3.885 \(\int \frac{1}{(e x)^{3/2} \left (a-b x^2\right ) \sqrt{c-d x^2}} \, dx\)

Optimal. Leaf size=379 \[ -\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}} \]

[Out]

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

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Rubi [A]  time = 1.94727, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}} \]

Antiderivative was successfully verified.

[In]  Int[1/((e*x)^(3/2)*(a - b*x^2)*Sqrt[c - d*x^2]),x]

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 0.632381, size = 338, normalized size = 0.89 \[ \frac{2 x \left (\frac{49 x^2 (b c-a d) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}-\frac{33 b d x^4 F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (b x^2-a\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}-\frac{21 \left (c-d x^2\right )}{a c}\right )}{21 (e x)^{3/2} \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((e*x)^(3/2)*(a - b*x^2)*Sqrt[c - d*x^2]),x]

[Out]

(2*x*((-21*(c - d*x^2))/(a*c) + (49*(b*c - a*d)*x^2*AppellF1[3/4, 1/2, 1, 7/4, (
d*x^2)/c, (b*x^2)/a])/((a - b*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]))) - (33*b*d*x^4*AppellF1[7/
4, 1/2, 1, 11/4, (d*x^2)/c, (b*x^2)/a])/((-a + b*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*AppellF1[11/4, 3/2, 1, 15/4, (d*x^2)/c, (b*x^2)/a])))))/
(21*(e*x)^(3/2)*Sqrt[c - d*x^2])

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Maple [B]  time = 0.041, size = 835, normalized size = 2.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*x^2 - a)*sqrt(-d*x^2 + c)*(e*x)^(3/2)),x, algorithm="maxima")

[Out]

-integrate(1/((b*x^2 - a)*sqrt(-d*x^2 + c)*(e*x)^(3/2)), 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(-1/((b*x^2 - a)*sqrt(-d*x^2 + c)*(e*x)^(3/2)),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(1/(e*x)**(3/2)/(-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. \[ \int -\frac{1}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b*x^2 - a)*sqrt(-d*x^2 + c)*(e*x)^(3/2)),x, algorithm="giac")

[Out]

integrate(-1/((b*x^2 - a)*sqrt(-d*x^2 + c)*(e*x)^(3/2)), x)