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

Optimal. Leaf size=283 \[ \frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a b \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a b \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{2 \sqrt [4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt{e} \sqrt{c-d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.990688, antiderivative size = 283, 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 \[ \frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a b \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a b \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{2 \sqrt [4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt{e} \sqrt{c-d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 164.329, size = 255, normalized size = 0.9 \[ \frac{2 \sqrt [4]{c} d^{\frac{3}{4}} \sqrt{1 - \frac{d x^{2}}{c}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | -1\right )}{b \sqrt{e} \sqrt{c - d x^{2}}} - \frac{\sqrt [4]{c} \sqrt{1 - \frac{d x^{2}}{c}} \left (a d - b c\right ) \Pi \left (- \frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}}; \operatorname{asin}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | -1\right )}{a b \sqrt [4]{d} \sqrt{e} \sqrt{c - d x^{2}}} - \frac{\sqrt [4]{c} \sqrt{1 - \frac{d x^{2}}{c}} \left (a d - b c\right ) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}}; \operatorname{asin}{\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}} \right )}\middle | -1\right )}{a b \sqrt [4]{d} \sqrt{e} \sqrt{c - d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c**(1/4)*d**(3/4)*sqrt(1 - d*x**2/c)*elliptic_f(asin(d**(1/4)*sqrt(e*x)/(c**(1
/4)*sqrt(e))), -1)/(b*sqrt(e)*sqrt(c - d*x**2)) - c**(1/4)*sqrt(1 - d*x**2/c)*(a
*d - b*c)*elliptic_pi(-sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)), asin(d**(1/4)*sqrt(e*x
)/(c**(1/4)*sqrt(e))), -1)/(a*b*d**(1/4)*sqrt(e)*sqrt(c - d*x**2)) - c**(1/4)*sq
rt(1 - d*x**2/c)*(a*d - b*c)*elliptic_pi(sqrt(b)*sqrt(c)/(sqrt(a)*sqrt(d)), asin
(d**(1/4)*sqrt(e*x)/(c**(1/4)*sqrt(e))), -1)/(a*b*d**(1/4)*sqrt(e)*sqrt(c - d*x*
*2))

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Mathematica [C]  time = 0.343848, size = 162, normalized size = 0.57 \[ -\frac{10 a c x \sqrt{c-d x^2} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\sqrt{e x} \left (a-b x^2\right ) \left (2 x^2 \left (a d F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-2 b c F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )-5 a c F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-10*a*c*x*Sqrt[c - d*x^2]*AppellF1[1/4, -1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a])/(S
qrt[e*x]*(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, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])))

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Maple [B]  time = 0.046, size = 651, normalized size = 2.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(sqrt(c - d*x**2)/(-a*sqrt(e*x) + b*x**2*sqrt(e*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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