Optimal. Leaf size=136 \[ -\frac{b^6 (a+b x)^{11}}{136136 a^7 x^{11}}+\frac{b^5 (a+b x)^{11}}{12376 a^6 x^{12}}-\frac{3 b^4 (a+b x)^{11}}{6188 a^5 x^{13}}+\frac{b^3 (a+b x)^{11}}{476 a^4 x^{14}}-\frac{b^2 (a+b x)^{11}}{136 a^3 x^{15}}+\frac{3 b (a+b x)^{11}}{136 a^2 x^{16}}-\frac{(a+b x)^{11}}{17 a x^{17}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0487079, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ -\frac{b^6 (a+b x)^{11}}{136136 a^7 x^{11}}+\frac{b^5 (a+b x)^{11}}{12376 a^6 x^{12}}-\frac{3 b^4 (a+b x)^{11}}{6188 a^5 x^{13}}+\frac{b^3 (a+b x)^{11}}{476 a^4 x^{14}}-\frac{b^2 (a+b x)^{11}}{136 a^3 x^{15}}+\frac{3 b (a+b x)^{11}}{136 a^2 x^{16}}-\frac{(a+b x)^{11}}{17 a x^{17}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^{10}}{x^{18}} \, dx &=-\frac{(a+b x)^{11}}{17 a x^{17}}-\frac{(6 b) \int \frac{(a+b x)^{10}}{x^{17}} \, dx}{17 a}\\ &=-\frac{(a+b x)^{11}}{17 a x^{17}}+\frac{3 b (a+b x)^{11}}{136 a^2 x^{16}}+\frac{\left (15 b^2\right ) \int \frac{(a+b x)^{10}}{x^{16}} \, dx}{136 a^2}\\ &=-\frac{(a+b x)^{11}}{17 a x^{17}}+\frac{3 b (a+b x)^{11}}{136 a^2 x^{16}}-\frac{b^2 (a+b x)^{11}}{136 a^3 x^{15}}-\frac{b^3 \int \frac{(a+b x)^{10}}{x^{15}} \, dx}{34 a^3}\\ &=-\frac{(a+b x)^{11}}{17 a x^{17}}+\frac{3 b (a+b x)^{11}}{136 a^2 x^{16}}-\frac{b^2 (a+b x)^{11}}{136 a^3 x^{15}}+\frac{b^3 (a+b x)^{11}}{476 a^4 x^{14}}+\frac{\left (3 b^4\right ) \int \frac{(a+b x)^{10}}{x^{14}} \, dx}{476 a^4}\\ &=-\frac{(a+b x)^{11}}{17 a x^{17}}+\frac{3 b (a+b x)^{11}}{136 a^2 x^{16}}-\frac{b^2 (a+b x)^{11}}{136 a^3 x^{15}}+\frac{b^3 (a+b x)^{11}}{476 a^4 x^{14}}-\frac{3 b^4 (a+b x)^{11}}{6188 a^5 x^{13}}-\frac{\left (3 b^5\right ) \int \frac{(a+b x)^{10}}{x^{13}} \, dx}{3094 a^5}\\ &=-\frac{(a+b x)^{11}}{17 a x^{17}}+\frac{3 b (a+b x)^{11}}{136 a^2 x^{16}}-\frac{b^2 (a+b x)^{11}}{136 a^3 x^{15}}+\frac{b^3 (a+b x)^{11}}{476 a^4 x^{14}}-\frac{3 b^4 (a+b x)^{11}}{6188 a^5 x^{13}}+\frac{b^5 (a+b x)^{11}}{12376 a^6 x^{12}}+\frac{b^6 \int \frac{(a+b x)^{10}}{x^{12}} \, dx}{12376 a^6}\\ &=-\frac{(a+b x)^{11}}{17 a x^{17}}+\frac{3 b (a+b x)^{11}}{136 a^2 x^{16}}-\frac{b^2 (a+b x)^{11}}{136 a^3 x^{15}}+\frac{b^3 (a+b x)^{11}}{476 a^4 x^{14}}-\frac{3 b^4 (a+b x)^{11}}{6188 a^5 x^{13}}+\frac{b^5 (a+b x)^{11}}{12376 a^6 x^{12}}-\frac{b^6 (a+b x)^{11}}{136136 a^7 x^{11}}\\ \end{align*}
