Optimal. Leaf size=117 \[ -\frac {1}{a^7 x}-\frac {b}{6 a^2 (a+b x)^6}-\frac {2 b}{5 a^3 (a+b x)^5}-\frac {3 b}{4 a^4 (a+b x)^4}-\frac {4 b}{3 a^5 (a+b x)^3}-\frac {5 b}{2 a^6 (a+b x)^2}-\frac {6 b}{a^7 (a+b x)}-\frac {7 b \log (x)}{a^8}+\frac {7 b \log (a+b x)}{a^8} \]
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Rubi [A]
time = 0.06, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {46}
\begin {gather*} -\frac {7 b \log (x)}{a^8}+\frac {7 b \log (a+b x)}{a^8}-\frac {6 b}{a^7 (a+b x)}-\frac {1}{a^7 x}-\frac {5 b}{2 a^6 (a+b x)^2}-\frac {4 b}{3 a^5 (a+b x)^3}-\frac {3 b}{4 a^4 (a+b x)^4}-\frac {2 b}{5 a^3 (a+b x)^5}-\frac {b}{6 a^2 (a+b x)^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)^7} \, dx &=\int \left (\frac {1}{a^7 x^2}-\frac {7 b}{a^8 x}+\frac {b^2}{a^2 (a+b x)^7}+\frac {2 b^2}{a^3 (a+b x)^6}+\frac {3 b^2}{a^4 (a+b x)^5}+\frac {4 b^2}{a^5 (a+b x)^4}+\frac {5 b^2}{a^6 (a+b x)^3}+\frac {6 b^2}{a^7 (a+b x)^2}+\frac {7 b^2}{a^8 (a+b x)}\right ) \, dx\\ &=-\frac {1}{a^7 x}-\frac {b}{6 a^2 (a+b x)^6}-\frac {2 b}{5 a^3 (a+b x)^5}-\frac {3 b}{4 a^4 (a+b x)^4}-\frac {4 b}{3 a^5 (a+b x)^3}-\frac {5 b}{2 a^6 (a+b x)^2}-\frac {6 b}{a^7 (a+b x)}-\frac {7 b \log (x)}{a^8}+\frac {7 b \log (a+b x)}{a^8}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 97, normalized size = 0.83 \begin {gather*} -\frac {\frac {a \left (60 a^6+1029 a^5 b x+3654 a^4 b^2 x^2+5985 a^3 b^3 x^3+5180 a^2 b^4 x^4+2310 a b^5 x^5+420 b^6 x^6\right )}{x (a+b x)^6}+420 b \log (x)-420 b \log (a+b x)}{60 a^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 108, normalized size = 0.92
method | result | size |
risch | \(\frac {-\frac {7 b^{6} x^{6}}{a^{7}}-\frac {77 b^{5} x^{5}}{2 a^{6}}-\frac {259 b^{4} x^{4}}{3 a^{5}}-\frac {399 b^{3} x^{3}}{4 a^{4}}-\frac {609 b^{2} x^{2}}{10 a^{3}}-\frac {343 b x}{20 a^{2}}-\frac {1}{a}}{x \left (b x +a \right )^{6}}-\frac {7 b \ln \left (x \right )}{a^{8}}+\frac {7 b \ln \left (-b x -a \right )}{a^{8}}\) | \(104\) |
norman | \(\frac {-\frac {1}{a}+\frac {42 b^{2} x^{2}}{a^{3}}+\frac {315 b^{3} x^{3}}{2 a^{4}}+\frac {770 b^{4} x^{4}}{3 a^{5}}+\frac {875 b^{5} x^{5}}{4 a^{6}}+\frac {959 b^{6} x^{6}}{10 a^{7}}+\frac {343 b^{7} x^{7}}{20 a^{8}}}{x \left (b x +a \right )^{6}}-\frac {7 b \ln \left (x \right )}{a^{8}}+\frac {7 b \ln \left (b x +a \right )}{a^{8}}\) | \(105\) |
default | \(-\frac {1}{a^{7} x}-\frac {b}{6 a^{2} \left (b x +a \right )^{6}}-\frac {2 b}{5 a^{3} \left (b x +a \right )^{5}}-\frac {3 b}{4 a^{4} \left (b x +a \right )^{4}}-\frac {4 b}{3 a^{5} \left (b x +a \right )^{3}}-\frac {5 b}{2 a^{6} \left (b x +a \right )^{2}}-\frac {6 b}{a^{7} \left (b x +a \right )}-\frac {7 b \ln \left (x \right )}{a^{8}}+\frac {7 b \ln \left (b x +a \right )}{a^{8}}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 157, normalized size = 1.34 \begin {gather*} -\frac {420 \, b^{6} x^{6} + 2310 \, a b^{5} x^{5} + 5180 \, a^{2} b^{4} x^{4} + 5985 \, a^{3} b^{3} x^{3} + 3654 \, a^{4} b^{2} x^{2} + 1029 \, a^{5} b x + 60 \, a^{6}}{60 \, {\left (a^{7} b^{6} x^{7} + 6 \, a^{8} b^{5} x^{6} + 15 \, a^{9} b^{4} x^{5} + 20 \, a^{10} b^{3} x^{4} + 15 \, a^{11} b^{2} x^{3} + 6 \, a^{12} b x^{2} + a^{13} x\right )}} + \frac {7 \, b \log \left (b x + a\right )}{a^{8}} - \frac {7 \, b \log \left (x\right )}{a^{8}} \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. 285 vs.
