Optimal. Leaf size=50 \[ \frac {b x^{-6 n} \left (a+b x^n\right )^6}{42 a^2 n}-\frac {x^{-7 n} \left (a+b x^n\right )^6}{7 a n} \]
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Rubi [A] time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {266, 45, 37} \[ \frac {b x^{-6 n} \left (a+b x^n\right )^6}{42 a^2 n}-\frac {x^{-7 n} \left (a+b x^n\right )^6}{7 a n} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 266
Rubi steps
\begin {align*} \int x^{-1-7 n} \left (a+b x^n\right )^5 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^8} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-7 n} \left (a+b x^n\right )^6}{7 a n}-\frac {b \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,x^n\right )}{7 a n}\\ &=-\frac {x^{-7 n} \left (a+b x^n\right )^6}{7 a n}+\frac {b x^{-6 n} \left (a+b x^n\right )^6}{42 a^2 n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.66 \[ \frac {x^{-7 n} \left (b x^n-6 a\right ) \left (a+b x^n\right )^6}{42 a^2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 74, normalized size = 1.48 \[ -\frac {21 \, b^{5} x^{5 \, n} + 70 \, a b^{4} x^{4 \, n} + 105 \, a^{2} b^{3} x^{3 \, n} + 84 \, a^{3} b^{2} x^{2 \, n} + 35 \, a^{4} b x^{n} + 6 \, a^{5}}{42 \, n x^{7 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 74, normalized size = 1.48 \[ -\frac {21 \, b^{5} x^{5 \, n} + 70 \, a b^{4} x^{4 \, n} + 105 \, a^{2} b^{3} x^{3 \, n} + 84 \, a^{3} b^{2} x^{2 \, n} + 35 \, a^{4} b x^{n} + 6 \, a^{5}}{42 \, n x^{7 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 88, normalized size = 1.76 \[ -\frac {a^{5} x^{-7 n}}{7 n}-\frac {5 a^{4} b \,x^{-6 n}}{6 n}-\frac {2 a^{3} b^{2} x^{-5 n}}{n}-\frac {5 a^{2} b^{3} x^{-4 n}}{2 n}-\frac {5 a \,b^{4} x^{-3 n}}{3 n}-\frac {b^{5} x^{-2 n}}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 99, normalized size = 1.98 \[ -\frac {a^{5}}{7 \, n x^{7 \, n}} - \frac {5 \, a^{4} b}{6 \, n x^{6 \, n}} - \frac {2 \, a^{3} b^{2}}{n x^{5 \, n}} - \frac {5 \, a^{2} b^{3}}{2 \, n x^{4 \, n}} - \frac {5 \, a b^{4}}{3 \, n x^{3 \, n}} - \frac {b^{5}}{2 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 99, normalized size = 1.98 \[ -\frac {a^5}{7\,n\,x^{7\,n}}-\frac {b^5}{2\,n\,x^{2\,n}}-\frac {5\,a^2\,b^3}{2\,n\,x^{4\,n}}-\frac {2\,a^3\,b^2}{n\,x^{5\,n}}-\frac {5\,a\,b^4}{3\,n\,x^{3\,n}}-\frac {5\,a^4\,b}{6\,n\,x^{6\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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