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

Optimal. Leaf size=83 \[ \frac{x \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \]

[Out]

(x*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3
)/a), -((d*x^3)/c)])/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3])

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Rubi [A]  time = 0.165912, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{x \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3
)/a), -((d*x^3)/c)])/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3])

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Rubi in Sympy [A]  time = 26.2513, size = 71, normalized size = 0.86 \[ \frac{x \sqrt{a + b x^{3}} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{4}{3},- \frac{b x^{3}}{a},- \frac{d x^{3}}{c} \right )}}{a c \sqrt{1 + \frac{b x^{3}}{a}} \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(a + b*x**3)*sqrt(c + d*x**3)*appellf1(1/3, 1/2, 1/2, 4/3, -b*x**3/a, -d*x
**3/c)/(a*c*sqrt(1 + b*x**3/a)*sqrt(1 + d*x**3/c))

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Mathematica [B]  time = 0.427193, size = 170, normalized size = 2.05 \[ -\frac{8 a c x F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{\sqrt{a+b x^3} \sqrt{c+d x^3} \left (3 x^3 \left (a d F_1\left (\frac{4}{3};\frac{1}{2},\frac{3}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+b c F_1\left (\frac{4}{3};\frac{3}{2},\frac{1}{2};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-8 a c F_1\left (\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-8*a*c*x*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(Sqrt[a + b*
x^3]*Sqrt[c + d*x^3]*(-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)])))

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt{b{x}^{3}+a}}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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