\(2 (107) = 214\).
time = 1.23, size = 285, normalized size = 2.44 \begin {gather*} -\frac {420 \, a b^{6} x^{6} + 2310 \, a^{2} b^{5} x^{5} + 5180 \, a^{3} b^{4} x^{4} + 5985 \, a^{4} b^{3} x^{3} + 3654 \, a^{5} b^{2} x^{2} + 1029 \, a^{6} b x + 60 \, a^{7} - 420 \, {\left (b^{7} x^{7} + 6 \, a b^{6} x^{6} + 15 \, a^{2} b^{5} x^{5} + 20 \, a^{3} b^{4} x^{4} + 15 \, a^{4} b^{3} x^{3} + 6 \, a^{5} b^{2} x^{2} + a^{6} b x\right )} \log \left (b x + a\right ) + 420 \, {\left (b^{7} x^{7} + 6 \, a b^{6} x^{6} + 15 \, a^{2} b^{5} x^{5} + 20 \, a^{3} b^{4} x^{4} + 15 \, a^{4} b^{3} x^{3} + 6 \, a^{5} b^{2} x^{2} + a^{6} b x\right )} \log \left (x\right )}{60 \, {\left (a^{8} b^{6} x^{7} + 6 \, a^{9} b^{5} x^{6} + 15 \, a^{10} b^{4} x^{5} + 20 \, a^{11} b^{3} x^{4} + 15 \, a^{12} b^{2} x^{3} + 6 \, a^{13} b x^{2} + a^{14} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 162, normalized size = 1.38 \begin {gather*} \frac {- 60 a^{6} - 1029 a^{5} b x - 3654 a^{4} b^{2} x^{2} - 5985 a^{3} b^{3} x^{3} - 5180 a^{2} b^{4} x^{4} - 2310 a b^{5} x^{5} - 420 b^{6} x^{6}}{60 a^{13} x + 360 a^{12} b x^{2} + 900 a^{11} b^{2} x^{3} + 1200 a^{10} b^{3} x^{4} + 900 a^{9} b^{4} x^{5} + 360 a^{8} b^{5} x^{6} + 60 a^{7} b^{6} x^{7}} + \frac {7 b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.77, size = 104, normalized size = 0.89 \begin {gather*} \frac {7 \, b \log \left ({\left | b x + a \right |}\right )}{a^{8}} - \frac {7 \, b \log \left ({\left | x \right |}\right )}{a^{8}} - \frac {420 \, a b^{6} x^{6} + 2310 \, a^{2} b^{5} x^{5} + 5180 \, a^{3} b^{4} x^{4} + 5985 \, a^{4} b^{3} x^{3} + 3654 \, a^{5} b^{2} x^{2} + 1029 \, a^{6} b x + 60 \, a^{7}}{60 \, {\left (b x + a\right )}^{6} a^{8} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 151, normalized size = 1.29 \begin {gather*} \frac {14\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^8}-\frac {\frac {1}{a}+\frac {609\,b^2\,x^2}{10\,a^3}+\frac {399\,b^3\,x^3}{4\,a^4}+\frac {259\,b^4\,x^4}{3\,a^5}+\frac {77\,b^5\,x^5}{2\,a^6}+\frac {7\,b^6\,x^6}{a^7}+\frac {343\,b\,x}{20\,a^2}}{a^6\,x+6\,a^5\,b\,x^2+15\,a^4\,b^2\,x^3+20\,a^3\,b^3\,x^4+15\,a^2\,b^4\,x^5+6\,a\,b^5\,x^6+b^6\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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