Optimal. Leaf size=157 \[ -\frac{56 b^3}{a^9 (a+b x)}-\frac{35 b^3}{2 a^8 (a+b x)^2}-\frac{20 b^3}{3 a^7 (a+b x)^3}-\frac{5 b^3}{2 a^6 (a+b x)^4}-\frac{4 b^3}{5 a^5 (a+b x)^5}-\frac{b^3}{6 a^4 (a+b x)^6}-\frac{28 b^2}{a^9 x}-\frac{84 b^3 \log (x)}{a^{10}}+\frac{84 b^3 \log (a+b x)}{a^{10}}+\frac{7 b}{2 a^8 x^2}-\frac{1}{3 a^7 x^3} \]
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Rubi [A] time = 0.102711, antiderivative size = 157, 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 = {44} \[ -\frac{56 b^3}{a^9 (a+b x)}-\frac{35 b^3}{2 a^8 (a+b x)^2}-\frac{20 b^3}{3 a^7 (a+b x)^3}-\frac{5 b^3}{2 a^6 (a+b x)^4}-\frac{4 b^3}{5 a^5 (a+b x)^5}-\frac{b^3}{6 a^4 (a+b x)^6}-\frac{28 b^2}{a^9 x}-\frac{84 b^3 \log (x)}{a^{10}}+\frac{84 b^3 \log (a+b x)}{a^{10}}+\frac{7 b}{2 a^8 x^2}-\frac{1}{3 a^7 x^3} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^4 (a+b x)^7} \, dx &=\int \left (\frac{1}{a^7 x^4}-\frac{7 b}{a^8 x^3}+\frac{28 b^2}{a^9 x^2}-\frac{84 b^3}{a^{10} x}+\frac{b^4}{a^4 (a+b x)^7}+\frac{4 b^4}{a^5 (a+b x)^6}+\frac{10 b^4}{a^6 (a+b x)^5}+\frac{20 b^4}{a^7 (a+b x)^4}+\frac{35 b^4}{a^8 (a+b x)^3}+\frac{56 b^4}{a^9 (a+b x)^2}+\frac{84 b^4}{a^{10} (a+b x)}\right ) \, dx\\ &=-\frac{1}{3 a^7 x^3}+\frac{7 b}{2 a^8 x^2}-\frac{28 b^2}{a^9 x}-\frac{b^3}{6 a^4 (a+b x)^6}-\frac{4 b^3}{5 a^5 (a+b x)^5}-\frac{5 b^3}{2 a^6 (a+b x)^4}-\frac{20 b^3}{3 a^7 (a+b x)^3}-\frac{35 b^3}{2 a^8 (a+b x)^2}-\frac{56 b^3}{a^9 (a+b x)}-\frac{84 b^3 \log (x)}{a^{10}}+\frac{84 b^3 \log (a+b x)}{a^{10}}\\ \end{align*}
Mathematica [A] time = 0.236212, size = 123, normalized size = 0.78 \[ -\frac{\frac{a \left (360 a^6 b^2 x^2+6174 a^5 b^3 x^3+21924 a^4 b^4 x^4+35910 a^3 b^5 x^5+31080 a^2 b^6 x^6-45 a^7 b x+10 a^8+13860 a b^7 x^7+2520 b^8 x^8\right )}{x^3 (a+b x)^6}-2520 b^3 \log (a+b x)+2520 b^3 \log (x)}{30 a^{10}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 144, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{a}^{7}{x}^{3}}}+{\frac{7\,b}{2\,{a}^{8}{x}^{2}}}-28\,{\frac{{b}^{2}}{{a}^{9}x}}-{\frac{{b}^{3}}{6\,{a}^{4} \left ( bx+a \right ) ^{6}}}-{\frac{4\,{b}^{3}}{5\,{a}^{5} \left ( bx+a \right ) ^{5}}}-{\frac{5\,{b}^{3}}{2\,{a}^{6} \left ( bx+a \right ) ^{4}}}-{\frac{20\,{b}^{3}}{3\,{a}^{7} \left ( bx+a \right ) ^{3}}}-{\frac{35\,{b}^{3}}{2\,{a}^{8} \left ( bx+a \right ) ^{2}}}-56\,{\frac{{b}^{3}}{{a}^{9} \left ( bx+a \right ) }}-84\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{10}}}+84\,{\frac{{b}^{3}\ln \left ( bx+a \right ) }{{a}^{10}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1492, size = 250, normalized size = 1.59 \begin{align*} -\frac{2520 \, b^{8} x^{8} + 