3.886 \(\int \frac{1}{(e x)^{5/2} \left (a-b x^2\right ) \sqrt{c-d x^2}} \, dx\)

Optimal. Leaf size=297 \[ \frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{2 d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 a c^{3/4} e^{5/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{3 a c e (e x)^{3/2}} \]

[Out]

(-2*Sqrt[c - d*x^2])/(3*a*c*e*(e*x)^(3/2)) + (2*d^(3/4)*Sqrt[1 - (d*x^2)/c]*Elli
pticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(3*a*c^(3/4)*e^(5/2)*S
qrt[c - d*x^2]) + (b*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^2*d^(
1/4)*e^(5/2)*Sqrt[c - d*x^2]) + (b*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^2*d^(1/4)*e^(5/2)*Sqrt[c - d*x^2])

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Rubi [A]  time = 1.30622, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{b \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{2 d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 a c^{3/4} e^{5/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{3 a c e (e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c - d*x^2])/(3*a*c*e*(e*x)^(3/2)) + (2*d^(3/4)*Sqrt[1 - (d*x^2)/c]*Elli
pticF[ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])], -1])/(3*a*c^(3/4)*e^(5/2)*S
qrt[c - d*x^2]) + (b*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^2*d^(
1/4)*e^(5/2)*Sqrt[c - d*x^2]) + (b*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^2*d^(1/4)*e^(5/2)*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(1/(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 = 0.561811, size = 338, normalized size = 1.14 \[ \frac{2 x \left (\frac{25 x^2 (a d+3 b c) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{9 b d x^4 F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (b x^2-a\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}-\frac{5 \left (c-d x^2\right )}{a c}\right )}{15 (e x)^{5/2} \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*((-5*(c - d*x^2))/(a*c) + (25*(3*b*c + a*d)*x^2*AppellF1[1/4, 1/2, 1, 5/4,
(d*x^2)/c, (b*x^2)/a])/((a - b*x^2)*(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c,
(b*x^2)/a] + 2*x^2*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, (d*x^2)/c, (b*x^2)/a] + a*
d*AppellF1[5/4, 3/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a]))) + (9*b*d*x^4*AppellF1[5/4,
1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/((-a + b*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])))))/(15*(e*
x)^(5/2)*Sqrt[c - d*x^2])

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Maple [B]  time = 0.04, size = 740, normalized size = 2.5 \[{\frac{bd}{6\,cxa \left ( d{x}^{2}-c \right ){e}^{2}} \left ( 2\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},1/2\,\sqrt{2} \right ) \sqrt{2}xad\sqrt{ab}\sqrt{cd}\sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{{\frac{-dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{-{\frac{dx}{\sqrt{cd}}}}-2\,{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},1/2\,\sqrt{2} \right ) \sqrt{2}xbc\sqrt{ab}\sqrt{cd}\sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{{\frac{-dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{-{\frac{dx}{\sqrt{cd}}}}-3\,{\it EllipticPi} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},{\frac{\sqrt{cd}b}{\sqrt{ab}d+\sqrt{cd}b}},1/2\,\sqrt{2} \right ) \sqrt{2}x{b}^{2}{c}^{2}\sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{{\frac{-dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{-{\frac{dx}{\sqrt{cd}}}}+3\,{\it EllipticPi} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},{\frac{\sqrt{cd}b}{\sqrt{ab}d+\sqrt{cd}b}},1/2\,\sqrt{2} \right ) \sqrt{2}xbc\sqrt{ab}\sqrt{cd}\sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{{\frac{-dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{-{\frac{dx}{\sqrt{cd}}}}+3\,{\it EllipticPi} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},{\frac{\sqrt{cd}b}{\sqrt{cd}b-\sqrt{ab}d}},1/2\,\sqrt{2} \right ) \sqrt{2}x{b}^{2}{c}^{2}\sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{{\frac{-dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{-{\frac{dx}{\sqrt{cd}}}}+3\,{\it EllipticPi} \left ( \sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}},{\frac{\sqrt{cd}b}{\sqrt{cd}b-\sqrt{ab}d}},1/2\,\sqrt{2} \right ) \sqrt{2}xbc\sqrt{ab}\sqrt{cd}\sqrt{{\frac{dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{{\frac{-dx+\sqrt{cd}}{\sqrt{cd}}}}\sqrt{-{\frac{dx}{\sqrt{cd}}}}+4\,{x}^{2}a{d}^{2}\sqrt{ab}-4\,{x}^{2}bcd\sqrt{ab}-4\,acd\sqrt{ab}+4\,b{c}^{2}\sqrt{ab} \right ) \sqrt{-d{x}^{2}+c} \left ( \sqrt{cd}b-\sqrt{ab}d \right ) ^{-1} \left ( \sqrt{ab}d+\sqrt{cd}b \right ) ^{-1}{\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ex}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b*d*(2*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*
x*a*d*(a*b)^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)-2*EllipticF(((d*x+(c*d)^(1/2
))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x*b*c*(a*b)^(1/2)*(c*d)^(1/2)*((d*x+(
c*d)^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d
)^(1/2))^(1/2)-3*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))*2^(1/2)*x*b^2*c^2*((d*x+(c*d)^(1/2))/
(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/
2)+3*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))*2^(1/2)*x*b*c*(a*b)^(1/2)*(c*d)^(1/2)*((d*x+(c*d)
^(1/2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1
/2))^(1/2)+3*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))*2^(1/2)*x*b^2*c^2*((d*x+(c*d)^(1/2))/(c*d
)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)+3
*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))*2^(1/2)*x*b*c*(a*b)^(1/2)*(c*d)^(1/2)*((d*x+(c*d)^(1/
2))/(c*d)^(1/2))^(1/2)*((-d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))
^(1/2)+4*x^2*a*d^2*(a*b)^(1/2)-4*x^2*b*c*d*(a*b)^(1/2)-4*a*c*d*(a*b)^(1/2)+4*b*c
^2*(a*b)^(1/2))*(-d*x^2+c)^(1/2)/x/c/a/((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^2/(e*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*x^2 - a)*sqrt(-d*x^2 + c)*(e*x)^(5/2)),x, algorithm="maxima")

[Out]

-integrate(1/((b*x^2 - a)*sqrt(-d*x^2 + c)*(e*x)^(5/2)), 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(-1/((b*x^2 - a)*sqrt(-d*x^2 + c)*(e*x)^(5/2)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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(1/(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. \[ \int -\frac{1}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*x^2 - a)*sqrt(-d*x^2 + c)*(e*x)^(5/2)),x, algorithm="giac")

[Out]

integrate(-1/((b*x^2 - a)*sqrt(-d*x^2 + c)*(e*x)^(5/2)), x)