Mathematica [A] time = 0.017388, size = 126, normalized size = 0.93 \[ -\frac{3 a^8 b^2}{x^{15}}-\frac{60 a^7 b^3}{7 x^{14}}-\frac{210 a^6 b^4}{13 x^{13}}-\frac{21 a^5 b^5}{x^{12}}-\frac{210 a^4 b^6}{11 x^{11}}-\frac{12 a^3 b^7}{x^{10}}-\frac{5 a^2 b^8}{x^9}-\frac{5 a^9 b}{8 x^{16}}-\frac{a^{10}}{17 x^{17}}-\frac{5 a b^9}{4 x^8}-\frac{b^{10}}{7 x^7} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 113, normalized size = 0.8 \begin{align*} -12\,{\frac{{a}^{3}{b}^{7}}{{x}^{10}}}-21\,{\frac{{a}^{5}{b}^{5}}{{x}^{12}}}-3\,{\frac{{a}^{8}{b}^{2}}{{x}^{15}}}-{\frac{210\,{a}^{4}{b}^{6}}{11\,{x}^{11}}}-{\frac{{a}^{10}}{17\,{x}^{17}}}-{\frac{5\,a{b}^{9}}{4\,{x}^{8}}}-{\frac{210\,{a}^{6}{b}^{4}}{13\,{x}^{13}}}-{\frac{{b}^{10}}{7\,{x}^{7}}}-{\frac{5\,{a}^{9}b}{8\,{x}^{16}}}-5\,{\frac{{a}^{2}{b}^{8}}{{x}^{9}}}-{\frac{60\,{a}^{7}{b}^{3}}{7\,{x}^{14}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02435, size = 151, normalized size = 1.11 \begin{align*} -\frac{19448 \, b^{10} x^{10} + 170170 \, a b^{9} x^{9} + 680680 \, a^{2} b^{8} x^{8} + 1633632 \, a^{3} b^{7} x^{7} + 2598960 \, a^{4} b^{6} x^{6} + 2858856 \, a^{5} b^{5} x^{5} + 2199120 \, a^{6} b^{4} x^{4} + 1166880 \, a^{7} b^{3} x^{3} + 408408 \, a^{8} b^{2} x^{2} + 85085 \, a^{9} b x + 8008 \, a^{10}}{136136 \, x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.45369, size = 315, normalized size = 2.32 \begin{align*} -\frac{19448 \, b^{10} x^{10} + 170170 \, a b^{9} x^{9} + 680680 \, a^{2} b^{8} x^{8} + 1633632 \, a^{3} b^{7} x^{7} + 2598960 \, a^{4} b^{6} x^{6} + 2858856 \, a^{5} b^{5} x^{5} + 2199120 \, a^{6} b^{4} x^{4} + 1166880 \, a^{7} b^{3} x^{3} + 408408 \, a^{8} b^{2} x^{2} + 85085 \, a^{9} b x + 8008 \, a^{10}}{136136 \, x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.92296, size = 121, normalized size = 0.89 \begin{align*} - \frac{8008 a^{10} + 85085 a^{9} b x + 408408 a^{8} b^{2} x^{2} + 1166880 a^{7} b^{3} x^{3} + 2199120 a^{6} b^{4} x^{4} + 2858856 a^{5} b^{5} x^{5} + 2598960 a^{4} b^{6} x^{6} + 1633632 a^{3} b^{7} x^{7} + 680680 a^{2} b^{8} x^{8} + 170170 a b^{9} x^{9} + 19448 b^{10} x^{10}}{136136 x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1591, size = 151, normalized size = 1.11 \begin{align*} -\frac{19448 \, b^{10} x^{10} + 170170 \, a b^{9} x^{9} + 680680 \, a^{2} b^{8} x^{8} + 1633632 \, a^{3} b^{7} x^{7} + 2598960 \, a^{4} b^{6} x^{6} + 2858856 \, a^{5} b^{5} x^{5} + 2199120 \, a^{6} b^{4} x^{4} + 1166880 \, a^{7} b^{3} x^{3} + 408408 \, a^{8} b^{2} x^{2} + 85085 \, a^{9} b x + 8008 \, a^{10}}{136136 \, x^{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]