13860 \, a b^{7} x^{7} + 31080 \, a^{2} b^{6} x^{6} + 35910 \, a^{3} b^{5} x^{5} + 21924 \, a^{4} b^{4} x^{4} + 6174 \, a^{5} b^{3} x^{3} + 360 \, a^{6} b^{2} x^{2} - 45 \, a^{7} b x + 10 \, a^{8}}{30 \,{\left (a^{9} b^{6} x^{9} + 6 \, a^{10} b^{5} x^{8} + 15 \, a^{11} b^{4} x^{7} + 20 \, a^{12} b^{3} x^{6} + 15 \, a^{13} b^{2} x^{5} + 6 \, a^{14} b x^{4} + a^{15} x^{3}\right )}} + \frac{84 \, b^{3} \log \left (b x + a\right )}{a^{10}} - \frac{84 \, b^{3} \log \left (x\right )}{a^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62415, size = 701, normalized size = 4.46 \begin{align*} -\frac{2520 \, a b^{8} x^{8} + 13860 \, a^{2} b^{7} x^{7} + 31080 \, a^{3} b^{6} x^{6} + 35910 \, a^{4} b^{5} x^{5} + 21924 \, a^{5} b^{4} x^{4} + 6174 \, a^{6} b^{3} x^{3} + 360 \, a^{7} b^{2} x^{2} - 45 \, a^{8} b x + 10 \, a^{9} - 2520 \,{\left (b^{9} x^{9} + 6 \, a b^{8} x^{8} + 15 \, a^{2} b^{7} x^{7} + 20 \, a^{3} b^{6} x^{6} + 15 \, a^{4} b^{5} x^{5} + 6 \, a^{5} b^{4} x^{4} + a^{6} b^{3} x^{3}\right )} \log \left (b x + a\right ) + 2520 \,{\left (b^{9} x^{9} + 6 \, a b^{8} x^{8} + 15 \, a^{2} b^{7} x^{7} + 20 \, a^{3} b^{6} x^{6} + 15 \, a^{4} b^{5} x^{5} + 6 \, a^{5} b^{4} x^{4} + a^{6} b^{3} x^{3}\right )} \log \left (x\right )}{30 \,{\left (a^{10} b^{6} x^{9} + 6 \, a^{11} b^{5} x^{8} + 15 \, a^{12} b^{4} x^{7} + 20 \, a^{13} b^{3} x^{6} + 15 \, a^{14} b^{2} x^{5} + 6 \, a^{15} b x^{4} + a^{16} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.70848, size = 187, normalized size = 1.19 \begin{align*} - \frac{10 a^{8} - 45 a^{7} b x + 360 a^{6} b^{2} x^{2} + 6174 a^{5} b^{3} x^{3} + 21924 a^{4} b^{4} x^{4} + 35910 a^{3} b^{5} x^{5} + 31080 a^{2} b^{6} x^{6} + 13860 a b^{7} x^{7} + 2520 b^{8} x^{8}}{30 a^{15} x^{3} + 180 a^{14} b x^{4} + 450 a^{13} b^{2} x^{5} + 600 a^{12} b^{3} x^{6} + 450 a^{11} b^{4} x^{7} + 180 a^{10} b^{5} x^{8} + 30 a^{9} b^{6} x^{9}} + \frac{84 b^{3} \left (- \log{\left (x \right )} + \log{\left (\frac{a}{b} + x \right )}\right )}{a^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19823, size = 176, normalized size = 1.12 \begin{align*} \frac{84 \, b^{3} \log \left ({\left | b x + a \right |}\right )}{a^{10}} - \frac{84 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{10}} - \frac{2520 \, a b^{8} x^{8} + 13860 \, a^{2} b^{7} x^{7} + 31080 \, a^{3} b^{6} x^{6} + 35910 \, a^{4} b^{5} x^{5} + 21924 \, a^{5} b^{4} x^{4} + 6174 \, a^{6} b^{3} x^{3} + 360 \, a^{7} b^{2} x^{2} - 45 \, a^{8} b x + 10 \, a^{9}}{30 \,{\left (b x + a\right )}^{6} a^{10} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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