Optimal. Leaf size=51 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+n+1}}{m+n+1}+\frac{b^2 x^{m+2 n+1}}{m+2 n+1} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0517128, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+n+1}}{m+n+1}+\frac{b^2 x^{m+2 n+1}}{m+2 n+1} \]
Antiderivative was successfully verified.
[In] Int[x^m*(a + b*x^n)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.99589, size = 46, normalized size = 0.9 \[ \frac{a^{2} x^{m + 1}}{m + 1} + \frac{2 a b x^{m + n + 1}}{m + n + 1} + \frac{b^{2} x^{m + 2 n + 1}}{m + 2 n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(a+b*x**n)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.048814, size = 46, normalized size = 0.9 \[ x^{m+1} \left (\frac{a^2}{m+1}+\frac{2 a b x^n}{m+n+1}+\frac{b^2 x^{2 n}}{m+2 n+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^m*(a + b*x^n)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.023, size = 63, normalized size = 1.2 \[{\frac{x{a}^{2}{{\rm e}^{m\ln \left ( x \right ) }}}{1+m}}+{\frac{{b}^{2}x{{\rm e}^{m\ln \left ( x \right ) }} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+m+2\,n}}+2\,{\frac{abx{{\rm e}^{m\ln \left ( x \right ) }}{{\rm e}^{n\ln \left ( x \right ) }}}{1+m+n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(a+b*x^n)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^m,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.238254, size = 204, normalized size = 4. \[ \frac{{\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2} +{\left (b^{2} m + b^{2}\right )} n\right )} x x^{m} x^{2 \, n} + 2 \,{\left (a b m^{2} + 2 \, a b m + a b + 2 \,{\left (a b m + a b\right )} n\right )} x x^{m} x^{n} +{\left (a^{2} m^{2} + 2 \, a^{2} n^{2} + 2 \, a^{2} m + a^{2} + 3 \,{\left (a^{2} m + a^{2}\right )} n\right )} x x^{m}}{m^{3} + 2 \,{\left (m + 1\right )} n^{2} + 3 \, m^{2} + 3 \,{\left (m^{2} + 2 \, m + 1\right )} n + 3 \, m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^m,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(x**m*(a+b*x**n)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.222618, size = 385, normalized size = 7.55 \[ \frac{b^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + b^{2} m n x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + 2 \, a b m^{2} x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + 4 \, a b m n x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + a^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, a^{2} m n x e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, a^{2} n^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, b^{2} m x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + b^{2} n x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + 4 \, a b m x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + 4 \, a b n x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + 2 \, a^{2} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, a^{2} n x e^{\left (m{\rm ln}\left (x\right )\right )} + b^{2} x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + 2 \, a b x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + a^{2} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{3} + 3 \, m^{2} n + 2 \, m n^{2} + 3 \, m^{2} + 6 \, m n + 2 \, n^{2} + 3 \, m + 3 \, n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^m,x, algorithm="giac")
[Out]