∖int √ ____ c−dx2 _____________________________________√ex∖left(a−bx2 ∖right)2∖,dx

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

Optimal. Leaf size=335 \[ \frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 b \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 b \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a b \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt{e x} \sqrt{c-d x^2}}{2 a e \left (a-b x^2\right )} \]

[Out]

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

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

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Rubi [A] time = 1.23237, antiderivative size = 335, 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 \[ \frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 b \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (3 b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^2 b \sqrt [4]{d} \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 a b \sqrt{e} \sqrt{c-d x^2}}+\frac{\sqrt{e x} \sqrt{c-d x^2}}{2 a e \left (a-b x^2\right )} \]

Antiderivative was successfully veriﬁed.

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

[Out]

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

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

[Out]

Timed out

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Mathematica [C] time = 0.314595, size = 317, normalized size = 0.95 \[ \frac{x \left (-\frac{75 c^2 F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}+\frac{9 c d x^2 F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}-\frac{5 \left (c-d x^2\right )}{a}\right )}{10 \sqrt{e x} \left (b x^2-a\right ) \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*((-5*(c - d*x^2))/a - (75*c^2*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*Appell

F1[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*c*d*x^2*AppellF1[5/4, 1/2, 1, 9/4, (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*Appell

F1[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]))))/(10*Sqrt[e*x]*(-a + b*x^2)*Sqrt[c - d*x^2])

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Maple [B] time = 0.037, size = 2251, normalized size = 6.7 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(-d*x^2+c)^(1/2)*d*(2*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^

(1/2))*2^(1/2)*x^2*a*b*d*((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)*(a*b)^(1/2)*(c*d)^(1/2)-2*Ellipti

cF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^2*b^2*c*((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)*(a*b)^(1/2)*(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)*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))*x^2*a*b^2*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)*(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)^(1/2)*x^2*a*b*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))*x^2*b^3*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)*(a*b)^(1/2)*Elliptic

Pi(((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)^(1/2)*x^2*b^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))*x^2*a*b^2*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)*(a*b)^(1/2)*Elliptic

Pi(((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*d)^(1/2)*x^2*a*b*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)*Elliptic

Pi(((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))*x^2*b^3*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)*(a*b)^(1/2)*Ellipt

icPi(((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*d)^(1/2)*x^2*b^2*c-2*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(

1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*a^2*d*((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)*(a*b)^(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))*2^(1/2)*a*b*

c*((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)*(a*b)^(1/2)*(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)*El

lipticPi(((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+((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)*Elli

pticPi(((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)^(1/2)*a^2*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)*Elliptic

Pi(((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^2-3*((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)*(a*b)^(1/2)*Elliptic

Pi(((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)^(1/2)*a*b*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))*a^2*b*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)*(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*d)^(1/2)*a^2*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))*a*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)*(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*d)^(1/2)*a*b*c+4*x^3*a*b*d^2*(a*b)^(1/2)-4*x^3*b^2*c*d*(a*b)^(1/2)-4*x*

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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