Optimal. Leaf size=65 \[ \frac{a^3 x^2}{2}+\frac{3 a^2 b x^{n+2}}{n+2}+\frac{3 a b^2 x^{2 (n+1)}}{2 (n+1)}+\frac{b^3 x^{3 n+2}}{3 n+2} \]
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Rubi [A] time = 0.0740201, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a^3 x^2}{2}+\frac{3 a^2 b x^{n+2}}{n+2}+\frac{3 a b^2 x^{2 (n+1)}}{2 (n+1)}+\frac{b^3 x^{3 n+2}}{3 n+2} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x^n)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ a^{3} \int x\, dx + \frac{3 a^{2} b x^{n + 2}}{n + 2} + \frac{3 a b^{2} x^{2 n + 2}}{2 \left (n + 1\right )} + \frac{b^{3} x^{3 n + 2}}{3 n + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(a+b*x**n)**3,x)
[Out]
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Mathematica [A] time = 0.0608854, size = 58, normalized size = 0.89 \[ \frac{1}{2} x^2 \left (a^3+\frac{6 a^2 b x^n}{n+2}+\frac{3 a b^2 x^{2 n}}{n+1}+\frac{2 b^3 x^{3 n}}{3 n+2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x^n)^3,x]
[Out]
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Maple [A] time = 0.018, size = 71, normalized size = 1.1 \[{\frac{{b}^{3}{x}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2+3\,n}}+{\frac{{x}^{2}{a}^{3}}{2}}+{\frac{3\,a{b}^{2}{x}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2+2\,n}}+3\,{\frac{{a}^{2}b{x}^{2}{{\rm e}^{n\ln \left ( x \right ) }}}{2+n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(a+b*x^n)^3,x)
[Out]
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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)^3*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.242931, size = 196, normalized size = 3.02 \[ \frac{2 \,{\left (b^{3} n^{2} + 3 \, b^{3} n + 2 \, b^{3}\right )} x^{2} x^{3 \, n} + 3 \,{\left (3 \, a b^{2} n^{2} + 8 \, a b^{2} n + 4 \, a b^{2}\right )} x^{2} x^{2 \, n} + 6 \,{\left (3 \, a^{2} b n^{2} + 5 \, a^{2} b n + 2 \, a^{2} b\right )} x^{2} x^{n} +{\left (3 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 12 \, a^{3} n + 4 \, a^{3}\right )} x^{2}}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^3*x,x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(a+b*x**n)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.217834, size = 270, normalized size = 4.15 \[ \frac{3 \, a^{3} n^{3} x^{2} + 2 \, b^{3} n^{2} x^{2} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 9 \, a b^{2} n^{2} x^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 18 \, a^{2} b n^{2} x^{2} e^{\left (n{\rm ln}\left (x\right )\right )} + 11 \, a^{3} n^{2} x^{2} + 6 \, b^{3} n x^{2} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 24 \, a b^{2} n x^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 30 \, a^{2} b n x^{2} e^{\left (n{\rm ln}\left (x\right )\right )} + 12 \, a^{3} n x^{2} + 4 \, b^{3} x^{2} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 12 \, a b^{2} x^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 12 \, a^{2} b x^{2} e^{\left (n{\rm ln}\left (x\right )\right )} + 4 \, a^{3} x^{2}}{2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 12 \, n + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^3*x,x, algorithm="giac")
[Out]