Optimal. Leaf size=100 \[ -\frac{2 a^3 \left (a+b \sqrt{x}\right )^{p+1}}{b^4 (p+1)}+\frac{6 a^2 \left (a+b \sqrt{x}\right )^{p+2}}{b^4 (p+2)}-\frac{6 a \left (a+b \sqrt{x}\right )^{p+3}}{b^4 (p+3)}+\frac{2 \left (a+b \sqrt{x}\right )^{p+4}}{b^4 (p+4)} \]
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Rubi [A] time = 0.0507882, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{2 a^3 \left (a+b \sqrt{x}\right )^{p+1}}{b^4 (p+1)}+\frac{6 a^2 \left (a+b \sqrt{x}\right )^{p+2}}{b^4 (p+2)}-\frac{6 a \left (a+b \sqrt{x}\right )^{p+3}}{b^4 (p+3)}+\frac{2 \left (a+b \sqrt{x}\right )^{p+4}}{b^4 (p+4)} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt{x}\right )^p x \, dx &=2 \operatorname{Subst}\left (\int x^3 (a+b x)^p \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^p}{b^3}+\frac{3 a^2 (a+b x)^{1+p}}{b^3}-\frac{3 a (a+b x)^{2+p}}{b^3}+\frac{(a+b x)^{3+p}}{b^3}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 a^3 \left (a+b \sqrt{x}\right )^{1+p}}{b^4 (1+p)}+\frac{6 a^2 \left (a+b \sqrt{x}\right )^{2+p}}{b^4 (2+p)}-\frac{6 a \left (a+b \sqrt{x}\right )^{3+p}}{b^4 (3+p)}+\frac{2 \left (a+b \sqrt{x}\right )^{4+p}}{b^4 (4+p)}\\ \end{align*}
Mathematica [A] time = 0.0542191, size = 95, normalized size = 0.95 \[ \frac{2 \left (a+b \sqrt{x}\right )^{p+1} \left (6 a^2 b (p+1) \sqrt{x}-6 a^3-3 a b^2 \left (p^2+3 p+2\right ) x+b^3 \left (p^3+6 p^2+11 p+6\right ) x^{3/2}\right )}{b^4 (p+1) (p+2) (p+3) (p+4)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.018, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\sqrt{x} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01214, size = 140, normalized size = 1.4 \begin{align*} \frac{2 \,{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{4} x^{2} +{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a b^{3} x^{\frac{3}{2}} - 3 \,{\left (p^{2} + p\right )} a^{2} b^{2} x + 6 \, a^{3} b p \sqrt{x} - 6 \, a^{4}\right )}{\left (b \sqrt{x} + a\right )}^{p}}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48636, size = 309, normalized size = 3.09 \begin{align*} -\frac{2 \,{\left (6 \, a^{4} -{\left (b^{4} p^{3} + 6 \, b^{4} p^{2} + 11 \, b^{4} p + 6 \, b^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x -{\left (6 \, a^{3} b p +{\left (a b^{3} p^{3} + 3 \, a b^{3} p^{2} + 2 \, a b^{3} p\right )} x\right )} \sqrt{x}\right )}{\left (b \sqrt{x} + a\right )}^{p}}{b^{4} p^{4} + 10 \, b^{4} p^{3} + 35 \, b^{4} p^{2} + 50 \, b^{4} p + 24 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13545, size = 554, normalized size = 5.54 \begin{align*} \frac{2 \,{\left ({\left (b \sqrt{x} + a\right )}^{4}{\left (b \sqrt{x} + a\right )}^{p} p^{3} - 3 \,{\left (b \sqrt{x} + a\right )}^{3}{\left (b \sqrt{x} + a\right )}^{p} a p^{3} + 3 \,{\left (b \sqrt{x} + a\right )}^{2}{\left (b \sqrt{x} + a\right )}^{p} a^{2} p^{3} -{\left (b \sqrt{x} + a\right )}{\left (b \sqrt{x} + a\right )}^{p} a^{3} p^{3} + 6 \,{\left (b \sqrt{x} + a\right )}^{4}{\left (b \sqrt{x} + a\right )}^{p} p^{2} - 21 \,{\left (b \sqrt{x} + a\right )}^{3}{\left (b \sqrt{x} + a\right )}^{p} a p^{2} + 24 \,{\left (b \sqrt{x} + a\right )}^{2}{\left (b \sqrt{x} + a\right )}^{p} a^{2} p^{2} - 9 \,{\left (b \sqrt{x} + a\right )}{\left (b \sqrt{x} + a\right )}^{p} a^{3} p^{2} + 11 \,{\left (b \sqrt{x} + a\right )}^{4}{\left (b \sqrt{x} + a\right )}^{p} p - 42 \,{\left (b \sqrt{x} + a\right )}^{3}{\left (b \sqrt{x} + a\right )}^{p} a p + 57 \,{\left (b \sqrt{x} + a\right )}^{2}{\left (b \sqrt{x} + a\right )}^{p} a^{2} p - 26 \,{\left (b \sqrt{x} + a\right )}{\left (b \sqrt{x} + a\right )}^{p} a^{3} p + 6 \,{\left (b \sqrt{x} + a\right )}^{4}{\left (b \sqrt{x} + a\right )}^{p} - 24 \,{\left (b \sqrt{x} + a\right )}^{3}{\left (b \sqrt{x} + a\right )}^{p} a + 36 \,{\left (b \sqrt{x} + a\right )}^{2}{\left (b \sqrt{x} + a\right )}^{p} a^{2} - 24 \,{\left (b \sqrt{x} + a\right )}{\left (b \sqrt{x} + a\right )}^{p} a^{3}\right )}}{{\left (b^{2} p^{4} + 10 \, b^{2} p^{3} + 35 \, b^{2} p^{2} + 50 \, b^{2} p + 24 \, b^{2}\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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