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

Optimal. Leaf size=420 \[ -\frac{d (e x)^{3/2}}{c e \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{b} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 2.06945, antiderivative size = 420, 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 \[ -\frac{d (e x)^{3/2}}{c e \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{b} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 0.798072, size = 356, normalized size = 0.85 \[ \frac{x \sqrt{e x} \left (\frac{33 a b d x^2 F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) (a d-b c) \left (2 x^2 \left (2 b c F_1\left (\frac{11}{4};\frac{1}{2},2;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{11}{4};\frac{3}{2},1;\frac{15}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+11 a c F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}-\frac{49 a (a d+2 b c) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) (a d-b c) \left (2 x^2 \left (2 b c F_1\left (\frac{7}{4};\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{7}{4};\frac{3}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+7 a c F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}-\frac{21 d}{b c^2-a c d}\right )}{21 \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [B]  time = 0.038, size = 830, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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