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 = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^{p+1}}{2 b (p+1)} \end {gather*}
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*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.72 \begin {gather*} \frac {\left ((a+b x)^2\right )^{p+1}}{2 b (p+1)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.44, size = 43, normalized size = 1.34 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 30, normalized size = 0.94 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p + 1}}{2 \, b {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 1.12 \begin {gather*} \frac {\left (b x +a \right )^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{2 \left (p +1\right ) b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 30, normalized size = 0.94 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p + 1}}{2 \, b {\left (p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.07, size = 53, normalized size = 1.66 \begin {gather*} \left (\frac {a^2}{2\,b\,\left (p+1\right )}+\frac {a\,x}{p+1}+\frac {b\,x^2}{2\,\left (p+1\right )}\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 119, normalized size = 3.72 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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