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

Optimal. Leaf size=88 \[ -\frac{\sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 x^2 \sqrt{a+b x^3} \sqrt{c+d x^3}} \]

[Out]

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

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Rubi [A]  time = 0.37888, antiderivative size = 88, 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 \[ -\frac{\sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (-\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{1}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 x^2 \sqrt{a+b x^3} \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 27.8334, size = 78, normalized size = 0.89 \[ - \frac{\sqrt{a + b x^{3}} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (- \frac{2}{3},\frac{1}{2},\frac{1}{2},\frac{1}{3},- \frac{b x^{3}}{a},- \frac{d x^{3}}{c} \right )}}{2 a c x^{2} \sqrt{1 + \frac{b x^{3}}{a}} \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Mathematica [B]  time = 0.502535, size = 357, normalized size = 4.06 \[ \frac{\frac{4 x^3 (a d+b c) F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{3 x^3 \left (a d F_1\left (\frac{4}{3};\frac{1}{2},\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+b c F_1\left (\frac{4}{3};\frac{3}{2},\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-8 a c F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}+\frac{7 b d x^6 F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{28 a c F_1\left (\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-6 x^3 \left (a d F_1\left (\frac{7}{3};\frac{1}{2},\frac{3}{2};\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+b c F_1\left (\frac{7}{3};\frac{3}{2},\frac{1}{2};\frac{10}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )}-\frac{\left (a+b x^3\right ) \left (c+d x^3\right )}{a c}}{2 x^2 \sqrt{a+b x^3} \sqrt{c+d x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-(((a + b*x^3)*(c + d*x^3))/(a*c)) + (4*(b*c + a*d)*x^3*AppellF1[1/3, 1/2, 1/2,
4/3, -((b*x^3)/a), -((d*x^3)/c)])/(-8*a*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3
)/a), -((d*x^3)/c)] + 3*x^3*(a*d*AppellF1[4/3, 1/2, 3/2, 7/3, -((b*x^3)/a), -((d
*x^3)/c)] + b*c*AppellF1[4/3, 3/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)])) + (7*
b*d*x^6*AppellF1[4/3, 1/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/(28*a*c*Appell
F1[4/3, 1/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] - 6*x^3*(a*d*AppellF1[7/3, 1/
2, 3/2, 10/3, -((b*x^3)/a), -((d*x^3)/c)] + b*c*AppellF1[7/3, 3/2, 1/2, 10/3, -(
(b*x^3)/a), -((d*x^3)/c)])))/(2*x^2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(1/x^3/(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. \[ \int \frac{1}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*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. \[ \int \frac{1}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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