Optimal. Leaf size=21 \[ -\frac {x^{-7 n}}{7 n \left (b+c x^n\right )^7} \]
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Rubi [A] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {446, 74} \[ -\frac {x^{-7 n}}{7 n \left (b+c x^n\right )^7} \]
Antiderivative was successfully verified.
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Rule 74
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{-1-7 n} \left (b+2 c x^n\right )}{\left (b+c x^n\right )^8} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b+2 c x}{x^8 (b+c x)^8} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-7 n}}{7 n \left (b+c x^n\right )^7}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 21, normalized size = 1.00 \[ -\frac {x^{-7 n}}{7 n \left (b+c x^n\right )^7} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 105, normalized size = 5.00 \[ -\frac {1}{7 \, {\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 21 \, b^{2} c^{5} n x^{12 \, n} + 35 \, b^{3} c^{4} n x^{11 \, n} + 35 \, b^{4} c^{3} n x^{10 \, n} + 21 \, b^{5} c^{2} n x^{9 \, n} + 7 \, b^{6} c n x^{8 \, n} + b^{7} n x^{7 \, n}\right )}} \]
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 \[ \int \frac {{\left (2 \, c x^{n} + b\right )} x^{-7 \, n - 1}}{{\left (c x^{n} + b\right )}^{8}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 203, normalized size = 9.67 \[ -\frac {x^{-7 n}}{7 b^{7} n}+\frac {c \,x^{-6 n}}{b^{8} n}-\frac {4 c^{2} x^{-5 n}}{b^{9} n}+\frac {12 c^{3} x^{-4 n}}{b^{10} n}-\frac {30 c^{4} x^{-3 n}}{b^{11} n}+\frac {66 c^{5} x^{-2 n}}{b^{12} n}+\frac {\left (9009 b^{5} c \,x^{n}+20020 b^{4} c^{2} x^{2 n}+24024 b^{3} c^{3} x^{3 n}+16380 b^{2} c^{4} x^{4 n}+6006 b \,c^{5} x^{5 n}+924 c^{6} x^{6 n}+1716 b^{6}\right ) c^{7}}{7 \left (c \,x^{n}+b \right )^{7} b^{13} n}-\frac {132 c^{6} x^{-n}}{b^{13} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.79, size = 612, normalized size = 29.14 \[ -\frac {1}{105} \, b {\left (\frac {360360 \, c^{13} x^{13 \, n} + 2342340 \, b c^{12} x^{12 \, n} + 6426420 \, b^{2} c^{11} x^{11 \, n} + 9579570 \, b^{3} c^{10} x^{10 \, n} + 8270262 \, b^{4} c^{9} x^{9 \, n} + 4018014 \, b^{5} c^{8} x^{8 \, n} + 934362 \, b^{6} c^{7} x^{7 \, n} + 45045 \, b^{7} c^{6} x^{6 \, n} - 5005 \, b^{8} c^{5} x^{5 \, n} + 1001 \, b^{9} c^{4} x^{4 \, n} - 273 \, b^{10} c^{3} x^{3 \, n} + 91 \, b^{11} c^{2} x^{2 \, n} - 35 \, b^{12} c x^{n} + 15 \, b^{13}}{b^{14} c^{7} n x^{14 \, n} + 7 \, b^{15} c^{6} n x^{13 \, n} + 21 \, b^{16} c^{5} n x^{12 \, n} + 35 \, b^{17} c^{4} n x^{11 \, n} + 35 \, b^{18} c^{3} n x^{10 \, n} + 21 \, b^{19} c^{2} n x^{9 \, n} + 7 \, b^{20} c n x^{8 \, n} + b^{21} n x^{7 \, n}} + \frac {360360 \, c^{7} \log \relax (x)}{b^{15}} - \frac {360360 \, c^{7} \log \left (\frac {c x^{n} + b}{c}\right )}{b^{15} n}\right )} + \frac {1}{105} \, c {\left (\frac {360360 \, c^{12} x^{12 \, n} + 2342340 \, b c^{11} x^{11 \, n} + 6426420 \, b^{2} c^{10} x^{10 \, n} + 9579570 \, b^{3} c^{9} x^{9 \, n} + 8270262 \, b^{4} c^{8} x^{8 \, n} + 4018014 \, b^{5} c^{7} x^{7 \, n} + 934362 \, b^{6} c^{6} x^{6 \, n} + 45045 \, b^{7} c^{5} x^{5 \, n} - 5005 \, b^{8} c^{4} x^{4 \, n} + 1001 \, b^{9} c^{3} x^{3 \, n} - 273 \, b^{10} c^{2} x^{2 \, n} + 91 \, b^{11} c x^{n} - 35 \, b^{12}}{b^{13} c^{7} n x^{13 \, n} + 7 \, b^{14} c^{6} n x^{12 \, n} + 21 \, b^{15} c^{5} n x^{11 \, n} + 35 \, b^{16} c^{4} n x^{10 \, n} + 35 \, b^{17} c^{3} n x^{9 \, n} + 21 \, b^{18} c^{2} n x^{8 \, n} + 7 \, b^{19} c n x^{7 \, n} + b^{20} n x^{6 \, n}} + \frac {360360 \, c^{6} \log \relax (x)}{b^{14}} - \frac {360360 \, c^{6} \log \left (\frac {c x^{n} + b}{c}\right )}{b^{14} n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.99, size = 105, normalized size = 5.00 \[ -\frac {1}{7\,x^{7\,n}\,\left (b^7\,n+c^7\,n\,x^{7\,n}+7\,b^6\,c\,n\,x^n+7\,b\,c^6\,n\,x^{6\,n}+21\,b^5\,c^2\,n\,x^{2\,n}+35\,b^4\,c^3\,n\,x^{3\,n}+35\,b^3\,c^4\,n\,x^{4\,n}+21\,b^2\,c^5\,n\,x^{5\,n}\right )} \]
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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