Optimal. Leaf size=65 \[ \frac{a^3 x^4}{4}+\frac{3 a^2 b x^{n+4}}{n+4}+\frac{3 a b^2 x^{2 (n+2)}}{2 (n+2)}+\frac{b^3 x^{3 n+4}}{3 n+4} \]
[Out]
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Rubi [A] time = 0.081155, antiderivative size = 65, 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 \[ \frac{a^3 x^4}{4}+\frac{3 a^2 b x^{n+4}}{n+4}+\frac{3 a b^2 x^{2 (n+2)}}{2 (n+2)}+\frac{b^3 x^{3 n+4}}{3 n+4} \]
Antiderivative was successfully verified.
[In] Int[x^3*(a + b*x^n)^3,x]
[Out]
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Rubi in Sympy [A] time = 11.8536, size = 56, normalized size = 0.86 \[ \frac{a^{3} x^{4}}{4} + \frac{3 a^{2} b x^{n + 4}}{n + 4} + \frac{3 a b^{2} x^{2 n + 4}}{2 \left (n + 2\right )} + \frac{b^{3} x^{3 n + 4}}{3 n + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(a+b*x**n)**3,x)
[Out]
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Mathematica [A] time = 0.0693182, size = 58, normalized size = 0.89 \[ \frac{1}{4} x^4 \left (a^3+\frac{12 a^2 b x^n}{n+4}+\frac{6 a b^2 x^{2 n}}{n+2}+\frac{4 b^3 x^{3 n}}{3 n+4}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^3*(a + b*x^n)^3,x]
[Out]
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Maple [A] time = 0.017, size = 65, normalized size = 1. \[{\frac{{a}^{3}{x}^{4}}{4}}+{\frac{{b}^{3}{x}^{4} \left ({x}^{n} \right ) ^{3}}{4+3\,n}}+{\frac{3\,a{b}^{2}{x}^{4} \left ({x}^{n} \right ) ^{2}}{4+2\,n}}+3\,{\frac{{a}^{2}b{x}^{4}{x}^{n}}{4+n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(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^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.243073, size = 196, normalized size = 3.02 \[ \frac{4 \,{\left (b^{3} n^{2} + 6 \, b^{3} n + 8 \, b^{3}\right )} x^{4} x^{3 \, n} + 6 \,{\left (3 \, a b^{2} n^{2} + 16 \, a b^{2} n + 16 \, a b^{2}\right )} x^{4} x^{2 \, n} + 12 \,{\left (3 \, a^{2} b n^{2} + 10 \, a^{2} b n + 8 \, a^{2} b\right )} x^{4} x^{n} +{\left (3 \, a^{3} n^{3} + 22 \, a^{3} n^{2} + 48 \, a^{3} n + 32 \, a^{3}\right )} x^{4}}{4 \,{\left (3 \, n^{3} + 22 \, n^{2} + 48 \, n + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^3*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 41.5301, size = 507, normalized size = 7.8 \[ \begin{cases} \frac{a^{3} x^{4}}{4} + 3 a^{2} b \log{\left (x \right )} - \frac{3 a b^{2}}{4 x^{4}} - \frac{b^{3}}{8 x^{8}} & \text{for}\: n = -4 \\\frac{a^{3} x^{4}}{4} + \frac{3 a^{2} b x^{2}}{2} + 3 a b^{2} \log{\left (x \right )} - \frac{b^{3}}{2 x^{2}} & \text{for}\: n = -2 \\\frac{a^{3} x^{4}}{4} + \frac{9 a^{2} b x^{\frac{8}{3}}}{8} + \frac{9 a b^{2} x^{\frac{4}{3}}}{4} + b^{3} \log{\left (x \right )} & \text{for}\: n = - \frac{4}{3} \\\frac{3 a^{3} n^{3} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{22 a^{3} n^{2} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{48 a^{3} n x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{32 a^{3} x^{4}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{36 a^{2} b n^{2} x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{120 a^{2} b n x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{96 a^{2} b x^{4} x^{n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{18 a b^{2} n^{2} x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{96 a b^{2} n x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{96 a b^{2} x^{4} x^{2 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{4 b^{3} n^{2} x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{24 b^{3} n x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} + \frac{32 b^{3} x^{4} x^{3 n}}{12 n^{3} + 88 n^{2} + 192 n + 128} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(a+b*x**n)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.218156, size = 270, normalized size = 4.15 \[ \frac{3 \, a^{3} n^{3} x^{4} + 4 \, b^{3} n^{2} x^{4} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 18 \, a b^{2} n^{2} x^{4} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 36 \, a^{2} b n^{2} x^{4} e^{\left (n{\rm ln}\left (x\right )\right )} + 22 \, a^{3} n^{2} x^{4} + 24 \, b^{3} n x^{4} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 96 \, a b^{2} n x^{4} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 120 \, a^{2} b n x^{4} e^{\left (n{\rm ln}\left (x\right )\right )} + 48 \, a^{3} n x^{4} + 32 \, b^{3} x^{4} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 96 \, a b^{2} x^{4} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 96 \, a^{2} b x^{4} e^{\left (n{\rm ln}\left (x\right )\right )} + 32 \, a^{3} x^{4}}{4 \,{\left (3 \, n^{3} + 22 \, n^{2} + 48 \, n + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^3*x^3,x, algorithm="giac")
[Out]