3.366 \(\int \frac{\sqrt{c+d x^3}}{x^2 \left (a+b x^3\right )} \, dx\)

Optimal. Leaf size=62 \[ -\frac{\sqrt{c+d x^3} F_1\left (-\frac{1}{3};1,-\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a x \sqrt{\frac{d x^3}{c}+1}} \]

[Out]

-((Sqrt[c + d*x^3]*AppellF1[-1/3, 1, -1/2, 2/3, -((b*x^3)/a), -((d*x^3)/c)])/(a*
x*Sqrt[1 + (d*x^3)/c]))

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Rubi [A]  time = 0.184608, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\sqrt{c+d x^3} F_1\left (-\frac{1}{3};1,-\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{a x \sqrt{\frac{d x^3}{c}+1}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[c + d*x^3]*AppellF1[-1/3, 1, -1/2, 2/3, -((b*x^3)/a), -((d*x^3)/c)])/(a*
x*Sqrt[1 + (d*x^3)/c]))

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Rubi in Sympy [A]  time = 24.5923, size = 51, normalized size = 0.82 \[ - \frac{\sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (- \frac{1}{3},- \frac{1}{2},1,\frac{2}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{a x \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c + d*x**3)*appellf1(-1/3, -1/2, 1, 2/3, -d*x**3/c, -b*x**3/a)/(a*x*sqrt(1
+ d*x**3/c))

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Mathematica [B]  time = 1.01524, size = 344, normalized size = 5.55 \[ \frac{\frac{25 c x^3 (2 b c-3 a d) F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{\left (a+b x^3\right ) \left (3 x^3 \left (2 b c F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-10 a c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}-\frac{16 b c d x^6 F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )}{\left (a+b x^3\right ) \left (3 x^3 \left (2 b c F_1\left (\frac{8}{3};\frac{1}{2},2;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )+a d F_1\left (\frac{8}{3};\frac{3}{2},1;\frac{11}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )-16 a c F_1\left (\frac{5}{3};\frac{1}{2},1;\frac{8}{3};-\frac{d x^3}{c},-\frac{b x^3}{a}\right )\right )}-\frac{10 \left (c+d x^3\right )}{a}}{10 x \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-10*(c + d*x^3))/a + (25*c*(2*b*c - 3*a*d)*x^3*AppellF1[2/3, 1/2, 1, 5/3, -((d
*x^3)/c), -((b*x^3)/a)])/((a + b*x^3)*(-10*a*c*AppellF1[2/3, 1/2, 1, 5/3, -((d*x
^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[5/3, 1/2, 2, 8/3, -((d*x^3)/c), -(
(b*x^3)/a)] + a*d*AppellF1[5/3, 3/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)]))) - (1
6*b*c*d*x^6*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)])/((a + b*x^3)
*(-16*a*c*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*
AppellF1[8/3, 1/2, 2, 11/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[8/3, 3/2,
1, 11/3, -((d*x^3)/c), -((b*x^3)/a)]))))/(10*x*Sqrt[c + d*x^3])

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Maple [C]  time = 0.016, size = 1314, normalized size = 21.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/a*(-(d*x^3+c)^(1/2)/x-I*3^(1/2)*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*
I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/
3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(
-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d
*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*EllipticE(
1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d
/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*
3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+1/d*(-c*d^2)^(1/3)*EllipticF(1/3*3^(1/2)*(I*(x
+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^
(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2
)^(1/3)))^(1/2))))-b/a*(-2/3*I/b*3^(1/2)*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/
3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d
^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+
1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(
1/2)/(d*x^3+c)^(1/2)*((-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*Ell
ipticE(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^
(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)
+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+1/d*(-c*d^2)^(1/3)*EllipticF(1/3*3^(1/2
)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^
(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*
(-c*d^2)^(1/3)))^(1/2)))+1/3*I/b/d^2*2^(1/2)*sum(1/_alpha*(-c*d^2)^(1/3)*(1/2*I*
d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d*
(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2)*(-1/2
*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(
d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d+2*_alpha^2*d^2-I*3^(1/2)*(-c*d
^2)^(2/3)-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1
/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1
/2),1/2*b/d*(2*I*_alpha^2*(-c*d^2)^(1/3)*3^(1/2)*d-I*_alpha*(-c*d^2)^(2/3)*3^(1/
2)+I*3^(1/2)*c*d-3*_alpha*(-c*d^2)^(2/3)-3*c*d)/(a*d-b*c),(I*3^(1/2)/d*(-c*d^2)^
(1/3)/(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=Root
Of(_Z^3*b+a)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(d*x^3 + c)/((b*x^3 + a)*x^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(d*x^3 + c)/((b*x^3 + a)*x^2),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c + d*x**3)/(x**2*(a + b*x**3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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