Optimal. Leaf size=96 \[ -\frac{b^4 (a+b x)^{11}}{15015 a^5 x^{11}}+\frac{b^3 (a+b x)^{11}}{1365 a^4 x^{12}}-\frac{2 b^2 (a+b x)^{11}}{455 a^3 x^{13}}+\frac{2 b (a+b x)^{11}}{105 a^2 x^{14}}-\frac{(a+b x)^{11}}{15 a x^{15}} \]
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Rubi [A] time = 0.0250919, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ -\frac{b^4 (a+b x)^{11}}{15015 a^5 x^{11}}+\frac{b^3 (a+b x)^{11}}{1365 a^4 x^{12}}-\frac{2 b^2 (a+b x)^{11}}{455 a^3 x^{13}}+\frac{2 b (a+b x)^{11}}{105 a^2 x^{14}}-\frac{(a+b x)^{11}}{15 a x^{15}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^{10}}{x^{16}} \, dx &=-\frac{(a+b x)^{11}}{15 a x^{15}}-\frac{(4 b) \int \frac{(a+b x)^{10}}{x^{15}} \, dx}{15 a}\\ &=-\frac{(a+b x)^{11}}{15 a x^{15}}+\frac{2 b (a+b x)^{11}}{105 a^2 x^{14}}+\frac{\left (2 b^2\right ) \int \frac{(a+b x)^{10}}{x^{14}} \, dx}{35 a^2}\\ &=-\frac{(a+b x)^{11}}{15 a x^{15}}+\frac{2 b (a+b x)^{11}}{105 a^2 x^{14}}-\frac{2 b^2 (a+b x)^{11}}{455 a^3 x^{13}}-\frac{\left (4 b^3\right ) \int \frac{(a+b x)^{10}}{x^{13}} \, dx}{455 a^3}\\ &=-\frac{(a+b x)^{11}}{15 a x^{15}}+\frac{2 b (a+b x)^{11}}{105 a^2 x^{14}}-\frac{2 b^2 (a+b x)^{11}}{455 a^3 x^{13}}+\frac{b^3 (a+b x)^{11}}{1365 a^4 x^{12}}+\frac{b^4 \int \frac{(a+b x)^{10}}{x^{12}} \, dx}{1365 a^4}\\ &=-\frac{(a+b x)^{11}}{15 a x^{15}}+\frac{2 b (a+b x)^{11}}{105 a^2 x^{14}}-\frac{2 b^2 (a+b x)^{11}}{455 a^3 x^{13}}+\frac{b^3 (a+b x)^{11}}{1365 a^4 x^{12}}-\frac{b^4 (a+b x)^{11}}{15015 a^5 x^{11}}\\ \end{align*}
Mathematica [A] time = 0.0136676, size = 130, normalized size = 1.35 \[ -\frac{45 a^8 b^2}{13 x^{13}}-\frac{10 a^7 b^3}{x^{12}}-\frac{210 a^6 b^4}{11 x^{11}}-\frac{126 a^5 b^5}{5 x^{10}}-\frac{70 a^4 b^6}{3 x^9}-\frac{15 a^3 b^7}{x^8}-\frac{45 a^2 b^8}{7 x^7}-\frac{5 a^9 b}{7 x^{14}}-\frac{a^{10}}{15 x^{15}}-\frac{5 a b^9}{3 x^6}-\frac{b^{10}}{5 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 113, normalized size = 1.2 \begin{align*} -{\frac{126\,{a}^{5}{b}^{5}}{5\,{x}^{10}}}-10\,{\frac{{a}^{7}{b}^{3}}{{x}^{12}}}-{\frac{{b}^{10}}{5\,{x}^{5}}}-{\frac{{a}^{10}}{15\,{x}^{15}}}-{\frac{210\,{a}^{6}{b}^{4}}{11\,{x}^{11}}}-{\frac{5\,a{b}^{9}}{3\,{x}^{6}}}-15\,{\frac{{a}^{3}{b}^{7}}{{x}^{8}}}-{\frac{45\,{a}^{8}{b}^{2}}{13\,{x}^{13}}}-{\frac{45\,{a}^{2}{b}^{8}}{7\,{x}^{7}}}-{\frac{70\,{a}^{4}{b}^{6}}{3\,{x}^{9}}}-{\frac{5\,{a}^{9}b}{7\,{x}^{14}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08695, size = 151, normalized size = 1.57 \begin{align*} -\frac{3003 \, b^{10} x^{10} + 25025 \, a b^{9} x^{9} + 96525 \, a^{2} b^{8} x^{8} + 225225 \, a^{3} b^{7} x^{7} + 350350 \, a^{4} b^{6} x^{6} + 378378 \, a^{5} b^{5} x^{5} + 286650 \, a^{6} b^{4} x^{4} + 150150 \, a^{7} b^{3} x^{3} + 51975 \, a^{8} b^{2} x^{2} + 10725 \, a^{9} b x + 1001 \, a^{10}}{15015 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47006, size = 301, normalized size = 3.14 \begin{align*} -\frac{3003 \, b^{10} x^{10} + 25025 \, a b^{9} x^{9} + 96525 \, a^{2} b^{8} x^{8} + 225225 \, a^{3} b^{7} x^{7} + 350350 \, a^{4} b^{6} x^{6} + 378378 \, a^{5} b^{5} x^{5} + 286650 \, a^{6} b^{4} x^{4} + 150150 \, a^{7} b^{3} x^{3} + 51975 \, a^{8} b^{2} x^{2} + 10725 \, a^{9} b x + 1001 \, a^{10}}{15015 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.84367, size = 121, normalized size = 1.26 \begin{align*} - \frac{1001 a^{10} + 10725 a^{9} b x + 51975 a^{8} b^{2} x^{2} + 150150 a^{7} b^{3} x^{3} + 286650 a^{6} b^{4} x^{4} + 378378 a^{5} b^{5} x^{5} + 350350 a^{4} b^{6} x^{6} + 225225 a^{3} b^{7} x^{7} + 96525 a^{2} b^{8} x^{8} + 25025 a b^{9} x^{9} + 3003 b^{10} x^{10}}{15015 x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15674, size = 151, normalized size = 1.57 \begin{align*} -\frac{3003 \, b^{10} x^{10} + 25025 \, a b^{9} x^{9} + 96525 \, a^{2} b^{8} x^{8} + 225225 \, a^{3} b^{7} x^{7} + 350350 \, a^{4} b^{6} x^{6} + 378378 \, a^{5} b^{5} x^{5} + 286650 \, a^{6} b^{4} x^{4} + 150150 \, a^{7} b^{3} x^{3} + 51975 \, a^{8} b^{2} x^{2} + 10725 \, a^{9} b x + 1001 \, a^{10}}{15015 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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