3.514 \(\int \frac{1}{x^2 \sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx\)

Optimal. Leaf size=86 \[ -\frac{\sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{x \sqrt{a+b x^3} \sqrt{c+d x^3}} \]

[Out]

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

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Rubi [A]  time = 0.378873, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{\sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{2}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{x \sqrt{a+b x^3} \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.0139, size = 75, normalized size = 0.87 \[ - \frac{\sqrt{a + b x^{3}} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (- \frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{2}{3},- \frac{b x^{3}}{a},- \frac{d x^{3}}{c} \right )}}{a c x \sqrt{1 + \frac{b x^{3}}{a}} \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.566252, size = 357, normalized size = 4.15 \[ \frac{-\frac{25 x^3 (a d+b c) F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{3 x^3 \left (a d F_1\left (\frac{5}{3};\frac{1}{2},\frac{3}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+b c F_1\left (\frac{5}{3};\frac{3}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-10 a c F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}-\frac{64 b d x^6 F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{3 x^3 \left (a d F_1\left (\frac{8}{3};\frac{1}{2},\frac{3}{2};\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+b c F_1\left (\frac{8}{3};\frac{3}{2},\frac{1}{2};\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-16 a c F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}-\frac{10 \left (a+b x^3\right ) \left (c+d x^3\right )}{a c}}{10 x \sqrt{a+b x^3} \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-10*(a + b*x^3)*(c + d*x^3))/(a*c) - (25*(b*c + a*d)*x^3*AppellF1[2/3, 1/2, 1/
2, 5/3, -((b*x^3)/a), -((d*x^3)/c)])/(-10*a*c*AppellF1[2/3, 1/2, 1/2, 5/3, -((b*
x^3)/a), -((d*x^3)/c)] + 3*x^3*(a*d*AppellF1[5/3, 1/2, 3/2, 8/3, -((b*x^3)/a), -
((d*x^3)/c)] + b*c*AppellF1[5/3, 3/2, 1/2, 8/3, -((b*x^3)/a), -((d*x^3)/c)])) -
(64*b*d*x^6*AppellF1[5/3, 1/2, 1/2, 8/3, -((b*x^3)/a), -((d*x^3)/c)])/(-16*a*c*A
ppellF1[5/3, 1/2, 1/2, 8/3, -((b*x^3)/a), -((d*x^3)/c)] + 3*x^3*(a*d*AppellF1[8/
3, 1/2, 3/2, 11/3, -((b*x^3)/a), -((d*x^3)/c)] + b*c*AppellF1[8/3, 3/2, 1/2, 11/
3, -((b*x^3)/a), -((d*x^3)/c)])))/(10*x*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}{\frac{1}{\sqrt{b{x}^{3}+a}}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^2),x, algorithm="fricas")

[Out]

integral(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{a + b x^{3}} \sqrt{c + d x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)*x^2), x)