Optimal. Leaf size=66 \[ \frac{a^3 x^3}{3}+\frac{3 a^2 b x^{n+3}}{n+3}+\frac{3 a b^2 x^{2 n+3}}{2 n+3}+\frac{b^3 x^{3 (n+1)}}{3 (n+1)} \]
[Out]
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Rubi [A] time = 0.0846406, antiderivative size = 66, 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 \[ \frac{a^3 x^3}{3}+\frac{3 a^2 b x^{n+3}}{n+3}+\frac{3 a b^2 x^{2 n+3}}{2 n+3}+\frac{b^3 x^{3 (n+1)}}{3 (n+1)} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*x^n)^3,x]
[Out]
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Rubi in Sympy [A] time = 12.2373, size = 56, normalized size = 0.85 \[ \frac{a^{3} x^{3}}{3} + \frac{3 a^{2} b x^{n + 3}}{n + 3} + \frac{3 a b^{2} x^{2 n + 3}}{2 n + 3} + \frac{b^{3} x^{3 n + 3}}{3 \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(a+b*x**n)**3,x)
[Out]
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Mathematica [A] time = 0.0599207, size = 57, normalized size = 0.86 \[ \frac{1}{3} x^3 \left (a^3+\frac{9 a^2 b x^n}{n+3}+\frac{9 a b^2 x^{2 n}}{2 n+3}+\frac{b^3 x^{3 n}}{n+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a + b*x^n)^3,x]
[Out]
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Maple [A] time = 0.019, size = 72, normalized size = 1.1 \[{\frac{{a}^{3}{x}^{3}}{3}}+{\frac{{b}^{3}{x}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3+3\,n}}+3\,{\frac{a{b}^{2}{x}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{3+2\,n}}+3\,{\frac{{a}^{2}b{x}^{3}{{\rm e}^{n\ln \left ( x \right ) }}}{3+n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(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^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238913, size = 194, normalized size = 2.94 \[ \frac{{\left (2 \, b^{3} n^{2} + 9 \, b^{3} n + 9 \, b^{3}\right )} x^{3} x^{3 \, n} + 9 \,{\left (a b^{2} n^{2} + 4 \, a b^{2} n + 3 \, a b^{2}\right )} x^{3} x^{2 \, n} + 9 \,{\left (2 \, a^{2} b n^{2} + 5 \, a^{2} b n + 3 \, a^{2} b\right )} x^{3} x^{n} +{\left (2 \, a^{3} n^{3} + 11 \, a^{3} n^{2} + 18 \, a^{3} n + 9 \, a^{3}\right )} x^{3}}{3 \,{\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^3*x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 11.7765, size = 500, normalized size = 7.58 \[ \begin{cases} \frac{a^{3} x^{3}}{3} + 3 a^{2} b \log{\left (x \right )} - \frac{a b^{2}}{x^{3}} - \frac{b^{3}}{6 x^{6}} & \text{for}\: n = -3 \\\frac{a^{3} x^{3}}{3} + 2 a^{2} b x^{\frac{3}{2}} + 3 a b^{2} \log{\left (x \right )} - \frac{2 b^{3}}{3 x^{\frac{3}{2}}} & \text{for}\: n = - \frac{3}{2} \\\frac{a^{3} x^{3}}{3} + \frac{3 a^{2} b x^{2}}{2} + 3 a b^{2} x + b^{3} \log{\left (x \right )} & \text{for}\: n = -1 \\\frac{2 a^{3} n^{3} x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{11 a^{3} n^{2} x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{18 a^{3} n x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{9 a^{3} x^{3}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{18 a^{2} b n^{2} x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{45 a^{2} b n x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{27 a^{2} b x^{3} x^{n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{9 a b^{2} n^{2} x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{36 a b^{2} n x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{27 a b^{2} x^{3} x^{2 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{2 b^{3} n^{2} x^{3} x^{3 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{9 b^{3} n x^{3} x^{3 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} + \frac{9 b^{3} x^{3} x^{3 n}}{6 n^{3} + 33 n^{2} + 54 n + 27} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(a+b*x**n)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.217295, size = 270, normalized size = 4.09 \[ \frac{2 \, a^{3} n^{3} x^{3} + 2 \, b^{3} n^{2} x^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 9 \, a b^{2} n^{2} x^{3} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 18 \, a^{2} b n^{2} x^{3} e^{\left (n{\rm ln}\left (x\right )\right )} + 11 \, a^{3} n^{2} x^{3} + 9 \, b^{3} n x^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 36 \, a b^{2} n x^{3} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 45 \, a^{2} b n x^{3} e^{\left (n{\rm ln}\left (x\right )\right )} + 18 \, a^{3} n x^{3} + 9 \, b^{3} x^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 27 \, a b^{2} x^{3} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 27 \, a^{2} b x^{3} e^{\left (n{\rm ln}\left (x\right )\right )} + 9 \, a^{3} x^{3}}{3 \,{\left (2 \, n^{3} + 11 \, n^{2} + 18 \, n + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^3*x^2,x, algorithm="giac")
[Out]