Optimal. Leaf size=61 \[ \frac{3 a^2 b x^{m+5}}{m+5}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+9}}{m+9}+\frac{b^3 x^{m+13}}{m+13} \]
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Rubi [A] time = 0.0202695, antiderivative size = 61, 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, Rules used = {270} \[ \frac{3 a^2 b x^{m+5}}{m+5}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+9}}{m+9}+\frac{b^3 x^{m+13}}{m+13} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin{align*} \int x^m \left (a+b x^4\right )^3 \, dx &=\int \left (a^3 x^m+3 a^2 b x^{4+m}+3 a b^2 x^{8+m}+b^3 x^{12+m}\right ) \, dx\\ &=\frac{a^3 x^{1+m}}{1+m}+\frac{3 a^2 b x^{5+m}}{5+m}+\frac{3 a b^2 x^{9+m}}{9+m}+\frac{b^3 x^{13+m}}{13+m}\\ \end{align*}
Mathematica [A] time = 0.0293832, size = 56, normalized size = 0.92 \[ x^{m+1} \left (\frac{3 a^2 b x^4}{m+5}+\frac{a^3}{m+1}+\frac{3 a b^2 x^8}{m+9}+\frac{b^3 x^{12}}{m+13}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 178, normalized size = 2.9 \begin{align*}{\frac{{x}^{1+m} \left ({b}^{3}{m}^{3}{x}^{12}+15\,{b}^{3}{m}^{2}{x}^{12}+59\,{b}^{3}m{x}^{12}+45\,{b}^{3}{x}^{12}+3\,a{b}^{2}{m}^{3}{x}^{8}+57\,a{b}^{2}{m}^{2}{x}^{8}+249\,a{b}^{2}m{x}^{8}+195\,a{b}^{2}{x}^{8}+3\,{a}^{2}b{m}^{3}{x}^{4}+69\,{a}^{2}b{m}^{2}{x}^{4}+417\,{a}^{2}bm{x}^{4}+351\,{a}^{2}b{x}^{4}+{a}^{3}{m}^{3}+27\,{a}^{3}{m}^{2}+227\,{a}^{3}m+585\,{a}^{3} \right ) }{ \left ( 13+m \right ) \left ( 9+m \right ) \left ( 5+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 = 1.53122, size = 355, normalized size = 5.82 \begin{align*} \frac{{\left ({\left (b^{3} m^{3} + 15 \, b^{3} m^{2} + 59 \, b^{3} m + 45 \, b^{3}\right )} x^{13} + 3 \,{\left (a b^{2} m^{3} + 19 \, a b^{2} m^{2} + 83 \, a b^{2} m + 65 \, a b^{2}\right )} x^{9} + 3 \,{\left (a^{2} b m^{3} + 23 \, a^{2} b m^{2} + 139 \, a^{2} b m + 117 \, a^{2} b\right )} x^{5} +{\left (a^{3} m^{3} + 27 \, a^{3} m^{2} + 227 \, a^{3} m + 585 \, a^{3}\right )} x\right )} x^{m}}{m^{4} + 28 \, m^{3} + 254 \, m^{2} + 812 \, m + 585} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.72517, size = 683, normalized size = 11.2 \begin{align*} \begin{cases} - \frac{a^{3}}{12 x^{12}} - \frac{3 a^{2} b}{8 x^{8}} - \frac{3 a b^{2}}{4 x^{4}} + b^{3} \log{\left (x \right )} & \text{for}\: m = -13 \\- \frac{a^{3}}{8 x^{8}} - \frac{3 a^{2} b}{4 x^{4}} + 3 a b^{2} \log{\left (x \right )} + \frac{b^{3} x^{4}}{4} & \text{for}\: m = -9 \\- \frac{a^{3}}{4 x^{4}} + 3 a^{2} b \log{\left (x \right )} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{8}}{8} & \text{for}\: m = -5 \\a^{3} \log{\left (x \right )} + \frac{3 a^{2} b x^{4}}{4} + \frac{3 a b^{2} x^{8}}{8} + \frac{b^{3} x^{12}}{12} & \text{for}\: m = -1 \\\frac{a^{3} m^{3} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{27 a^{3} m^{2} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{227 a^{3} m x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{585 a^{3} x x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{3 a^{2} b m^{3} x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{69 a^{2} b m^{2} x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{417 a^{2} b m x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{351 a^{2} b x^{5} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{3 a b^{2} m^{3} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{57 a b^{2} m^{2} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{249 a b^{2} m x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{195 a b^{2} x^{9} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{b^{3} m^{3} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{15 b^{3} m^{2} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{59 b^{3} m x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} + \frac{45 b^{3} x^{13} x^{m}}{m^{4} + 28 m^{3} + 254 m^{2} + 812 m + 585} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15096, size = 302, normalized size = 4.95 \begin{align*} \frac{b^{3} m^{3} x^{13} x^{m} + 15 \, b^{3} m^{2} x^{13} x^{m} + 59 \, b^{3} m x^{13} x^{m} + 45 \, b^{3} x^{13} x^{m} + 3 \, a b^{2} m^{3} x^{9} x^{m} + 57 \, a b^{2} m^{2} x^{9} x^{m} + 249 \, a b^{2} m x^{9} x^{m} + 195 \, a b^{2} x^{9} x^{m} + 3 \, a^{2} b m^{3} x^{5} x^{m} + 69 \, a^{2} b m^{2} x^{5} x^{m} + 417 \, a^{2} b m x^{5} x^{m} + 351 \, a^{2} b x^{5} x^{m} + a^{3} m^{3} x x^{m} + 27 \, a^{3} m^{2} x x^{m} + 227 \, a^{3} m x x^{m} + 585 \, a^{3} x x^{m}}{m^{4} + 28 \, m^{3} + 254 \, m^{2} + 812 \, m + 585} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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