Optimal. Leaf size=109 \[ -\frac {a^6}{6 b^7 (a+b x)^6}+\frac {6 a^5}{5 b^7 (a+b x)^5}-\frac {15 a^4}{4 b^7 (a+b x)^4}+\frac {20 a^3}{3 b^7 (a+b x)^3}-\frac {15 a^2}{2 b^7 (a+b x)^2}+\frac {6 a}{b^7 (a+b x)}+\frac {\log (a+b x)}{b^7} \]
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Rubi [A]
time = 0.04, antiderivative size = 109, 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 = {45}
\begin {gather*} -\frac {a^6}{6 b^7 (a+b x)^6}+\frac {6 a^5}{5 b^7 (a+b x)^5}-\frac {15 a^4}{4 b^7 (a+b x)^4}+\frac {20 a^3}{3 b^7 (a+b x)^3}-\frac {15 a^2}{2 b^7 (a+b x)^2}+\frac {6 a}{b^7 (a+b x)}+\frac {\log (a+b x)}{b^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^6}{(a+b x)^7} \, dx &=\int \left (\frac {a^6}{b^6 (a+b x)^7}-\frac {6 a^5}{b^6 (a+b x)^6}+\frac {15 a^4}{b^6 (a+b x)^5}-\frac {20 a^3}{b^6 (a+b x)^4}+\frac {15 a^2}{b^6 (a+b x)^3}-\frac {6 a}{b^6 (a+b x)^2}+\frac {1}{b^6 (a+b x)}\right ) \, dx\\ &=-\frac {a^6}{6 b^7 (a+b x)^6}+\frac {6 a^5}{5 b^7 (a+b x)^5}-\frac {15 a^4}{4 b^7 (a+b x)^4}+\frac {20 a^3}{3 b^7 (a+b x)^3}-\frac {15 a^2}{2 b^7 (a+b x)^2}+\frac {6 a}{b^7 (a+b x)}+\frac {\log (a+b x)}{b^7}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 77, normalized size = 0.71 \begin {gather*} \frac {\frac {a \left (147 a^5+822 a^4 b x+1875 a^3 b^2 x^2+2200 a^2 b^3 x^3+1350 a b^4 x^4+360 b^5 x^5\right )}{(a+b x)^6}+60 \log (a+b x)}{60 b^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 100, normalized size = 0.92
method | result | size |
norman | \(\frac {\frac {49 a^{6}}{20 b^{7}}+\frac {6 a \,x^{5}}{b^{2}}+\frac {45 a^{2} x^{4}}{2 b^{3}}+\frac {125 a^{4} x^{2}}{4 b^{5}}+\frac {137 a^{5} x}{10 b^{6}}+\frac {110 a^{3} x^{3}}{3 b^{4}}}{\left (b x +a \right )^{6}}+\frac {\ln \left (b x +a \right )}{b^{7}}\) | \(80\) |
risch | \(\frac {\frac {49 a^{6}}{20 b^{7}}+\frac {6 a \,x^{5}}{b^{2}}+\frac {45 a^{2} x^{4}}{2 b^{3}}+\frac {125 a^{4} x^{2}}{4 b^{5}}+\frac {137 a^{5} x}{10 b^{6}}+\frac {110 a^{3} x^{3}}{3 b^{4}}}{\left (b x +a \right )^{6}}+\frac {\ln \left (b x +a \right )}{b^{7}}\) | \(80\) |
default | \(-\frac {a^{6}}{6 b^{7} \left (b x +a \right )^{6}}+\frac {6 a^{5}}{5 b^{7} \left (b x +a \right )^{5}}-\frac {15 a^{4}}{4 b^{7} \left (b x +a \right )^{4}}+\frac {20 a^{3}}{3 b^{7} \left (b x +a \right )^{3}}-\frac {15 a^{2}}{2 b^{7} \left (b x +a \right )^{2}}+\frac {6 a}{b^{7} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{7}}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 136, normalized size = 1.25 \begin {gather*} \frac {360 \, a b^{5} x^{5} + 1350 \, a^{2} b^{4} x^{4} + 2200 \, a^{3} b^{3} x^{3} + 1875 \, a^{4} b^{2} x^{2} + 822 \, a^{5} b x + 147 \, a^{6}}{60 \, {\left (b^{13} x^{6} + 6 \, a b^{12} x^{5} + 15 \, a^{2} b^{11} x^{4} + 20 \, a^{3} b^{10} x^{3} + 15 \, a^{4} b^{9} x^{2} + 6 \, a^{5} b^{8} x + a^{6} b^{7}\right )}} + \frac {\log \left (b x + a\right )}{b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.91, size = 193, normalized size = 1.77 \begin {gather*} \frac {360 \, a b^{5} x^{5} + 1350 \, a^{2} b^{4} x^{4} + 2200 \, a^{3} b^{3} x^{3} + 1875 \, a^{4} b^{2} x^{2} + 822 \, a^{5} b x + 147 \, a^{6} + 60 \, {\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{60 \, {\left (b^{13} x^{6} + 6 \, a b^{12} x^{5} + 15 \, a^{2} b^{11} x^{4} + 20 \, a^{3} b^{10} x^{3} + 15 \, a^{4} b^{9} x^{2} + 6 \, a^{5} b^{8} x + a^{6} b^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 141, normalized size = 1.29 \begin {gather*} \frac {147 a^{6} + 822 a^{5} b x + 1875 a^{4} b^{2} x^{2} + 2200 a^{3} b^{3} x^{3} + 1350 a^{2} b^{4} x^{4} + 360 a b^{5} x^{5}}{60 a^{6} b^{7} + 360 a^{5} b^{8} x + 900 a^{4} b^{9} x^{2} + 1200 a^{3} b^{10} x^{3} + 900 a^{2} b^{11} x^{4} + 360 a b^{12} x^{5} + 60 b^{13} x^{6}} + \frac {\log {\left (a + b x \right )}}{b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.23, size = 79, normalized size = 0.72 \begin {gather*} \frac {\log \left ({\left | b x + a \right |}\right )}{b^{7}} + \frac {360 \, a b^{4} x^{5} + 1350 \, a^{2} b^{3} x^{4} + 2200 \, a^{3} b^{2} x^{3} + 1875 \, a^{4} b x^{2} + 822 \, a^{5} x + \frac {147 \, a^{6}}{b}}{60 \, {\left (b x + a\right )}^{6} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 81, normalized size = 0.74 \begin {gather*} \frac {\ln \left (a+b\,x\right )+\frac {6\,a}{a+b\,x}-\frac {15\,a^2}{2\,{\left (a+b\,x\right )}^2}+\frac {20\,a^3}{3\,{\left (a+b\,x\right )}^3}-\frac {15\,a^4}{4\,{\left (a+b\,x\right )}^4}+\frac {6\,a^5}{5\,{\left (a+b\,x\right )}^5}-\frac {a^6}{6\,{\left (a+b\,x\right )}^6}}{b^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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