Optimal. Leaf size=84 \[ -\frac {a^3 \left (a+b x^n\right )^9}{9 b^4 n}+\frac {3 a^2 \left (a+b x^n\right )^{10}}{10 b^4 n}+\frac {\left (a+b x^n\right )^{12}}{12 b^4 n}-\frac {3 a \left (a+b x^n\right )^{11}}{11 b^4 n} \]
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Rubi [A] time = 0.05, antiderivative size = 84, 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} \[ \frac {3 a^2 \left (a+b x^n\right )^{10}}{10 b^4 n}-\frac {a^3 \left (a+b x^n\right )^9}{9 b^4 n}+\frac {\left (a+b x^n\right )^{12}}{12 b^4 n}-\frac {3 a \left (a+b x^n\right )^{11}}{11 b^4 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1+4 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int x^3 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a^3 (a+b x)^8}{b^3}+\frac {3 a^2 (a+b x)^9}{b^3}-\frac {3 a (a+b x)^{10}}{b^3}+\frac {(a+b x)^{11}}{b^3}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^3 \left (a+b x^n\right )^9}{9 b^4 n}+\frac {3 a^2 \left (a+b x^n\right )^{10}}{10 b^4 n}-\frac {3 a \left (a+b x^n\right )^{11}}{11 b^4 n}+\frac {\left (a+b x^n\right )^{12}}{12 b^4 n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 53, normalized size = 0.63 \[ -\frac {\left (a+b x^n\right )^9 \left (a^3-9 a^2 b x^n+45 a b^2 x^{2 n}-165 b^3 x^{3 n}\right )}{1980 b^4 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 113, normalized size = 1.35 \[ \frac {165 \, b^{8} x^{12 \, n} + 1440 \, a b^{7} x^{11 \, n} + 5544 \, a^{2} b^{6} x^{10 \, n} + 12320 \, a^{3} b^{5} x^{9 \, n} + 17325 \, a^{4} b^{4} x^{8 \, n} + 15840 \, a^{5} b^{3} x^{7 \, n} + 9240 \, a^{6} b^{2} x^{6 \, n} + 3168 \, a^{7} b x^{5 \, n} + 495 \, a^{8} x^{4 \, n}}{1980 \, n} \]
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 \[ \int {\left (b x^{n} + a\right )}^{8} x^{4 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 136, normalized size = 1.62 \[ \frac {a^{8} x^{4 n}}{4 n}+\frac {8 a^{7} b \,x^{5 n}}{5 n}+\frac {14 a^{6} b^{2} x^{6 n}}{3 n}+\frac {8 a^{5} b^{3} x^{7 n}}{n}+\frac {35 a^{4} b^{4} x^{8 n}}{4 n}+\frac {56 a^{3} b^{5} x^{9 n}}{9 n}+\frac {14 a^{2} b^{6} x^{10 n}}{5 n}+\frac {8 a \,b^{7} x^{11 n}}{11 n}+\frac {b^{8} x^{12 n}}{12 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 135, normalized size = 1.61 \[ \frac {b^{8} x^{12 \, n}}{12 \, n} + \frac {8 \, a b^{7} x^{11 \, n}}{11 \, n} + \frac {14 \, a^{2} b^{6} x^{10 \, n}}{5 \, n} + \frac {56 \, a^{3} b^{5} x^{9 \, n}}{9 \, n} + \frac {35 \, a^{4} b^{4} x^{8 \, n}}{4 \, n} + \frac {8 \, a^{5} b^{3} x^{7 \, n}}{n} + \frac {14 \, a^{6} b^{2} x^{6 \, n}}{3 \, n} + \frac {8 \, a^{7} b x^{5 \, n}}{5 \, n} + \frac {a^{8} x^{4 \, n}}{4 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 135, normalized size = 1.61 \[ \frac {a^8\,x^{4\,n}}{4\,n}+\frac {b^8\,x^{12\,n}}{12\,n}+\frac {14\,a^6\,b^2\,x^{6\,n}}{3\,n}+\frac {8\,a^5\,b^3\,x^{7\,n}}{n}+\frac {35\,a^4\,b^4\,x^{8\,n}}{4\,n}+\frac {56\,a^3\,b^5\,x^{9\,n}}{9\,n}+\frac {14\,a^2\,b^6\,x^{10\,n}}{5\,n}+\frac {8\,a^7\,b\,x^{5\,n}}{5\,n}+\frac {8\,a\,b^7\,x^{11\,n}}{11\,n} \]
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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