Optimal. Leaf size=34 \[ \frac {\left (a x+\frac {b x^2}{2}\right )^{n+1}}{n+1}+a x+\frac {b x^2}{2} \]
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Rubi [A] time = 0.01, antiderivative size = 34, 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} \[ \frac {\left (a x+\frac {b x^2}{2}\right )^{n+1}}{n+1}+a x+\frac {b x^2}{2} \]
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 )^n\right ) \, dx &=\operatorname {Subst}\left (\int \left (1+x^n\right ) \, dx,x,a x+\frac {b x^2}{2}\right )\\ &=a x+\frac {b x^2}{2}+\frac {\left (a x+\frac {b x^2}{2}\right )^{1+n}}{1+n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 34, normalized size = 1.00 \[ \frac {x (2 a+b x) \left (\left (a x+\frac {b x^2}{2}\right )^n+n+1\right )}{2 (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 48, normalized size = 1.41 \[ \frac {{\left (b n + b\right )} x^{2} + {\left (b x^{2} + 2 \, a x\right )} {\left (\frac {1}{2} \, b x^{2} + a x\right )}^{n} + 2 \, {\left (a n + a\right )} x}{2 \, {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 30, normalized size = 0.88 \[ \frac {1}{2} \, b x^{2} + a x + \frac {{\left (\frac {1}{2} \, b x^{2} + a x\right )}^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.91 \[ \frac {b \,x^{2}}{2}+a x +\frac {\left (\frac {1}{2} b \,x^{2}+a x \right )^{n +1}}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 52, normalized size = 1.53 \[ \frac {1}{2} \, b x^{2} + a x + \frac {{\left (b x^{2} + 2 \, a x\right )} e^{\left (n \log \left (b x + 2 \, a\right ) + n \log \relax (x)\right )}}{2^{n + 1} n + 2^{n + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.12, size = 31, normalized size = 0.91 \[ \frac {x\,\left (2\,a+b\,x\right )\,\left (n+{\left (\frac {b\,x^2}{2}+a\,x\right )}^n+1\right )}{2\,\left (n+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 50.75, size = 230, normalized size = 6.76 \[ \begin {cases} a \left (x + \frac {\log {\relax (x )}}{a}\right ) & \text {for}\: b = 0 \wedge n = -1 \\a \left (\frac {a^{n} x x^{n}}{n + 1} + \frac {n x}{n + 1} + \frac {x}{n + 1}\right ) & \text {for}\: b = 0 \\a x + \frac {b x^{2}}{2} + \log {\relax (x )} + \log {\left (\frac {2 a}{b} + x \right )} & \text {for}\: n = -1 \\\frac {2 \cdot 2^{n} a b n x}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac {2 \cdot 2^{n} a b x}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac {2^{n} b^{2} n x^{2}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac {2^{n} b^{2} x^{2}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac {2 a b x \left (2 a x + b x^{2}\right )^{n}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} + \frac {b^{2} x^{2} \left (2 a x + b x^{2}\right )^{n}}{2 \cdot 2^{n} b n + 2 \cdot 2^{n} b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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