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

Optimal. Leaf size=88 \[ \frac{x^2 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 \sqrt{a+b x^3} \sqrt{c+d x^3}} \]

[Out]

(x^2*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1/2, 5/3, -((b*x
^3)/a), -((d*x^3)/c)])/(2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])

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Rubi [A]  time = 0.297622, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{x^2 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{2 \sqrt{a+b x^3} \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(x^2*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1/2, 5/3, -((b*x
^3)/a), -((d*x^3)/c)])/(2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])

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Rubi in Sympy [A]  time = 22.6115, size = 75, normalized size = 0.85 \[ \frac{x^{2} \sqrt{a + b x^{3}} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},\frac{1}{2},\frac{5}{3},- \frac{b x^{3}}{a},- \frac{d x^{3}}{c} \right )}}{2 a c \sqrt{1 + \frac{b x^{3}}{a}} \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(a + b*x**3)*sqrt(c + d*x**3)*appellf1(2/3, 1/2, 1/2, 5/3, -b*x**3/a, -
d*x**3/c)/(2*a*c*sqrt(1 + b*x**3/a)*sqrt(1 + d*x**3/c))

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Mathematica [A]  time = 0.453151, size = 172, normalized size = 1.95 \[ -\frac{5 a c x^2 F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{\sqrt{a+b x^3} \sqrt{c+d x^3} \left (3 x^3 \left (a d F_1\left (\frac{5}{3};\frac{1}{2},\frac{3}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+b c F_1\left (\frac{5}{3};\frac{3}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-10 a c F_1\left (\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-5*a*c*x^2*AppellF1[2/3, 1/2, 1/2, 5/3, -((b*x^3)/a), -((d*x^3)/c)])/(Sqrt[a +
b*x^3]*Sqrt[c + d*x^3]*(-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)])))

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{x{\frac{1}{\sqrt{b{x}^{3}+a}}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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