Optimal. Leaf size=34 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b (2 p+1)} \]
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Rubi [A] time = 0.0056333, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {609} \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 609
Rubi steps
\begin{align*} \int \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b (1+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0111934, size = 23, normalized size = 0.68 \[ \frac{(a+b x) \left ((a+b x)^2\right )^p}{2 b p+b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 35, normalized size = 1. \begin{align*}{\frac{ \left ( bx+a \right ) \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{b \left ( 1+2\,p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11127, size = 34, normalized size = 1. \begin{align*} \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{2 \, p}}{b{\left (2 \, p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60567, size = 69, normalized size = 2.03 \begin{align*} \frac{{\left (b x + a\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \, b p + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16335, size = 69, normalized size = 2.03 \begin{align*} \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b x +{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a}{2 \, b p + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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