd-{\it EllipticPi} \left ( \sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}},{b\sqrt{cd} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1}},{\frac{\sqrt{2}}{2}} \right ) abcd \right ) \sqrt{-{dx{\frac{1}{\sqrt{cd}}}}}\sqrt{{1 \left ( -dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}}\sqrt{{1 \left ( dx+\sqrt{cd} \right ){\frac{1}{\sqrt{cd}}}}}\sqrt{-d{x}^{2}+c}\sqrt{ex} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(5/2)/(-b*x^2+a)/(-d*x^2+c)^(1/2),x)

[Out]

1/2*(4*EllipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c*d-4*El
lipticE(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^2-2*EllipticF((
(d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*a*b*c*d+2*EllipticF(((d*x+(c*d
)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*b^2*c^2+(c*d)^(1/2)*(a*b)^(1/2)*Ellipti
cPi(((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-(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))*a*d-Ell
ipticPi(((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-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)*(-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^2*(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)/(d*x^2-c)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{\left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x)^(5/2)/((b*x^2 - a)*sqrt(-d*x^2 + c)),x, algorithm="maxima")

[Out]

-integrate((e*x)^(5/2)/((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)^(5/2)/((b*x^2 - a)*sqrt(-d*x^2 + c)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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)**(5/2)/(-b*x**2+a)/(-d*x**2+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{\left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x)^(5/2)/((b*x^2 - a)*sqrt(-d*x^2 + c)),x, algorithm="giac")

[Out]

integrate(-(e*x)^(5/2)/((b*x^2 - a)*sqrt(-d*x^2 + c)), x)