Optimal. Leaf size=75 \[ \frac{a^2 \left (a+b x^n\right )^{p+1}}{b^3 n (p+1)}-\frac{2 a \left (a+b x^n\right )^{p+2}}{b^3 n (p+2)}+\frac{\left (a+b x^n\right )^{p+3}}{b^3 n (p+3)} \]
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Rubi [A] time = 0.0448444, antiderivative size = 75, 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} \[ \frac{a^2 \left (a+b x^n\right )^{p+1}}{b^3 n (p+1)}-\frac{2 a \left (a+b x^n\right )^{p+2}}{b^3 n (p+2)}+\frac{\left (a+b x^n\right )^{p+3}}{b^3 n (p+3)} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1+3 n} \left (a+b x^n\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int x^2 (a+b x)^p \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^p}{b^2}-\frac{2 a (a+b x)^{1+p}}{b^2}+\frac{(a+b x)^{2+p}}{b^2}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{a^2 \left (a+b x^n\right )^{1+p}}{b^3 n (1+p)}-\frac{2 a \left (a+b x^n\right )^{2+p}}{b^3 n (2+p)}+\frac{\left (a+b x^n\right )^{3+p}}{b^3 n (3+p)}\\ \end{align*}
Mathematica [A] time = 0.0370347, size = 66, normalized size = 0.88 \[ \frac{\left (a+b x^n\right )^{p+1} \left (2 a^2-2 a b (p+1) x^n+b^2 \left (p^2+3 p+2\right ) x^{2 n}\right )}{b^3 n (p+1) (p+2) (p+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 105, normalized size = 1.4 \begin{align*}{\frac{ \left ({b}^{3}{p}^{2} \left ({x}^{n} \right ) ^{3}+a{b}^{2}{p}^{2} \left ({x}^{n} \right ) ^{2}+3\,{b}^{3}p \left ({x}^{n} \right ) ^{3}+ap \left ({x}^{n} \right ) ^{2}{b}^{2}+2\, \left ({x}^{n} \right ) ^{3}{b}^{3}-2\,{a}^{2}p{x}^{n}b+2\,{a}^{3} \right ) \left ( a+b{x}^{n} \right ) ^{p}}{ \left ( 2+p \right ) \left ( 3+p \right ) \left ( 1+p \right ) n{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02099, size = 107, normalized size = 1.43 \begin{align*} \frac{{\left ({\left (p^{2} + 3 \, p + 2\right )} b^{3} x^{3 \, n} +{\left (p^{2} + p\right )} a b^{2} x^{2 \, n} - 2 \, a^{2} b p x^{n} + 2 \, a^{3}\right )}{\left (b x^{n} + a\right )}^{p}}{{\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} b^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43983, size = 215, normalized size = 2.87 \begin{align*} -\frac{{\left (2 \, a^{2} b p x^{n} - 2 \, a^{3} -{\left (b^{3} p^{2} + 3 \, b^{3} p + 2 \, b^{3}\right )} x^{3 \, n} -{\left (a b^{2} p^{2} + a b^{2} p\right )} x^{2 \, n}\right )}{\left (b x^{n} + a\right )}^{p}}{b^{3} n p^{3} + 6 \, b^{3} n p^{2} + 11 \, b^{3} n p + 6 \, b^{3} 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^{3 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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