Optimal. Leaf size=103 \[ -\frac{a^3 \left (a+b x^n\right )^{p+1}}{b^4 n (p+1)}+\frac{3 a^2 \left (a+b x^n\right )^{p+2}}{b^4 n (p+2)}-\frac{3 a \left (a+b x^n\right )^{p+3}}{b^4 n (p+3)}+\frac{\left (a+b x^n\right )^{p+4}}{b^4 n (p+4)} \]
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Rubi [A] time = 0.0607738, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ -\frac{a^3 \left (a+b x^n\right )^{p+1}}{b^4 n (p+1)}+\frac{3 a^2 \left (a+b x^n\right )^{p+2}}{b^4 n (p+2)}-\frac{3 a \left (a+b x^n\right )^{p+3}}{b^4 n (p+3)}+\frac{\left (a+b x^n\right )^{p+4}}{b^4 n (p+4)} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1+4 n} \left (a+b x^n\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int x^3 (a+b x)^p \, dx,x,x^n\right )}{n}\\ &=\frac{\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,x^n\right )}{n}\\ &=-\frac{a^3 \left (a+b x^n\right )^{1+p}}{b^4 n (1+p)}+\frac{3 a^2 \left (a+b x^n\right )^{2+p}}{b^4 n (2+p)}-\frac{3 a \left (a+b x^n\right )^{3+p}}{b^4 n (3+p)}+\frac{\left (a+b x^n\right )^{4+p}}{b^4 n (4+p)}\\ \end{align*}
Mathematica [A] time = 0.0722685, size = 78, normalized size = 0.76 \[ \frac{\left (a+b x^n\right )^{p+1} \left (\frac{3 a^2 \left (a+b x^n\right )}{p+2}-\frac{a^3}{p+1}-\frac{3 a \left (a+b x^n\right )^2}{p+3}+\frac{\left (a+b x^n\right )^3}{p+4}\right )}{b^4 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 171, normalized size = 1.7 \begin{align*} -{\frac{ \left ( -{b}^{4}{p}^{3} \left ({x}^{n} \right ) ^{4}-a{b}^{3}{p}^{3} \left ({x}^{n} \right ) ^{3}-6\,{b}^{4}{p}^{2} \left ({x}^{n} \right ) ^{4}-3\,a{b}^{3}{p}^{2} \left ({x}^{n} \right ) ^{3}-11\,{b}^{4}p \left ({x}^{n} \right ) ^{4}+3\,{a}^{2}{b}^{2}{p}^{2} \left ({x}^{n} \right ) ^{2}-2\,ap \left ({x}^{n} \right ) ^{3}{b}^{3}-6\, \left ({x}^{n} \right ) ^{4}{b}^{4}+3\,{a}^{2}p \left ({x}^{n} \right ) ^{2}{b}^{2}-6\,{a}^{3}p{x}^{n}b+6\,{a}^{4} \right ) \left ( a+b{x}^{n} \right ) ^{p}}{ \left ( 3+p \right ) \left ( 4+p \right ) \left ( 2+p \right ) \left ( 1+p \right ) n{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00344, size = 154, normalized size = 1.5 \begin{align*} \frac{{\left ({\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{4} x^{4 \, n} +{\left (p^{3} + 3 \, p^{2} + 2 \, p\right )} a b^{3} x^{3 \, n} - 3 \,{\left (p^{2} + p\right )} a^{2} b^{2} x^{2 \, n} + 6 \, a^{3} b p x^{n} - 6 \, a^{4}\right )}{\left (b x^{n} + a\right )}^{p}}{{\left (p^{4} + 10 \, p^{3} + 35 \, p^{2} + 50 \, p + 24\right )} b^{4} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42323, size = 327, normalized size = 3.17 \begin{align*} \frac{{\left (6 \, a^{3} b p x^{n} - 6 \, a^{4} +{\left (b^{4} p^{3} + 6 \, b^{4} p^{2} + 11 \, b^{4} p + 6 \, b^{4}\right )} x^{4 \, n} +{\left (a b^{3} p^{3} + 3 \, a b^{3} p^{2} + 2 \, a b^{3} p\right )} x^{3 \, n} - 3 \,{\left (a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x^{2 \, n}\right )}{\left (b x^{n} + a\right )}^{p}}{b^{4} n p^{4} + 10 \, b^{4} n p^{3} + 35 \, b^{4} n p^{2} + 50 \, b^{4} n p + 24 \, b^{4} n} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{n} + a\right )}^{p} x^{4 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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