Optimal. Leaf size=61 \[ \frac{3 a^2 b x^{m+2}}{m+2}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+3}}{m+3}+\frac{b^3 x^{m+4}}{m+4} \]
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Rubi [A] time = 0.0182403, antiderivative size = 61, 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, Rules used = {43} \[ \frac{3 a^2 b x^{m+2}}{m+2}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+3}}{m+3}+\frac{b^3 x^{m+4}}{m+4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^m (a+b x)^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{1+m}+3 a b^2 x^{2+m}+b^3 x^{3+m}\right ) \, dx\\ &=\frac{a^3 x^{1+m}}{1+m}+\frac{3 a^2 b x^{2+m}}{2+m}+\frac{3 a b^2 x^{3+m}}{3+m}+\frac{b^3 x^{4+m}}{4+m}\\ \end{align*}
Mathematica [A] time = 0.0300431, size = 54, normalized size = 0.89 \[ x^{m+1} \left (\frac{3 a^2 b x}{m+2}+\frac{a^3}{m+1}+\frac{3 a b^2 x^2}{m+3}+\frac{b^3 x^3}{m+4}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0., size = 170, normalized size = 2.8 \begin{align*}{\frac{{x}^{1+m} \left ({b}^{3}{m}^{3}{x}^{3}+3\,a{b}^{2}{m}^{3}{x}^{2}+6\,{b}^{3}{m}^{2}{x}^{3}+3\,{a}^{2}b{m}^{3}x+21\,a{b}^{2}{m}^{2}{x}^{2}+11\,{b}^{3}m{x}^{3}+{a}^{3}{m}^{3}+24\,{a}^{2}b{m}^{2}x+42\,a{b}^{2}m{x}^{2}+6\,{b}^{3}{x}^{3}+9\,{a}^{3}{m}^{2}+57\,{a}^{2}bmx+24\,a{b}^{2}{x}^{2}+26\,{a}^{3}m+36\,{a}^{2}bx+24\,{a}^{3} \right ) }{ \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.58413, size = 336, normalized size = 5.51 \begin{align*} \frac{{\left ({\left (b^{3} m^{3} + 6 \, b^{3} m^{2} + 11 \, b^{3} m + 6 \, b^{3}\right )} x^{4} + 3 \,{\left (a b^{2} m^{3} + 7 \, a b^{2} m^{2} + 14 \, a b^{2} m + 8 \, a b^{2}\right )} x^{3} + 3 \,{\left (a^{2} b m^{3} + 8 \, a^{2} b m^{2} + 19 \, a^{2} b m + 12 \, a^{2} b\right )} x^{2} +{\left (a^{3} m^{3} + 9 \, a^{3} m^{2} + 26 \, a^{3} m + 24 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.769065, size = 663, normalized size = 10.87 \begin{align*} \begin{cases} - \frac{a^{3}}{3 x^{3}} - \frac{3 a^{2} b}{2 x^{2}} - \frac{3 a b^{2}}{x} + b^{3} \log{\left (x \right )} & \text{for}\: m = -4 \\- \frac{a^{3}}{2 x^{2}} - \frac{3 a^{2} b}{x} + 3 a b^{2} \log{\left (x \right )} + b^{3} x & \text{for}\: m = -3 \\- \frac{a^{3}}{x} + 3 a^{2} b \log{\left (x \right )} + 3 a b^{2} x + \frac{b^{3} x^{2}}{2} & \text{for}\: m = -2 \\a^{3} \log{\left (x \right )} + 3 a^{2} b x + \frac{3 a b^{2} x^{2}}{2} + \frac{b^{3} x^{3}}{3} & \text{for}\: m = -1 \\\frac{a^{3} m^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{9 a^{3} m^{2} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{26 a^{3} m x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a^{3} x x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{3 a^{2} b m^{3} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a^{2} b m^{2} x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{57 a^{2} b m x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{36 a^{2} b x^{2} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{3 a b^{2} m^{3} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{21 a b^{2} m^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{42 a b^{2} m x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{24 a b^{2} x^{3} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{b^{3} m^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{6 b^{3} m^{2} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{11 b^{3} m x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac{6 b^{3} x^{4} x^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07291, size = 302, normalized size = 4.95 \begin{align*} \frac{b^{3} m^{3} x^{4} x^{m} + 3 \, a b^{2} m^{3} x^{3} x^{m} + 6 \, b^{3} m^{2} x^{4} x^{m} + 3 \, a^{2} b m^{3} x^{2} x^{m} + 21 \, a b^{2} m^{2} x^{3} x^{m} + 11 \, b^{3} m x^{4} x^{m} + a^{3} m^{3} x x^{m} + 24 \, a^{2} b m^{2} x^{2} x^{m} + 42 \, a b^{2} m x^{3} x^{m} + 6 \, b^{3} x^{4} x^{m} + 9 \, a^{3} m^{2} x x^{m} + 57 \, a^{2} b m x^{2} x^{m} + 24 \, a b^{2} x^{3} x^{m} + 26 \, a^{3} m x x^{m} + 36 \, a^{2} b x^{2} x^{m} + 24 \, a^{3} x x^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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