Optimal. Leaf size=21 \[ -\frac {x^{-7 n}}{7 n \left (b+c x^n\right )^7} \]
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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1598, 457, 75}
\begin {gather*} -\frac {x^{-7 n}}{7 n \left (b+c x^n\right )^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{-7 (-1+n)} \left (b+2 c x^n\right )}{\left (b x+c x^{1+n}\right )^8} \, dx &=\int \frac {x^{-8-7 (-1+n)} \left (b+2 c x^n\right )}{\left (b+c x^n\right )^8} \, dx\\ &=\frac {\text {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 = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {x^{-7 n}}{7 n \left (b+c x^n\right )^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(202\) vs.
\(2(21)=42\).
time = 0.24, size = 203, normalized size = 9.67
method | result | size |
risch | \(-\frac {132 c^{6} x^{-n}}{b^{13} n}+\frac {66 c^{5} x^{-2 n}}{b^{12} n}-\frac {30 c^{4} x^{-3 n}}{b^{11} n}+\frac {12 c^{3} x^{-4 n}}{b^{10} n}-\frac {4 c^{2} x^{-5 n}}{b^{9} n}+\frac {c \,x^{-6 n}}{b^{8} n}-\frac {x^{-7 n}}{7 b^{7} n}+\frac {c^{7} \left (924 x^{6 n} c^{6}+6006 b \,c^{5} x^{5 n}+16380 b^{2} c^{4} x^{4 n}+24024 b^{3} c^{3} x^{3 n}+20020 b^{4} c^{2} x^{2 n}+9009 b^{5} c \,x^{n}+1716 b^{6}\right )}{7 b^{13} n \left (b +c \,x^{n}\right )^{7}}\) | \(203\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 612 vs.
\(2 (21) = 42\).
time = 0.34, size = 612, normalized size = 29.14 \begin {gather*} -\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 \left (x\right )}{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 \left (x\right )}{b^{14}} - \frac {360360 \, c^{6} \log \left (\frac {c x^{n} + b}{c}\right )}{b^{14} n}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (21) = 42\).
time = 0.53, size = 143, normalized size = 6.81 \begin {gather*} -\frac {x^{14}}{7 \, {\left (b^{7} n x^{7} x^{7 \, n + 7} + 7 \, b^{6} c n x^{6} x^{8 \, n + 8} + 21 \, b^{5} c^{2} n x^{5} x^{9 \, n + 9} + 35 \, b^{4} c^{3} n x^{4} x^{10 \, n + 10} + 35 \, b^{3} c^{4} n x^{3} x^{11 \, n + 11} + 21 \, b^{2} c^{5} n x^{2} x^{12 \, n + 12} + 7 \, b c^{6} n x x^{13 \, n + 13} + c^{7} n x^{14 \, n + 14}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {x^{7-7\,n}\,\left (b+2\,c\,x^n\right )}{{\left (b\,x+c\,x^{n+1}\right )}^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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