Optimal. Leaf size=191 \[ \frac{45 b^2}{a^{11} (a+b x)}+\frac{18 b^2}{a^{10} (a+b x)^2}+\frac{28 b^2}{3 a^9 (a+b x)^3}+\frac{21 b^2}{4 a^8 (a+b x)^4}+\frac{3 b^2}{a^7 (a+b x)^5}+\frac{5 b^2}{3 a^6 (a+b x)^6}+\frac{6 b^2}{7 a^5 (a+b x)^7}+\frac{3 b^2}{8 a^4 (a+b x)^8}+\frac{b^2}{9 a^3 (a+b x)^9}+\frac{55 b^2 \log (x)}{a^{12}}-\frac{55 b^2 \log (a+b x)}{a^{12}}+\frac{10 b}{a^{11} x}-\frac{1}{2 a^{10} x^2} \]
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Rubi [A] time = 0.141398, antiderivative size = 191, 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} \[ \frac{45 b^2}{a^{11} (a+b x)}+\frac{18 b^2}{a^{10} (a+b x)^2}+\frac{28 b^2}{3 a^9 (a+b x)^3}+\frac{21 b^2}{4 a^8 (a+b x)^4}+\frac{3 b^2}{a^7 (a+b x)^5}+\frac{5 b^2}{3 a^6 (a+b x)^6}+\frac{6 b^2}{7 a^5 (a+b x)^7}+\frac{3 b^2}{8 a^4 (a+b x)^8}+\frac{b^2}{9 a^3 (a+b x)^9}+\frac{55 b^2 \log (x)}{a^{12}}-\frac{55 b^2 \log (a+b x)}{a^{12}}+\frac{10 b}{a^{11} x}-\frac{1}{2 a^{10} x^2} \]
Antiderivative was successfully verified.
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Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 (a+b x)^{10}} \, dx &=\int \left (\frac{1}{a^{10} x^3}-\frac{10 b}{a^{11} x^2}+\frac{55 b^2}{a^{12} x}-\frac{b^3}{a^3 (a+b x)^{10}}-\frac{3 b^3}{a^4 (a+b x)^9}-\frac{6 b^3}{a^5 (a+b x)^8}-\frac{10 b^3}{a^6 (a+b x)^7}-\frac{15 b^3}{a^7 (a+b x)^6}-\frac{21 b^3}{a^8 (a+b x)^5}-\frac{28 b^3}{a^9 (a+b x)^4}-\frac{36 b^3}{a^{10} (a+b x)^3}-\frac{45 b^3}{a^{11} (a+b x)^2}-\frac{55 b^3}{a^{12} (a+b x)}\right ) \, dx\\ &=-\frac{1}{2 a^{10} x^2}+\frac{10 b}{a^{11} x}+\frac{b^2}{9 a^3 (a+b x)^9}+\frac{3 b^2}{8 a^4 (a+b x)^8}+\frac{6 b^2}{7 a^5 (a+b x)^7}+\frac{5 b^2}{3 a^6 (a+b x)^6}+\frac{3 b^2}{a^7 (a+b x)^5}+\frac{21 b^2}{4 a^8 (a+b x)^4}+\frac{28 b^2}{3 a^9 (a+b x)^3}+\frac{18 b^2}{a^{10} (a+b x)^2}+\frac{45 b^2}{a^{11} (a+b x)}+\frac{55 b^2 \log (x)}{a^{12}}-\frac{55 b^2 \log (a+b x)}{a^{12}}\\ \end{align*}
Mathematica [A] time = 0.239428, size = 145, normalized size = 0.76 \[ \frac{\frac{a \left (78419 a^8 b^2 x^2+456291 a^7 b^3 x^3+1326204 a^6 b^4 x^4+2318316 a^5 b^5 x^5+2604294 a^4 b^6 x^6+1905750 a^3 b^7 x^7+882420 a^2 b^8 x^8+2772 a^9 b x-252 a^{10}+235620 a b^9 x^9+27720 b^{10} x^{10}\right )}{x^2 (a+b x)^9}-27720 b^2 \log (a+b x)+27720 b^2 \log (x)}{504 a^{12}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 178, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{a}^{10}{x}^{2}}}+10\,{\frac{b}{{a}^{11}x}}+{\frac{{b}^{2}}{9\,{a}^{3} \left ( bx+a \right ) ^{9}}}+{\frac{3\,{b}^{2}}{8\,{a}^{4} \left ( bx+a \right ) ^{8}}}+{\frac{6\,{b}^{2}}{7\,{a}^{5} \left ( bx+a \right ) ^{7}}}+{\frac{5\,{b}^{2}}{3\,{a}^{6} \left ( bx+a \right ) ^{6}}}+3\,{\frac{{b}^{2}}{{a}^{7} \left ( bx+a \right ) ^{5}}}+{\frac{21\,{b}^{2}}{4\,{a}^{8} \left ( bx+a \right ) ^{4}}}+{\frac{28\,{b}^{2}}{3\,{a}^{9} \left ( bx+a \right ) ^{3}}}+18\,{\frac{{b}^{2}}{{a}^{10} \left ( bx+a \right ) ^{2}}}+45\,{\frac{{b}^{2}}{{a}^{11} \left ( bx+a \right ) }}+55\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{12}}}-55\,{\frac{{b}^{2}\ln \left ( bx+a \right ) }{{a}^{12}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17859, size = 324, normalized size = 1.7 \begin{align*} \frac{27720 \, b^{10} x^{10} + 235620 \, a b^{9} x^{9} + 882420 \, a^{2} b^{8} x^{8} + 1905750 \, a^{3} b^{7} x^{7} + 2604294 \, a^{4} b^{6} x^{6} + 2318316 \, a^{5} b^{5} x^{5} + 1326204 \, a^{6} b^{4} x^{4} + 456291 \, a^{7} b^{3} x^{3} + 78419 \, a^{8} b^{2} x^{2} + 2772 \, a^{9} b x - 252 \, a^{10}}{504 \,{\left (a^{11} b^{9} x^{11} + 9 \, a^{12} b^{8} x^{10} + 36 \, a^{13} b^{7} x^{9} + 84 \, a^{14} b^{6} x^{8} + 126 \, a^{15} b^{5} x^{7} + 126 \, a^{16} b^{4} x^{6} + 84 \, a^{17} b^{3} x^{5} + 36 \, a^{18} b^{2} x^{4} + 9 \, a^{19} b x^{3} + a^{20} x^{2}\right )}} - \frac{55 \, b^{2} \log \left (b x + a\right )}{a^{12}} + \frac{55 \, b^{2} \log \left (x\right )}{a^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.64254, size = 1019, normalized size = 5.34 \begin{align*} \frac{27720 \, a b^{10} x^{10} + 235620 \, a^{2} b^{9} x^{9} + 882420 \, a^{3} b^{8} x^{8} + 1905750 \, a^{4} b^{7} x^{7} + 2604294 \, a^{5} b^{6} x^{6} + 2318316 \, a^{6} b^{5} x^{5} + 1326204 \, a^{7} b^{4} x^{4} + 456291 \, a^{8} b^{3} x^{3} + 78419 \, a^{9} b^{2} x^{2} + 2772 \, a^{10} b x - 252 \, a^{11} - 27720 \,{\left (b^{11} x^{11} + 9 \, a b^{10} x^{10} + 36 \, a^{2} b^{9} x^{9} + 84 \, a^{3} b^{8} x^{8} + 126 \, a^{4} b^{7} x^{7} + 126 \, a^{5} b^{6} x^{6} + 84 \, a^{6} b^{5} x^{5} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{3} + a^{9} b^{2} x^{2}\right )} \log \left (b x + a\right ) + 27720 \,{\left (b^{11} x^{11} + 9 \, a b^{10} x^{10} + 36 \, a^{2} b^{9} x^{9} + 84 \, a^{3} b^{8} x^{8} + 126 \, a^{4} b^{7} x^{7} + 126 \, a^{5} b^{6} x^{6} + 84 \, a^{6} b^{5} x^{5} + 36 \, a^{7} b^{4} x^{4} + 9 \, a^{8} b^{3} x^{3} + a^{9} b^{2} x^{2}\right )} \log \left (x\right )}{504 \,{\left (a^{12} b^{9} x^{11} + 9 \, a^{13} b^{8} x^{10} + 36 \, a^{14} b^{7} x^{9} + 84 \, a^{15} b^{6} x^{8} + 126 \, a^{16} b^{5} x^{7} + 126 \, a^{17} b^{4} x^{6} + 84 \, a^{18} b^{3} x^{5} + 36 \, a^{19} b^{2} x^{4} + 9 \, a^{20} b x^{3} + a^{21} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.31266, size = 246, normalized size = 1.29 \begin{align*} \frac{- 252 a^{10} + 2772 a^{9} b x + 78419 a^{8} b^{2} x^{2} + 456291 a^{7} b^{3} x^{3} + 1326204 a^{6} b^{4} x^{4} + 2318316 a^{5} b^{5} x^{5} + 2604294 a^{4} b^{6} x^{6} + 1905750 a^{3} b^{7} x^{7} + 882420 a^{2} b^{8} x^{8} + 235620 a b^{9} x^{9} + 27720 b^{10} x^{10}}{504 a^{20} x^{2} + 4536 a^{19} b x^{3} + 18144 a^{18} b^{2} x^{4} + 42336 a^{17} b^{3} x^{5} + 63504 a^{16} b^{4} x^{6} + 63504 a^{15} b^{5} x^{7} + 42336 a^{14} b^{6} x^{8} + 18144 a^{13} b^{7} x^{9} + 4536 a^{12} b^{8} x^{10} + 504 a^{11} b^{9} x^{11}} + \frac{55 b^{2} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23231, size = 205, normalized size = 1.07 \begin{align*} -\frac{55 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{12}} + \frac{55 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{12}} + \frac{27720 \, a b^{10} x^{10} + 235620 \, a^{2} b^{9} x^{9} + 882420 \, a^{3} b^{8} x^{8} + 1905750 \, a^{4} b^{7} x^{7} + 2604294 \, a^{5} b^{6} x^{6} + 2318316 \, a^{6} b^{5} x^{5} + 1326204 \, a^{7} b^{4} x^{4} + 456291 \, a^{8} b^{3} x^{3} + 78419 \, a^{9} b^{2} x^{2} + 2772 \, a^{10} b x - 252 \, a^{11}}{504 \,{\left (b x + a\right )}^{9} a^{12} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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