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 = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b (2 p+1)} \end {gather*}
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*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.68 \begin {gather*} \frac {(a+b x) \left ((a+b x)^2\right )^p}{2 b p+b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \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 = 0.42, size = 32, normalized size = 0.94 \begin {gather*} \frac {{\left (b x + a\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \, b p + b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 51, normalized size = 1.50 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 35, normalized size = 1.03 \begin {gather*} \frac {\left (b x +a \right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{\left (2 p +1\right ) b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 25, normalized size = 0.74 \begin {gather*} \frac {{\left (b x + a\right )} {\left (b x + a\right )}^{2 \, p}}{b {\left (2 \, p + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.66, size = 41, normalized size = 1.21 \begin {gather*} \left (\frac {x}{2\,p+1}+\frac {a}{b\,\left (2\,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 [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {x}{\sqrt {a^{2}}} & \text {for}\: b = 0 \wedge p = - \frac {1}{2} \\x \left (a^{2}\right )^{p} & \text {for}\: b = 0 \\\int \frac {1}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2}}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {a \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 b p + b} + \frac {b x \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}}{2 b p + b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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