Optimal. Leaf size=84 \[ \frac{3 a^2 \left (a+b x^n\right )^7}{7 b^4 n}-\frac{a^3 \left (a+b x^n\right )^6}{6 b^4 n}+\frac{\left (a+b x^n\right )^9}{9 b^4 n}-\frac{3 a \left (a+b x^n\right )^8}{8 b^4 n} \]
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Rubi [A] time = 0.0466852, antiderivative size = 84, 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} \[ \frac{3 a^2 \left (a+b x^n\right )^7}{7 b^4 n}-\frac{a^3 \left (a+b x^n\right )^6}{6 b^4 n}+\frac{\left (a+b x^n\right )^9}{9 b^4 n}-\frac{3 a \left (a+b x^n\right )^8}{8 b^4 n} \]
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 )^5 \, dx &=\frac{\operatorname{Subst}\left (\int x^3 (a+b x)^5 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^5}{b^3}+\frac{3 a^2 (a+b x)^6}{b^3}-\frac{3 a (a+b x)^7}{b^3}+\frac{(a+b x)^8}{b^3}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^3 \left (a+b x^n\right )^6}{6 b^4 n}+\frac{3 a^2 \left (a+b x^n\right )^7}{7 b^4 n}-\frac{3 a \left (a+b x^n\right )^8}{8 b^4 n}+\frac{\left (a+b x^n\right )^9}{9 b^4 n}\\ \end{align*}
Mathematica [A] time = 0.0379688, size = 82, normalized size = 0.98 \[ \frac{\frac{5}{3} a^3 b^2 x^{6 n}+\frac{10}{7} a^2 b^3 x^{7 n}+a^4 b x^{5 n}+\frac{1}{4} a^5 x^{4 n}+\frac{5}{8} a b^4 x^{8 n}+\frac{1}{9} b^5 x^{9 n}}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 87, normalized size = 1. \begin{align*}{\frac{{b}^{5} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{5\,a{b}^{4} \left ({x}^{n} \right ) ^{8}}{8\,n}}+{\frac{10\,{a}^{2}{b}^{3} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{5\,{a}^{3}{b}^{2} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{{a}^{4}b \left ({x}^{n} \right ) ^{5}}{n}}+{\frac{{a}^{5} \left ({x}^{n} \right ) ^{4}}{4\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32311, size = 173, normalized size = 2.06 \begin{align*} \frac{56 \, b^{5} x^{9 \, n} + 315 \, a b^{4} x^{8 \, n} + 720 \, a^{2} b^{3} x^{7 \, n} + 840 \, a^{3} b^{2} x^{6 \, n} + 504 \, a^{4} b x^{5 \, n} + 126 \, a^{5} x^{4 \, n}}{504 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 167.599, size = 92, normalized size = 1.1 \begin{align*} \begin{cases} \frac{a^{5} x^{4 n}}{4 n} + \frac{a^{4} b x^{5 n}}{n} + \frac{5 a^{3} b^{2} x^{6 n}}{3 n} + \frac{10 a^{2} b^{3} x^{7 n}}{7 n} + \frac{5 a b^{4} x^{8 n}}{8 n} + \frac{b^{5} x^{9 n}}{9 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{5} \log{\left (x \right )} & \text{otherwise} \end{cases} \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 )}^{5} x^{4 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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