∖int √ ____ c−dx2 _________________________________________

3.902 \(\int \frac{\sqrt{c-d x^2}}{(e x)^{5/2} \left (a-b x^2\right )^2} \, dx\)

Optimal. Leaf size=355 \[ \frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{7 \sqrt [4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 a^2 e^{5/2} \sqrt{c-d x^2}}-\frac{7 \sqrt{c-d x^2}}{6 a^2 e (e x)^{3/2}}+\frac{\sqrt{c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right )} \]

[Out]

(-7*Sqrt[c - d*x^2])/(6*a^2*e*(e*x)^(3/2)) + Sqrt[c - d*x^2]/(2*a*e*(e*x)^(3/2)*

(a - b*x^2)) + (7*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])/(6*a^2*e^(5/2)*Sqrt[c - d*x^2]) + (c^(1/4)*(

7*b*c - 5*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^3*d^(1/4)*e^(5/2)

*Sqrt[c - d*x^2]) + (c^(1/4)*(7*b*c - 5*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqr

t[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])],

-1])/(4*a^3*d^(1/4)*e^(5/2)*Sqrt[c - d*x^2])

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Rubi [A] time = 1.77868, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} (7 b c-5 a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 a^3 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2}}+\frac{7 \sqrt [4]{c} d^{3/4} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{6 a^2 e^{5/2} \sqrt{c-d x^2}}-\frac{7 \sqrt{c-d x^2}}{6 a^2 e (e x)^{3/2}}+\frac{\sqrt{c-d x^2}}{2 a e (e x)^{3/2} \left (a-b x^2\right )} \]

Antiderivative was successfully veriﬁed.

[In] Int[Sqrt[c - d*x^2]/((e*x)^(5/2)*(a - b*x^2)^2),x]

[Out]

(-7*Sqrt[c - d*x^2])/(6*a^2*e*(e*x)^(3/2)) + Sqrt[c - d*x^2]/(2*a*e*(e*x)^(3/2)*

(a - b*x^2)) + (7*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])/(6*a^2*e^(5/2)*Sqrt[c - d*x^2]) + (c^(1/4)*(

7*b*c - 5*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^3*d^(1/4)*e^(5/2)

*Sqrt[c - d*x^2]) + (c^(1/4)*(7*b*c - 5*a*d)*Sqrt[1 - (d*x^2)/c]*EllipticPi[(Sqr

t[b]*Sqrt[c])/(Sqrt[a]*Sqrt[d]), ArcSin[(d^(1/4)*Sqrt[e*x])/(c^(1/4)*Sqrt[e])],

-1])/(4*a^3*d^(1/4)*e^(5/2)*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((-d*x**2+c)**(1/2)/(e*x)**(5/2)/(-b*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [C] time = 0.971014, size = 361, normalized size = 1.02 \[ \frac{x \left (\frac{25 a c x^2 (21 b c-8 a d) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{63 a b c d x^4 F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (b x^2-a\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{5 \left (4 a-7 b x^2\right ) \left (d x^2-c\right )}{a-b x^2}\right )}{30 a^2 (e x)^{5/2} \sqrt{c-d x^2}} \]

Warning: Unable to verify antiderivative.

[In] Integrate[Sqrt[c - d*x^2]/((e*x)^(5/2)*(a - b*x^2)^2),x]

[Out]

(x*((5*(4*a - 7*b*x^2)*(-c + d*x^2))/(a - b*x^2) + (25*a*c*(21*b*c - 8*a*d)*x^2*

AppellF1[1/4, 1/2, 1, 5/4, (d*x^2)/c, (b*x^2)/a])/((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, 3/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])))

+ (63*a*b*c*d*x^4*AppellF1[5/4, 1/2, 1, 9/4, (d*x^2)/c, (b*x^2)/a])/((-a + b*x^

2)*(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])))))/(30*a^2*(e*x)^(5/2)*Sqrt[c - d*x^2])

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Maple [B] time = 0.038, size = 2316, normalized size = 6.5 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((-d*x^2+c)^(1/2)/(e*x)^(5/2)/(-b*x^2+a)^2,x)

[Out]

1/24*(-d*x^2+c)^(1/2)*b*d*(-28*x^4*b^2*c*d*(a*b)^(1/2)-14*EllipticF(((d*x+(c*d)^

(1/2))/(c*d)^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^3*b^2*c*(c*d)^(1/2)*((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)+15*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))*2^(1/2)*x^3*a*b^2*c*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)+21*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))*2^(1/2)*x^3*b^2*c*(c*d)^(1/2)*((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)-15*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))*2^(1/2)*x^3*a*b^2*c*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)+21*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))*2^(1/2)*x^3*b^2*c*(c*d)^(1/2)

*((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)-14*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2))^

(1/2),1/2*2^(1/2))*2^(1/2)*x*a^2*d*(c*d)^(1/2)*((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)

-15*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))*2^(1/2)*x*a^2*b*c*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)+15*Ellip

ticPi(((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))*2^(1/2)*x*a^2*d*(c*d)^(1/2)*((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)+15*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))*2^(1/2)*x*a^2*b*c*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)+15*Ell

ipticPi(((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))*2^(1/2)*x*a^2*d*(c*d)^(1/2)*((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)-16*x^2*a^2*d^2*(a*b)^(1/2)+28*x^2*b^2*c^2*(a*b)^(1/2)+16*a^2*c*d*(a*b)^(1/2

)-16*a*b*c^2*(a*b)^(1/2)-21*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))*2^(1/2)*x^3*b^3*c^2*((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)+28*x^4*a*b*d^2*(a*b)^(1/2)+14*EllipticF(((d*x+(c*d)^(1/2))/(c*d)

^(1/2))^(1/2),1/2*2^(1/2))*2^(1/2)*x^3*a*b*d*(c*d)^(1/2)*((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)-15*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))*2^(1/2)*x^3*a*b*d*(c*d)^(1/2)*((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)-15*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))*2^(1/2)*x^3*a*b*d*(c*d)^(1/

2)*((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)+14*EllipticF(((d*x+(c*d)^(1/2))/(c*d)^(1/2)

)^(1/2),1/2*2^(1/2))*2^(1/2)*x*a*b*c*(c*d)^(1/2)*((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)-21*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))*2^(1/2)*x*a*b*c*(c*d)^(1/2)*((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)-21*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))*2^(1/2)*x*a*b*c*(c*d)^(1/2)*((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)+21*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))*2^(1/2)*x*a*b^2*c^2*((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)-21*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))*2^(1/2)*x*a*b^2*c^2*((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)-12*x^2*a*b*c*d*(a*b)^(1/2)+21*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))*2^(1/2)*x^3*b^3*c

^2*((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))/x/a^2/e^2/(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. \[ \int \frac{\sqrt{-d x^{2} + c}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(sqrt(-d*x^2 + c)/((b*x^2 - a)^2*(e*x)^(5/2)),x, algorithm="maxima")

[Out]

integrate(sqrt(-d*x^2 + c)/((b*x^2 - a)^2*(e*x)^(5/2)), x)

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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(sqrt(-d*x^2 + c)/((b*x^2 - a)^2*(e*x)^(5/2)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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)**(5/2)/(-b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-d x^{2} + c}}{{\left (b x^{2} - a\right )}^{2} \left (e x\right )^{\frac{5}{2}}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(sqrt(-d*x^2 + c)/((b*x^2 - a)^2*(e*x)^(5/2)),x, algorithm="giac")

[Out]

integrate(sqrt(-d*x^2 + c)/((b*x^2 - a)^2*(e*x)^(5/2)), x)

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