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 = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.43, size = 53, normalized size = 1.36 \[ \frac {{\left (a b p x + {\left (b^{2} p + b^{2}\right )} x^{2} - a^{2}\right )} {\left (b x + a\right )}^{p}}{b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 76, normalized size = 1.95 \[ \frac {{\left (b x + a\right )}^{p} b^{2} p x^{2} + {\left (b x + a\right )}^{p} a b p x + {\left (b x + a\right )}^{p} b^{2} x^{2} - {\left (b x + a\right )}^{p} a^{2}}{b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.92 \[ -\frac {\left (-x p b -b x +a \right ) \left (b x +a \right )^{p +1}}{\left (p^{2}+3 p +2\right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 42, normalized size = 1.08 \[ \frac {{\left (b^{2} {\left (p + 1\right )} x^{2} + a b p x - a^{2}\right )} {\left (b x + a\right )}^{p}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 94, normalized size = 2.41 \[ \left \{\begin {array}{cl} -\frac {a\,\ln \left (a+b\,x\right )-b\,x}{b^2} & \text {\ if\ \ }p=-1\\ \frac {\ln \left (a+b\,x\right )+\frac {a}{a+b\,x}}{b^2} & \text {\ if\ \ }p=-2\\ \frac {2\,\left (\frac {{\left (a+b\,x\right )}^{p+2}}{2\,p+4}-\frac {a\,{\left (a+b\,x\right )}^{p+1}}{2\,p+2}\right )}{b^2} & \text {\ if\ \ }p\neq -1\wedge p\neq -2 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.65, size = 201, normalized size = 5.15 \[ \begin {cases} \frac {a^{p} x^{2}}{2} & \text {for}\: b = 0 \\\frac {a \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac {a}{a b^{2} + b^{3} x} + \frac {b x \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text {for}\: p = -2 \\- \frac {a \log {\left (\frac {a}{b} + x \right )}}{b^{2}} + \frac {x}{b} & \text {for}\: p = -1 \\- \frac {a^{2} \left (a + b x\right )^{p}}{b^{2} p^{2} + 3 b^{2} p + 2 b^{2}} + \frac {a b p x \left (a + b x\right )^{p}}{b^{2} p^{2} + 3 b^{2} p + 2 b^{2}} + \frac {b^{2} p x^{2} \left (a + b x\right )^{p}}{b^{2} p^{2} + 3 b^{2} p + 2 b^{2}} + \frac {b^{2} x^{2} \left (a + b x\right )^{p}}{b^{2} p^{2} + 3 b^{2} p + 2 b^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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