Optimal. Leaf size=28 \[ \frac{1}{160} x^5 (2 a+b x)^5+a x+\frac{b x^2}{2} \]
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Rubi [A] time = 0.0270042, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1591} \[ \frac{1}{160} x^5 (2 a+b x)^5+a x+\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Rule 1591
Rubi steps
\begin{align*} \int (a+b x) \left (1+\left (a x+\frac{b x^2}{2}\right )^4\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^4\right ) \, dx,x,a x+\frac{b x^2}{2}\right )\\ &=a x+\frac{b x^2}{2}+\frac{1}{160} x^5 (2 a+b x)^5\\ \end{align*}
Mathematica [B] time = 0.0051486, size = 80, normalized size = 2.86 \[ \frac{1}{4} a^2 b^3 x^8+\frac{1}{2} a^3 b^2 x^7+\frac{1}{2} a^4 b x^6+\frac{a^5 x^5}{5}+\frac{1}{16} a b^4 x^9+a x+\frac{b^5 x^{10}}{160}+\frac{b x^2}{2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 67, normalized size = 2.4 \begin{align*}{\frac{{b}^{5}{x}^{10}}{160}}+{\frac{a{b}^{4}{x}^{9}}{16}}+{\frac{{a}^{2}{b}^{3}{x}^{8}}{4}}+{\frac{{a}^{3}{b}^{2}{x}^{7}}{2}}+{\frac{{a}^{4}b{x}^{6}}{2}}+{\frac{{a}^{5}{x}^{5}}{5}}+{\frac{b{x}^{2}}{2}}+ax \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00854, size = 89, normalized size = 3.18 \begin{align*} \frac{1}{160} \, b^{5} x^{10} + \frac{1}{16} \, a b^{4} x^{9} + \frac{1}{4} \, a^{2} b^{3} x^{8} + \frac{1}{2} \, a^{3} b^{2} x^{7} + \frac{1}{2} \, a^{4} b x^{6} + \frac{1}{5} \, a^{5} x^{5} + \frac{1}{2} \, b x^{2} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.11006, size = 158, normalized size = 5.64 \begin{align*} \frac{1}{160} x^{10} b^{5} + \frac{1}{16} x^{9} b^{4} a + \frac{1}{4} x^{8} b^{3} a^{2} + \frac{1}{2} x^{7} b^{2} a^{3} + \frac{1}{2} x^{6} b a^{4} + \frac{1}{5} x^{5} a^{5} + \frac{1}{2} x^{2} b + x a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.080792, size = 70, normalized size = 2.5 \begin{align*} \frac{a^{5} x^{5}}{5} + \frac{a^{4} b x^{6}}{2} + \frac{a^{3} b^{2} x^{7}}{2} + \frac{a^{2} b^{3} x^{8}}{4} + \frac{a b^{4} x^{9}}{16} + a x + \frac{b^{5} x^{10}}{160} + \frac{b x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23564, size = 89, normalized size = 3.18 \begin{align*} \frac{1}{160} \, b^{5} x^{10} + \frac{1}{16} \, a b^{4} x^{9} + \frac{1}{4} \, a^{2} b^{3} x^{8} + \frac{1}{2} \, a^{3} b^{2} x^{7} + \frac{1}{2} \, a^{4} b x^{6} + \frac{1}{5} \, a^{5} x^{5} + \frac{1}{2} \, b x^{2} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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