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

Optimal. Leaf size=328 \[ \frac{2 \sqrt [4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 b^2 \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a b^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a b^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{2 d \sqrt{e x} \sqrt{c-d x^2}}{3 b e} \]

[Out]

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

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Rubi [A]  time = 1.43485, antiderivative size = 328, 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 \[ \frac{2 \sqrt [4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} (5 b c-3 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 b^2 \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a b^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (b c-a d)^2 \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a b^2 \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{2 d \sqrt{e x} \sqrt{c-d x^2}}{3 b e} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [C]  time = 0.810809, size = 425, normalized size = 1.3 \[ \frac{2 x \left (\frac{25 a c^2 (a d-3 b c) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}+\frac{d \left (-10 x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )-9 a c \left (5 a c-2 a d x^2-10 b c x^2+5 b d x^4\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}{2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}\right )}{15 b \sqrt{e x} \left (b x^2-a\right ) \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*x*((25*a*c^2*(-3*b*c + a*d)*AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a])
/(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])) + (d*(-9*a*c*(5*a*c - 10*b*c*x^2 - 2*a*d*x^2 + 5*b*d*x^4)*Appe
llF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a] - 10*x^2*(a - b*x^2)*(c - d*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])))/(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*b*Sqrt[e*x]*(-a + b*x
^2)*Sqrt[c - d*x^2])

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Maple [B]  time = 0.033, size = 1721, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(-d*x^2+c)^(1/2)/b*d*(6*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2
*2^(1/2))*2^(1/2)*a^2*d^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)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)-16*Ellip
ticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a*b*c*d*((-d*x+(
c*d)^(1/2))/(c*d)^(1/2))^(1/2)*(-x*d/(c*d)^(1/2))^(1/2)*(a*b)^(1/2)*(c*d)^(1/2)*
((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2)+10*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2
))^(1/2),1/2*2^(1/2))*2^(1/2)*b^2*c^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)*(c*d)^(1/2)*((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1
/2)+3*((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))*a^2*b*c*d^2-3*((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))*(a*b)^(1/2)*(c*d)^(1/2)*a^2*
d^2-6*((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))*a*b^2*c^2*d+6*((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))*(a*b)^(1/2)*(c*d)^(1/2)*a*b*
c*d+3*((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^3*c^3-3*(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))*b^2*c^2-
3*((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))*a^2*b*c*d^2-3*((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))*(a*b)^(1/2)*(c*d)^(1/2)*a^2*d^2+
6*((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))*a*b^2*c^2*d+6*((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))*(a*b)^(1/2)*(c*d)^(1/2)*a*b*c*d-
3*((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^3*c^3-3*(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))*b^2*c^2+4*x^
3*a*b*d^3*(a*b)^(1/2)-4*x^3*b^2*c*d^2*(a*b)^(1/2)-4*x*a*b*c*d^2*(a*b)^(1/2)+4*x*
b^2*c^2*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. \[ -\int \frac{{\left (-d x^{2} + c\right )}^{\frac{3}{2}}}{{\left (b x^{2} - a\right )} \sqrt{e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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