3.46 \(\int \left (a+b x^3\right )^{-1-\frac{b c}{3 b c-3 a d}} \left (c+d x^3\right )^{-1+\frac{a d}{3 b c-3 a d}} \, dx\)

Optimal. Leaf size=53 \[ \frac{x \left (a+b x^3\right )^{-\frac{b c}{3 b c-3 a d}} \left (c+d x^3\right )^{\frac{a d}{3 b c-3 a d}}}{a c} \]

[Out]

(x*(c + d*x^3)^((a*d)/(3*b*c - 3*a*d)))/(a*c*(a + b*x^3)^((b*c)/(3*b*c - 3*a*d))
)

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Rubi [A]  time = 0.0608956, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.02 \[ \frac{x \left (a+b x^3\right )^{-\frac{b c}{3 b c-3 a d}} \left (c+d x^3\right )^{\frac{a d}{3 b c-3 a d}}}{a c} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c + d*x^3)^((a*d)/(3*b*c - 3*a*d)))/(a*c*(a + b*x^3)^((b*c)/(3*b*c - 3*a*d))
)

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Rubi in Sympy [A]  time = 10.5525, size = 44, normalized size = 0.83 \[ \frac{x \left (a + b x^{3}\right )^{\frac{b c}{3 \left (a d - b c\right )}} \left (c + d x^{3}\right )^{- \frac{a d}{3 \left (a d - b c\right )}}}{a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a + b*x**3)**(b*c/(3*(a*d - b*c)))*(c + d*x**3)**(-a*d/(3*(a*d - b*c)))/(a*c)

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Mathematica [C]  time = 2.43632, size = 594, normalized size = 11.21 \[ 4 a c x \left (a+b x^3\right )^{\frac{b c}{3 a d-3 b c}} \left (c+d x^3\right )^{\frac{a d}{3 b c-3 a d}} \left (\frac{b F_1\left (\frac{1}{3};\frac{b c}{3 b c-3 a d}+1,\frac{a d}{3 a d-3 b c};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{\left (a+b x^3\right ) \left (x^3 \left (a^2 d^2 F_1\left (\frac{4}{3};\frac{b c}{3 b c-3 a d}+1,\frac{a d}{3 a d-3 b c}+1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+b c (3 a d-4 b c) F_1\left (\frac{4}{3};\frac{b c}{3 b c-3 a d}+2,\frac{a d}{3 a d-3 b c};\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )+4 a c (b c-a d) F_1\left (\frac{1}{3};\frac{b c}{3 b c-3 a d}+1,\frac{a d}{3 a d-3 b c};\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )}+\frac{d F_1\left (\frac{1}{3};\frac{b c}{3 b c-3 a d},\frac{a d}{3 a d-3 b c}+1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{\left (c+d x^3\right ) \left (x^3 \left (b^2 c^2 F_1\left (\frac{4}{3};\frac{b c}{3 b c-3 a d}+1,\frac{a d}{3 a d-3 b c}+1;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )+a d (3 b c-4 a d) F_1\left (\frac{4}{3};\frac{b c}{3 b c-3 a d},\frac{a d}{3 a d-3 b c}+2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )+4 a c (a d-b c) F_1\left (\frac{1}{3};\frac{b c}{3 b c-3 a d},\frac{a d}{3 a d-3 b c}+1;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )}\right ) \]

Warning: Unable to verify antiderivative.

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

[Out]

4*a*c*x*(a + b*x^3)^((b*c)/(-3*b*c + 3*a*d))*(c + d*x^3)^((a*d)/(3*b*c - 3*a*d))
*((d*AppellF1[1/3, (b*c)/(3*b*c - 3*a*d), 1 + (a*d)/(-3*b*c + 3*a*d), 4/3, -((b*
x^3)/a), -((d*x^3)/c)])/((c + d*x^3)*(4*a*c*(-(b*c) + a*d)*AppellF1[1/3, (b*c)/(
3*b*c - 3*a*d), 1 + (a*d)/(-3*b*c + 3*a*d), 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x
^3*(a*d*(3*b*c - 4*a*d)*AppellF1[4/3, (b*c)/(3*b*c - 3*a*d), 2 + (a*d)/(-3*b*c +
3*a*d), 7/3, -((b*x^3)/a), -((d*x^3)/c)] + b^2*c^2*AppellF1[4/3, 1 + (b*c)/(3*b
*c - 3*a*d), 1 + (a*d)/(-3*b*c + 3*a*d), 7/3, -((b*x^3)/a), -((d*x^3)/c)]))) + (
b*AppellF1[1/3, 1 + (b*c)/(3*b*c - 3*a*d), (a*d)/(-3*b*c + 3*a*d), 4/3, -((b*x^3
)/a), -((d*x^3)/c)])/((a + b*x^3)*(4*a*c*(b*c - a*d)*AppellF1[1/3, 1 + (b*c)/(3*
b*c - 3*a*d), (a*d)/(-3*b*c + 3*a*d), 4/3, -((b*x^3)/a), -((d*x^3)/c)] + x^3*(a^
2*d^2*AppellF1[4/3, 1 + (b*c)/(3*b*c - 3*a*d), 1 + (a*d)/(-3*b*c + 3*a*d), 7/3,
-((b*x^3)/a), -((d*x^3)/c)] + b*c*(-4*b*c + 3*a*d)*AppellF1[4/3, 2 + (b*c)/(3*b*
c - 3*a*d), (a*d)/(-3*b*c + 3*a*d), 7/3, -((b*x^3)/a), -((d*x^3)/c)]))))

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Maple [A]  time = 0.006, size = 71, normalized size = 1.3 \[{\frac{x}{ac} \left ( b{x}^{3}+a \right ) ^{1-{\frac{3\,ad-4\,bc}{3\,ad-3\,bc}}} \left ( d{x}^{3}+c \right ) ^{1-{\frac{4\,ad-3\,bc}{3\,ad-3\,bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x^3+a)^(1-1/3*(3*a*d-4*b*c)/(a*d-b*c))*(d*x^3+c)^(1-1/3*(4*a*d-3*b*c)/(a*d-b*
c))/a/c*x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{-\frac{b c}{3 \,{\left (b c - a d\right )}} - 1}{\left (d x^{3} + c\right )}^{\frac{a d}{3 \,{\left (b c - a d\right )}} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.252313, size = 123, normalized size = 2.32 \[ \frac{b d x^{7} +{\left (b c + a d\right )} x^{4} + a c x}{{\left (b x^{3} + a\right )}^{\frac{4 \, b c - 3 \, a d}{3 \,{\left (b c - a d\right )}}}{\left (d x^{3} + c\right )}^{\frac{3 \, b c - 4 \, a d}{3 \,{\left (b c - a d\right )}}} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*d*x^7 + (b*c + a*d)*x^4 + a*c*x)/((b*x^3 + a)^(1/3*(4*b*c - 3*a*d)/(b*c - a*d
))*(d*x^3 + c)^(1/3*(3*b*c - 4*a*d)/(b*c - a*d))*a*c)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{3} + a\right )}^{-\frac{b c}{3 \,{\left (b c - a d\right )}} - 1}{\left (d x^{3} + c\right )}^{\frac{a d}{3 \,{\left (b c - a d\right )}} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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