Optimal. Leaf size=32 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{p+1}}{2 b (p+1)} \]
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Rubi [A] time = 0.0081101, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {629} \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{p+1}}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin{align*} \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\frac{\left (a^2+2 a b x+b^2 x^2\right )^{1+p}}{2 b (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0120167, size = 23, normalized size = 0.72 \[ \frac{\left ((a+b x)^2\right )^{p+1}}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 36, normalized size = 1.1 \begin{align*}{\frac{ \left ( bx+a \right ) ^{2} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{2\,b \left ( 1+p \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 [A] time = 1.54481, size = 93, normalized size = 2.91 \begin{align*} \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \,{\left (b p + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.478004, size = 119, normalized size = 3.72 \begin{align*} \begin{cases} \frac{x}{a} & \text{for}\: b = 0 \wedge p = -1 \\a x \left (a^{2}\right )^{p} & \text{for}\: b = 0 \\\frac{\log{\left (\frac{a}{b} + x \right )}}{b} & \text{for}\: p = -1 \\\frac{a^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 b p + 2 b} + \frac{2 a b x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 b p + 2 b} + \frac{b^{2} x^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 b p + 2 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1604, size = 108, normalized size = 3.38 \begin{align*} \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{2} x^{2} + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b x +{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2}}{2 \,{\left (b p + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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