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 = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, 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*}
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Mathematica [B] time = 0.01, size = 80, normalized size = 2.86 \[ \frac {a^5 x^5}{5}+\frac {1}{2} a^4 b x^6+\frac {1}{2} a^3 b^2 x^7+\frac {1}{4} a^2 b^3 x^8+\frac {1}{16} a b^4 x^9+a x+\frac {b^5 x^{10}}{160}+\frac {b x^2}{2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 66, normalized size = 2.36 \[ \frac {1}{160} x^{10} b^{5} + \frac {1}{16} x^{9} b^{4} a + \frac {1}{4} x^{8} b^{3} a^{2} + \frac {1}{2} x^{7} b^{2} a^{3} + \frac {1}{2} x^{6} b a^{4} + \frac {1}{5} x^{5} a^{5} + \frac {1}{2} x^{2} b + x a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 24, normalized size = 0.86 \[ \frac {1}{160} \, {\left (b x^{2} + 2 \, a x\right )}^{5} + \frac {1}{2} \, b x^{2} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 67, normalized size = 2.39 \[ \frac {1}{160} b^{5} x^{10}+\frac {1}{16} a \,b^{4} x^{9}+\frac {1}{4} a^{2} b^{3} x^{8}+\frac {1}{2} a^{3} b^{2} x^{7}+\frac {1}{2} a^{4} b \,x^{6}+\frac {1}{5} a^{5} x^{5}+\frac {1}{2} b \,x^{2}+a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 66, normalized size = 2.36 \[ \frac {1}{160} \, b^{5} x^{10} + \frac {1}{16} \, a b^{4} x^{9} + \frac {1}{4} \, a^{2} b^{3} x^{8} + \frac {1}{2} \, a^{3} b^{2} x^{7} + \frac {1}{2} \, a^{4} b x^{6} + \frac {1}{5} \, a^{5} x^{5} + \frac {1}{2} \, b x^{2} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 66, normalized size = 2.36 \[ \frac {a^5\,x^5}{5}+\frac {a^4\,b\,x^6}{2}+\frac {a^3\,b^2\,x^7}{2}+\frac {a^2\,b^3\,x^8}{4}+\frac {a\,b^4\,x^9}{16}+a\,x+\frac {b^5\,x^{10}}{160}+\frac {b\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 70, normalized size = 2.50 \[ \frac {a^{5} x^{5}}{5} + \frac {a^{4} b x^{6}}{2} + \frac {a^{3} b^{2} x^{7}}{2} + \frac {a^{2} b^{3} x^{8}}{4} + \frac {a b^{4} x^{9}}{16} + a x + \frac {b^{5} x^{10}}{160} + \frac {b x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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