Optimal. Leaf size=130 \[ -\frac{b^3 (a+b x)^{11} (4 A b-15 a B)}{60060 a^5 x^{11}}+\frac{b^2 (a+b x)^{11} (4 A b-15 a B)}{5460 a^4 x^{12}}-\frac{b (a+b x)^{11} (4 A b-15 a B)}{910 a^3 x^{13}}+\frac{(a+b x)^{11} (4 A b-15 a B)}{210 a^2 x^{14}}-\frac{A (a+b x)^{11}}{15 a x^{15}} \]
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Rubi [A] time = 0.0470627, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {78, 45, 37} \[ -\frac{b^3 (a+b x)^{11} (4 A b-15 a B)}{60060 a^5 x^{11}}+\frac{b^2 (a+b x)^{11} (4 A b-15 a B)}{5460 a^4 x^{12}}-\frac{b (a+b x)^{11} (4 A b-15 a B)}{910 a^3 x^{13}}+\frac{(a+b x)^{11} (4 A b-15 a B)}{210 a^2 x^{14}}-\frac{A (a+b x)^{11}}{15 a x^{15}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^{10} (A+B x)}{x^{16}} \, dx &=-\frac{A (a+b x)^{11}}{15 a x^{15}}+\frac{(-4 A b+15 a B) \int \frac{(a+b x)^{10}}{x^{15}} \, dx}{15 a}\\ &=-\frac{A (a+b x)^{11}}{15 a x^{15}}+\frac{(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}+\frac{(b (4 A b-15 a B)) \int \frac{(a+b x)^{10}}{x^{14}} \, dx}{70 a^2}\\ &=-\frac{A (a+b x)^{11}}{15 a x^{15}}+\frac{(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac{b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}-\frac{\left (b^2 (4 A b-15 a B)\right ) \int \frac{(a+b x)^{10}}{x^{13}} \, dx}{455 a^3}\\ &=-\frac{A (a+b x)^{11}}{15 a x^{15}}+\frac{(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac{b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}+\frac{b^2 (4 A b-15 a B) (a+b x)^{11}}{5460 a^4 x^{12}}+\frac{\left (b^3 (4 A b-15 a B)\right ) \int \frac{(a+b x)^{10}}{x^{12}} \, dx}{5460 a^4}\\ &=-\frac{A (a+b x)^{11}}{15 a x^{15}}+\frac{(4 A b-15 a B) (a+b x)^{11}}{210 a^2 x^{14}}-\frac{b (4 A b-15 a B) (a+b x)^{11}}{910 a^3 x^{13}}+\frac{b^2 (4 A b-15 a B) (a+b x)^{11}}{5460 a^4 x^{12}}-\frac{b^3 (4 A b-15 a B) (a+b x)^{11}}{60060 a^5 x^{11}}\\ \end{align*}
Mathematica [A] time = 0.052496, size = 202, normalized size = 1.55 \[ -\frac{17325 a^8 b^2 x^2 (12 A+13 B x)+54600 a^7 b^3 x^3 (11 A+12 B x)+114660 a^6 b^4 x^4 (10 A+11 B x)+168168 a^5 b^5 x^5 (9 A+10 B x)+175175 a^4 b^6 x^6 (8 A+9 B x)+128700 a^3 b^7 x^7 (7 A+8 B x)+64350 a^2 b^8 x^8 (6 A+7 B x)+3300 a^9 b x (13 A+14 B x)+286 a^{10} (14 A+15 B x)+20020 a b^9 x^9 (5 A+6 B x)+3003 b^{10} x^{10} (4 A+5 B x)}{60060 x^{15}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 208, normalized size = 1.6 \begin{align*} -{\frac{{b}^{9} \left ( Ab+10\,Ba \right ) }{5\,{x}^{5}}}-{\frac{5\,{a}^{7}{b}^{2} \left ( 8\,Ab+3\,Ba \right ) }{4\,{x}^{12}}}-{\frac{A{a}^{10}}{15\,{x}^{15}}}-{\frac{30\,{a}^{6}{b}^{3} \left ( 7\,Ab+4\,Ba \right ) }{11\,{x}^{11}}}-{\frac{B{b}^{10}}{4\,{x}^{4}}}-{\frac{15\,{a}^{3}{b}^{6} \left ( 4\,Ab+7\,Ba \right ) }{4\,{x}^{8}}}-{\frac{5\,{a}^{8}b \left ( 9\,Ab+2\,Ba \right ) }{13\,{x}^{13}}}-{\frac{5\,a{b}^{8} \left ( 2\,Ab+9\,Ba \right ) }{6\,{x}^{6}}}-{\frac{{a}^{9} \left ( 10\,Ab+Ba \right ) }{14\,{x}^{14}}}-{\frac{15\,{a}^{2}{b}^{7} \left ( 3\,Ab+8\,Ba \right ) }{7\,{x}^{7}}}-{\frac{14\,{a}^{4}{b}^{5} \left ( 5\,Ab+6\,Ba \right ) }{3\,{x}^{9}}}-{\frac{21\,{a}^{5}{b}^{4} \left ( 6\,Ab+5\,Ba \right ) }{5\,{x}^{10}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02173, size = 328, normalized size = 