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 = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 43
Rule 266
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*}
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Mathematica [A] time = 0.03, 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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fricas [B] time = 0.77, size = 74, normalized size = 1.85 \[ \frac {6 \, b^{5} x^{7 \, n} + 35 \, a b^{4} x^{6 \, n} + 84 \, a^{2} b^{3} x^{5 \, n} + 105 \, a^{3} b^{2} x^{4 \, n} + 70 \, a^{4} b x^{3 \, n} + 21 \, a^{5} x^{2 \, n}}{42 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{5} x^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 88, normalized size = 2.20 \[ \frac {a^{5} x^{2 n}}{2 n}+\frac {5 a^{4} b \,x^{3 n}}{3 n}+\frac {5 a^{3} b^{2} x^{4 n}}{2 n}+\frac {2 a^{2} b^{3} x^{5 n}}{n}+\frac {5 a \,b^{4} x^{6 n}}{6 n}+\frac {b^{5} x^{7 n}}{7 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.59, size = 87, normalized size = 2.18 \[ \frac {b^{5} x^{7 \, n}}{7 \, n} + \frac {5 \, a b^{4} x^{6 \, n}}{6 \, n} + \frac {2 \, a^{2} b^{3} x^{5 \, n}}{n} + \frac {5 \, a^{3} b^{2} x^{4 \, n}}{2 \, n} + \frac {5 \, a^{4} b x^{3 \, n}}{3 \, n} + \frac {a^{5} x^{2 \, n}}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 87, normalized size = 2.18 \[ \frac {a^5\,x^{2\,n}}{2\,n}+\frac {b^5\,x^{7\,n}}{7\,n}+\frac {5\,a^3\,b^2\,x^{4\,n}}{2\,n}+\frac {2\,a^2\,b^3\,x^{5\,n}}{n}+\frac {5\,a^4\,b\,x^{3\,n}}{3\,n}+\frac {5\,a\,b^4\,x^{6\,n}}{6\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 117.65, size = 94, normalized size = 2.35 \[ \begin {cases} \frac {a^{5} x^{2 n}}{2 n} + \frac {5 a^{4} b x^{3 n}}{3 n} + \frac {5 a^{3} b^{2} x^{4 n}}{2 n} + \frac {2 a^{2} b^{3} x^{5 n}}{n} + \frac {5 a b^{4} x^{6 n}}{6 n} + \frac {b^{5} x^{7 n}}{7 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{5} \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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