Optimal. Leaf size=48 \[ \frac{2 (a+b x)^{\frac{p+4}{2}}}{b^2 (p+4)}-\frac{2 a (a+b x)^{\frac{p+2}{2}}}{b^2 (p+2)} \]
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Rubi [A] time = 0.0142299, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{2 (a+b x)^{\frac{p+4}{2}}}{b^2 (p+4)}-\frac{2 a (a+b x)^{\frac{p+2}{2}}}{b^2 (p+2)} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x (a+b x)^{p/2} \, dx &=\int \left (\frac{(a+b x)^{1+\frac{p}{2}}}{b}-\frac{a (a+b x)^{p/2}}{b}\right ) \, dx\\ &=-\frac{2 a (a+b x)^{\frac{2+p}{2}}}{b^2 (2+p)}+\frac{2 (a+b x)^{\frac{4+p}{2}}}{b^2 (4+p)}\\ \end{align*}
Mathematica [A] time = 0.0199537, size = 38, normalized size = 0.79 \[ \frac{2 (a+b x)^{\frac{p}{2}+1} (b (p+2) x-2 a)}{b^2 (p+2) (p+4)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 43, normalized size = 0.9 \begin{align*} -2\,{\frac{ \left ( \sqrt{bx+a} \right ) ^{p} \left ( -xpb-2\,bx+2\,a \right ) \left ( bx+a \right ) }{{b}^{2} \left ({p}^{2}+6\,p+8 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967557, size = 61, normalized size = 1.27 \begin{align*} \frac{2 \,{\left (b^{2}{\left (p + 2\right )} x^{2} + a b p x - 2 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{1}{2} \, p}}{{\left (p^{2} + 6 \, p + 8\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81903, size = 117, normalized size = 2.44 \begin{align*} \frac{2 \,{\left (a b p x +{\left (b^{2} p + 2 \, b^{2}\right )} x^{2} - 2 \, a^{2}\right )} \sqrt{b x + a}^{p}}{b^{2} p^{2} + 6 \, b^{2} p + 8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.654637, size = 216, normalized size = 4.5 \begin{align*} \begin{cases} \frac{a^{\frac{p}{2}} 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 = -4 \\- \frac{a \log{\left (\frac{a}{b} + x \right )}}{b^{2}} + \frac{x}{b} & \text{for}\: p = -2 \\- \frac{4 a^{2} \left (a + b x\right )^{\frac{p}{2}}}{b^{2} p^{2} + 6 b^{2} p + 8 b^{2}} + \frac{2 a b p x \left (a + b x\right )^{\frac{p}{2}}}{b^{2} p^{2} + 6 b^{2} p + 8 b^{2}} + \frac{2 b^{2} p x^{2} \left (a + b x\right )^{\frac{p}{2}}}{b^{2} p^{2} + 6 b^{2} p + 8 b^{2}} + \frac{4 b^{2} x^{2} \left (a + b x\right )^{\frac{p}{2}}}{b^{2} p^{2} + 6 b^{2} p + 8 b^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06266, size = 116, normalized size = 2.42 \begin{align*} \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{1}{2} \, p} b^{2} p x^{2} +{\left (b x + a\right )}^{\frac{1}{2} \, p} a b p x + 2 \,{\left (b x + a\right )}^{\frac{1}{2} \, p} b^{2} x^{2} - 2 \,{\left (b x + a\right )}^{\frac{1}{2} \, p} a^{2}\right )}}{b^{2} p^{2} + 6 \, b^{2} p + 8 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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