Optimal. Leaf size=39 \[ \frac{(a+b x)^{p+2}}{b^2 (p+2)}-\frac{a (a+b x)^{p+1}}{b^2 (p+1)} \]
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Rubi [A] time = 0.0129684, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{(a+b x)^{p+2}}{b^2 (p+2)}-\frac{a (a+b x)^{p+1}}{b^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x (a+b x)^p \, dx &=\int \left (-\frac{a (a+b x)^p}{b}+\frac{(a+b x)^{1+p}}{b}\right ) \, dx\\ &=-\frac{a (a+b x)^{1+p}}{b^2 (1+p)}+\frac{(a+b x)^{2+p}}{b^2 (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0188862, size = 33, normalized size = 0.85 \[ \frac{(a+b x)^{p+1} (b (p+1) x-a)}{b^2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 36, normalized size = 0.9 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+p} \left ( -xpb-bx+a \right ) }{{b}^{2} \left ({p}^{2}+3\,p+2 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965327, size = 57, normalized size = 1.46 \begin{align*} \frac{{\left (b^{2}{\left (p + 1\right )} x^{2} + a b p x - a^{2}\right )}{\left (b x + a\right )}^{p}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99131, size = 104, normalized size = 2.67 \begin{align*} \frac{{\left (a b p x +{\left (b^{2} p + b^{2}\right )} x^{2} - a^{2}\right )}{\left (b x + a\right )}^{p}}{b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.605047, size = 201, normalized size = 5.15 \begin{align*} \begin{cases} \frac{a^{p} x^{2}}{2} & \text{for}\: b = 0 \\\frac{a \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac{a}{a b^{2} + b^{3} x} + \frac{b x \log{\left (\frac{a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text{for}\: p = -2 \\- \frac{a \log{\left (\frac{a}{b} + x \right )}}{b^{2}} + \frac{x}{b} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + b x\right )^{p}}{b^{2} p^{2} + 3 b^{2} p + 2 b^{2}} + \frac{a b p x \left (a + b x\right )^{p}}{b^{2} p^{2} + 3 b^{2} p + 2 b^{2}} + \frac{b^{2} p x^{2} \left (a + b x\right )^{p}}{b^{2} p^{2} + 3 b^{2} p + 2 b^{2}} + \frac{b^{2} x^{2} \left (a + b x\right )^{p}}{b^{2} p^{2} + 3 b^{2} p + 2 b^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07666, size = 103, normalized size = 2.64 \begin{align*} \frac{{\left (b x + a\right )}^{p} b^{2} p x^{2} +{\left (b x + a\right )}^{p} a b p x +{\left (b x + a\right )}^{p} b^{2} x^{2} -{\left (b x + a\right )}^{p} a^{2}}{b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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