Optimal. Leaf size=40 \[ \frac{\left (a+b x^n\right )^7}{7 b^2 n}-\frac{a \left (a+b x^n\right )^6}{6 b^2 n} \]
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Rubi [A] time = 0.0193388, antiderivative size = 40, 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} \[ \frac{\left (a+b x^n\right )^7}{7 b^2 n}-\frac{a \left (a+b x^n\right )^6}{6 b^2 n} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1+2 n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int x (a+b x)^5 \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^5}{b}+\frac{(a+b x)^6}{b}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a \left (a+b x^n\right )^6}{6 b^2 n}+\frac{\left (a+b x^n\right )^7}{7 b^2 n}\\ \end{align*}
Mathematica [A] time = 0.0219488, size = 27, normalized size = 0.68 \[ -\frac{\left (a-6 b x^n\right ) \left (a+b x^n\right )^6}{42 b^2 n} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.018, size = 88, normalized size = 2.2 \begin{align*}{\frac{{b}^{5} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{5\,a{b}^{4} \left ({x}^{n} \right ) ^{6}}{6\,n}}+2\,{\frac{{a}^{2}{b}^{3} \left ({x}^{n} \right ) ^{5}}{n}}+{\frac{5\,{a}^{3}{b}^{2} \left ({x}^{n} \right ) ^{4}}{2\,n}}+{\frac{5\,{a}^{4}b \left ({x}^{n} \right ) ^{3}}{3\,n}}+{\frac{{a}^{5} \left ({x}^{n} \right ) ^{2}}{2\,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 [B] time = 1.3048, size = 165, normalized size = 4.12 \begin{align*} \frac{6 \, b^{5} x^{7 \, n} + 35 \, a b^{4} x^{6 \, n} + 84 \, a^{2} b^{3} x^{5 \, n} + 105 \, a^{3} b^{2} x^{4 \, n} + 70 \, a^{4} b x^{3 \, n} + 21 \, a^{5} x^{2 \, n}}{42 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 166.398, size = 94, normalized size = 2.35 \begin{align*} \begin{cases} \frac{a^{5} x^{2 n}}{2 n} + \frac{5 a^{4} b x^{3 n}}{3 n} + \frac{5 a^{3} b^{2} x^{4 n}}{2 n} + \frac{2 a^{2} b^{3} x^{5 n}}{n} + \frac{5 a b^{4} x^{6 n}}{6 n} + \frac{b^{5} x^{7 n}}{7 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^{2 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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