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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]