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 = 0.0642931, antiderivative size = 109, normalized size of antiderivative = 1., 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 = {43} \[ -\frac{a^6}{6 b^7 (a+b x)^6}+\frac{6 a^5}{5 b^7 (a+b x)^5}-\frac{15 a^4}{4 b^7 (a+b x)^4}+\frac{20 a^3}{3 b^7 (a+b x)^3}-\frac{15 a^2}{2 b^7 (a+b x)^2}+\frac{6 a}{b^7 (a+b x)}+\frac{\log (a+b x)}{b^7} \]
Antiderivative was successfully verified.
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Rule 43
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*}
Mathematica [A] time = 0.0377205, size = 77, normalized size = 0.71 \[ \frac{\frac{a \left (1875 a^3 b^2 x^2+2200 a^2 b^3 x^3+822 a^4 b x+147 a^5+1350 a b^4 x^4+360 b^5 x^5\right )}{(a+b x)^6}+60 \log (a+b x)}{60 b^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 100, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{6\,{b}^{7} \left ( bx+a \right ) ^{6}}}+{\frac{6\,{a}^{5}}{5\,{b}^{7} \left ( bx+a \right ) ^{5}}}-{\frac{15\,{a}^{4}}{4\,{b}^{7} \left ( bx+a \right ) ^{4}}}+{\frac{20\,{a}^{3}}{3\,{b}^{7} \left ( bx+a \right ) ^{3}}}-{\frac{15\,{a}^{2}}{2\,{b}^{7} \left ( bx+a \right ) ^{2}}}+6\,{\frac{a}{{b}^{7} \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) }{{b}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08102, size = 184, normalized size = 1.69 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54159, size = 428, normalized size = 3.93 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.04979, size = 141, normalized size = 1.29 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20754, size = 107, normalized size = 0.98 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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