3.510 \(\int \frac{x^4}{\sqrt{a+b x^3} \sqrt{c+d x^3}} \, dx\)

Optimal. Leaf size=88 \[ \frac{x^5 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{5 \sqrt{a+b x^3} \sqrt{c+d x^3}} \]

[Out]

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

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Rubi [A]  time = 0.398808, 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 \[ \frac{x^5 \sqrt{\frac{b x^3}{a}+1} \sqrt{\frac{d x^3}{c}+1} F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{5 \sqrt{a+b x^3} \sqrt{c+d x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]

[Out]

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

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Rubi in Sympy [A]  time = 27.788, size = 75, normalized size = 0.85 \[ \frac{x^{5} \sqrt{a + b x^{3}} \sqrt{c + d x^{3}} \operatorname{appellf_{1}}{\left (\frac{5}{3},\frac{1}{2},\frac{1}{2},\frac{8}{3},- \frac{b x^{3}}{a},- \frac{d x^{3}}{c} \right )}}{5 a c \sqrt{1 + \frac{b x^{3}}{a}} \sqrt{1 + \frac{d x^{3}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)

[Out]

x**5*sqrt(a + b*x**3)*sqrt(c + d*x**3)*appellf1(5/3, 1/2, 1/2, 8/3, -b*x**3/a, -
d*x**3/c)/(5*a*c*sqrt(1 + b*x**3/a)*sqrt(1 + d*x**3/c))

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Mathematica [A]  time = 0.456362, size = 174, normalized size = 1.98 \[ -\frac{16 a c x^5 F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{5 \sqrt{a+b x^3} \sqrt{c+d x^3} \left (3 x^3 \left (a d F_1\left (\frac{8}{3};\frac{1}{2},\frac{3}{2};\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+b c F_1\left (\frac{8}{3};\frac{3}{2},\frac{1}{2};\frac{11}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-16 a c F_1\left (\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]

[Out]

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

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Maple [F]  time = 0.078, size = 0, normalized size = 0. \[ \int{{x}^{4}{\frac{1}{\sqrt{b{x}^{3}+a}}}{\frac{1}{\sqrt{d{x}^{3}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)

[Out]

int(x^4/(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. \[ \int \frac{x^{4}}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)),x, algorithm="maxima")

[Out]

integrate(x^4/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)),x, algorithm="fricas")

[Out]

integral(x^4/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{a + b x^{3}} \sqrt{c + d x^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)

[Out]

Integral(x**4/(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. \[ \int \frac{x^{4}}{\sqrt{b x^{3} + a} \sqrt{d x^{3} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)),x, algorithm="giac")

[Out]

integrate(x^4/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)), x)