2.52 \begin{align*} -\frac{15015 \, B b^{10} x^{11} + 4004 \, A a^{10} + 12012 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 50050 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 128700 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 225225 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 280280 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 252252 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 163800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 75075 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 23100 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4290 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60060 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.46514, size = 589, normalized size = 4.53 \begin{align*} -\frac{15015 \, B b^{10} x^{11} + 4004 \, A a^{10} + 12012 \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{10} + 50050 \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{9} + 128700 \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{8} + 225225 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{7} + 280280 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{6} + 252252 \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{5} + 163800 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{4} + 75075 \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{3} + 23100 \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{2} + 4290 \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x}{60060 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 146.726, size = 245, normalized size = 1.88 \begin{align*} - \frac{4004 A a^{10} + 15015 B b^{10} x^{11} + x^{10} \left (12012 A b^{10} + 120120 B a b^{9}\right ) + x^{9} \left (100100 A a b^{9} + 450450 B a^{2} b^{8}\right ) + x^{8} \left (386100 A a^{2} b^{8} + 1029600 B a^{3} b^{7}\right ) + x^{7} \left (900900 A a^{3} b^{7} + 1576575 B a^{4} b^{6}\right ) + x^{6} \left (1401400 A a^{4} b^{6} + 1681680 B a^{5} b^{5}\right ) + x^{5} \left (1513512 A a^{5} b^{5} + 1261260 B a^{6} b^{4}\right ) + x^{4} \left (1146600 A a^{6} b^{4} + 655200 B a^{7} b^{3}\right ) + x^{3} \left (600600 A a^{7} b^{3} + 225225 B a^{8} b^{2}\right ) + x^{2} \left (207900 A a^{8} b^{2} + 46200 B a^{9} b\right ) + x \left (42900 A a^{9} b + 4290 B a^{10}\right )}{60060 x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21491, size = 328, normalized size = 2.52 \begin{align*} -\frac{15015 \, B b^{10} x^{11} + 120120 \, B a b^{9} x^{10} + 12012 \, A b^{10} x^{10} + 450450 \, B a^{2} b^{8} x^{9} + 100100 \, A a b^{9} x^{9} + 1029600 \, B a^{3} b^{7} x^{8} + 386100 \, A a^{2} b^{8} x^{8} + 1576575 \, B a^{4} b^{6} x^{7} + 900900 \, A a^{3} b^{7} x^{7} + 1681680 \, B a^{5} b^{5} x^{6} + 1401400 \, A a^{4} b^{6} x^{6} + 1261260 \, B a^{6} b^{4} x^{5} + 1513512 \, A a^{5} b^{5} x^{5} + 655200 \, B a^{7} b^{3} x^{4} + 1146600 \, A a^{6} b^{4} x^{4} + 225225 \, B a^{8} b^{2} x^{3} + 600600 \, A a^{7} b^{3} x^{3} + 46200 \, B a^{9} b x^{2} + 207900 \, A a^{8} b^{2} x^{2} + 4290 \, B a^{10} x + 42900 \, A a^{9} b x + 4004 \, A a^{10}}{60060 \, x^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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