Optimal. Leaf size=57 \[ \frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} F_1\left (\frac{1}{3};-m,2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0716967, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x \left (a+b x^3\right )^m \left (\frac{b x^3}{a}+1\right )^{-m} F_1\left (\frac{1}{3};-m,2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{c^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^3)^m/(c + d*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 19.708, size = 44, normalized size = 0.77 \[ \frac{x \left (1 + \frac{b x^{3}}{a}\right )^{- m} \left (a + b x^{3}\right )^{m} \operatorname{appellf_{1}}{\left (\frac{1}{3},2,- m,\frac{4}{3},- \frac{d x^{3}}{c},- \frac{b x^{3}}{a} \right )}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**3+a)**m/(d*x**3+c)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.297776, size = 162, normalized size = 2.84 \[ -\frac{4 a c x \left (a+b x^3\right )^m F_1\left (\frac{1}{3};-m,2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )}{\left (c+d x^3\right )^2 \left (-3 x^3 \left (b c m F_1\left (\frac{4}{3};1-m,2;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )-2 a d F_1\left (\frac{4}{3};-m,3;\frac{7}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )-4 a c F_1\left (\frac{1}{3};-m,2;\frac{4}{3};-\frac{b x^3}{a},-\frac{d x^3}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x^3)^m/(c + d*x^3)^2,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{ \left ( b{x}^{3}+a \right ) ^{m}}{ \left ( d{x}^{3}+c \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x^3+a)^m/(d*x^3+c)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a\right )}^{m}}{{\left (d x^{3} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^m/(d*x^3 + c)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{3} + a\right )}^{m}}{d^{2} x^{6} + 2 \, c d x^{3} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^m/(d*x^3 + c)^2,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)**m/(d*x**3+c)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a\right )}^{m}}{{\left (d x^{3} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^3 + a)^m/(d*x^3 + c)^2,x, algorithm="giac")
[Out]