Optimal. Leaf size=83 \[ -\frac{a^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac{3 a^2 (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a (a+b x)^{n+3}}{b^4 (n+3)}+\frac{(a+b x)^{n+4}}{b^4 (n+4)} \]
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Rubi [A] time = 0.031316, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ -\frac{a^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac{3 a^2 (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a (a+b x)^{n+3}}{b^4 (n+3)}+\frac{(a+b x)^{n+4}}{b^4 (n+4)} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^3 (a+b x)^n \, dx &=\int \left (-\frac{a^3 (a+b x)^n}{b^3}+\frac{3 a^2 (a+b x)^{1+n}}{b^3}-\frac{3 a (a+b x)^{2+n}}{b^3}+\frac{(a+b x)^{3+n}}{b^3}\right ) \, dx\\ &=-\frac{a^3 (a+b x)^{1+n}}{b^4 (1+n)}+\frac{3 a^2 (a+b x)^{2+n}}{b^4 (2+n)}-\frac{3 a (a+b x)^{3+n}}{b^4 (3+n)}+\frac{(a+b x)^{4+n}}{b^4 (4+n)}\\ \end{align*}
Mathematica [A] time = 0.0496236, size = 67, normalized size = 0.81 \[ \frac{(a+b x)^{n+1} \left (\frac{3 a^2 (a+b x)}{n+2}-\frac{a^3}{n+1}-\frac{3 a (a+b x)^2}{n+3}+\frac{(a+b x)^3}{n+4}\right )}{b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 126, normalized size = 1.5 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{3}{n}^{3}{x}^{3}-6\,{b}^{3}{n}^{2}{x}^{3}+3\,a{b}^{2}{n}^{2}{x}^{2}-11\,{b}^{3}n{x}^{3}+9\,a{b}^{2}n{x}^{2}-6\,{b}^{3}{x}^{3}-6\,{a}^{2}bnx+6\,a{b}^{2}{x}^{2}-6\,{a}^{2}bx+6\,{a}^{3} \right ) }{{b}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04599, size = 136, normalized size = 1.64 \begin{align*} \frac{{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} +{\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \,{\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54057, size = 292, normalized size = 3.52 \begin{align*} \frac{{\left (6 \, a^{3} b n x +{\left (b^{4} n^{3} + 6 \, b^{4} n^{2} + 11 \, b^{4} n + 6 \, b^{4}\right )} x^{4} - 6 \, a^{4} +{\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 3 \,{\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2}\right )}{\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.80595, size = 1319, normalized size = 15.89 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05998, size = 305, normalized size = 3.67 \begin{align*} \frac{{\left (b x + a\right )}^{n} b^{4} n^{3} x^{4} +{\left (b x + a\right )}^{n} a b^{3} n^{3} x^{3} + 6 \,{\left (b x + a\right )}^{n} b^{4} n^{2} x^{4} + 3 \,{\left (b x + a\right )}^{n} a b^{3} n^{2} x^{3} + 11 \,{\left (b x + a\right )}^{n} b^{4} n x^{4} - 3 \,{\left (b x + a\right )}^{n} a^{2} b^{2} n^{2} x^{2} + 2 \,{\left (b x + a\right )}^{n} a b^{3} n x^{3} + 6 \,{\left (b x + a\right )}^{n} b^{4} x^{4} - 3 \,{\left (b x + a\right )}^{n} a^{2} b^{2} n x^{2} + 6 \,{\left (b x + a\right )}^{n} a^{3} b n x - 6 \,{\left (b x + a\right )}^{n} a^{4}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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