3.695 \(\int \frac{x^3}{\left (a+b x^6\right ) \sqrt{c+d x^6}} \, dx\)

Optimal. Leaf size=64 \[ \frac{x^4 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{2}{3};1,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a \sqrt{c+d x^6}} \]

[Out]

(x^4*Sqrt[1 + (d*x^6)/c]*AppellF1[2/3, 1, 1/2, 5/3, -((b*x^6)/a), -((d*x^6)/c)])
/(4*a*Sqrt[c + d*x^6])

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Rubi [A]  time = 0.235214, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x^4 \sqrt{\frac{d x^6}{c}+1} F_1\left (\frac{2}{3};1,\frac{1}{2};\frac{5}{3};-\frac{b x^6}{a},-\frac{d x^6}{c}\right )}{4 a \sqrt{c+d x^6}} \]

Antiderivative was successfully verified.

[In]  Int[x^3/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

(x^4*Sqrt[1 + (d*x^6)/c]*AppellF1[2/3, 1, 1/2, 5/3, -((b*x^6)/a), -((d*x^6)/c)])
/(4*a*Sqrt[c + d*x^6])

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Rubi in Sympy [A]  time = 27.3414, size = 51, normalized size = 0.8 \[ \frac{x^{4} \sqrt{c + d x^{6}} \operatorname{appellf_{1}}{\left (\frac{2}{3},\frac{1}{2},1,\frac{5}{3},- \frac{d x^{6}}{c},- \frac{b x^{6}}{a} \right )}}{4 a c \sqrt{1 + \frac{d x^{6}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

x**4*sqrt(c + d*x**6)*appellf1(2/3, 1/2, 1, 5/3, -d*x**6/c, -b*x**6/a)/(4*a*c*sq
rt(1 + d*x**6/c))

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Mathematica [B]  time = 0.283297, size = 165, normalized size = 2.58 \[ -\frac{5 a c x^4 F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )}{2 \left (a+b x^6\right ) \sqrt{c+d x^6} \left (3 x^6 \left (2 b c F_1\left (\frac{5}{3};\frac{1}{2},2;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )+a d F_1\left (\frac{5}{3};\frac{3}{2},1;\frac{8}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )-10 a c F_1\left (\frac{2}{3};\frac{1}{2},1;\frac{5}{3};-\frac{d x^6}{c},-\frac{b x^6}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^3/((a + b*x^6)*Sqrt[c + d*x^6]),x]

[Out]

(-5*a*c*x^4*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^6)/c), -((b*x^6)/a)])/(2*(a + b*x^
6)*Sqrt[c + d*x^6]*(-10*a*c*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^6)/c), -((b*x^6)/a
)] + 3*x^6*(2*b*c*AppellF1[5/3, 1/2, 2, 8/3, -((d*x^6)/c), -((b*x^6)/a)] + a*d*A
ppellF1[5/3, 3/2, 1, 8/3, -((d*x^6)/c), -((b*x^6)/a)])))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{b{x}^{6}+a}{\frac{1}{\sqrt{d{x}^{6}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x^6+a)/(d*x^6+c)^(1/2),x)

[Out]

int(x^3/(b*x^6+a)/(d*x^6+c)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (b x^{6} + a\right )} \sqrt{d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^6 + a)*sqrt(d*x^6 + c)),x, algorithm="maxima")

[Out]

integrate(x^3/((b*x^6 + a)*sqrt(d*x^6 + c)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^6 + a)*sqrt(d*x^6 + c)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (a + b x^{6}\right ) \sqrt{c + d x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x**6+a)/(d*x**6+c)**(1/2),x)

[Out]

Integral(x**3/((a + b*x**6)*sqrt(c + d*x**6)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (b x^{6} + a\right )} \sqrt{d x^{6} + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/((b*x^6 + a)*sqrt(d*x^6 + c)),x, algorithm="giac")

[Out]

integrate(x^3/((b*x^6 + a)*sqrt(d*x^6 + c)), x)