Optimal. Leaf size=44 \[ \frac{a^2 x^4}{4}+\frac{2 a b x^{n+4}}{n+4}+\frac{b^2 x^{2 (n+2)}}{2 (n+2)} \]
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Rubi [A] time = 0.0566562, antiderivative size = 44, 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 \[ \frac{a^2 x^4}{4}+\frac{2 a b x^{n+4}}{n+4}+\frac{b^2 x^{2 (n+2)}}{2 (n+2)} \]
Antiderivative was successfully verified.
[In] Int[x^3*(a + b*x^n)^2,x]
[Out]
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Rubi in Sympy [A] time = 8.14883, size = 36, normalized size = 0.82 \[ \frac{a^{2} x^{4}}{4} + \frac{2 a b x^{n + 4}}{n + 4} + \frac{b^{2} x^{2 n + 4}}{2 \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(a+b*x**n)**2,x)
[Out]
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Mathematica [A] time = 0.0581358, size = 38, normalized size = 0.86 \[ \frac{1}{4} x^4 \left (a^2+\frac{8 a b x^n}{n+4}+\frac{2 b^2 x^{2 n}}{n+2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(a + b*x^n)^2,x]
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Maple [A] time = 0.019, size = 47, normalized size = 1.1 \[{\frac{{x}^{4}{a}^{2}}{4}}+{\frac{{b}^{2}{x}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{4+2\,n}}+2\,{\frac{ab{x}^{4}{{\rm e}^{n\ln \left ( x \right ) }}}{4+n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(a+b*x^n)^2,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)^2*x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238972, size = 100, normalized size = 2.27 \[ \frac{2 \,{\left (b^{2} n + 4 \, b^{2}\right )} x^{4} x^{2 \, n} + 8 \,{\left (a b n + 2 \, a b\right )} x^{4} x^{n} +{\left (a^{2} n^{2} + 6 \, a^{2} n + 8 \, a^{2}\right )} x^{4}}{4 \,{\left (n^{2} + 6 \, n + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.74612, size = 202, normalized size = 4.59 \[ \begin{cases} \frac{a^{2} x^{4}}{4} + 2 a b \log{\left (x \right )} - \frac{b^{2}}{4 x^{4}} & \text{for}\: n = -4 \\\frac{a^{2} x^{4}}{4} + a b x^{2} + b^{2} \log{\left (x \right )} & \text{for}\: n = -2 \\\frac{a^{2} n^{2} x^{4}}{4 n^{2} + 24 n + 32} + \frac{6 a^{2} n x^{4}}{4 n^{2} + 24 n + 32} + \frac{8 a^{2} x^{4}}{4 n^{2} + 24 n + 32} + \frac{8 a b n x^{4} x^{n}}{4 n^{2} + 24 n + 32} + \frac{16 a b x^{4} x^{n}}{4 n^{2} + 24 n + 32} + \frac{2 b^{2} n x^{4} x^{2 n}}{4 n^{2} + 24 n + 32} + \frac{8 b^{2} x^{4} x^{2 n}}{4 n^{2} + 24 n + 32} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(a+b*x**n)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.216387, size = 127, normalized size = 2.89 \[ \frac{a^{2} n^{2} x^{4} + 2 \, b^{2} n x^{4} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 8 \, a b n x^{4} e^{\left (n{\rm ln}\left (x\right )\right )} + 6 \, a^{2} n x^{4} + 8 \, b^{2} x^{4} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 16 \, a b x^{4} e^{\left (n{\rm ln}\left (x\right )\right )} + 8 \, a^{2} x^{4}}{4 \,{\left (n^{2} + 6 \, n + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^2*x^3,x, algorithm="giac")
[Out]