Optimal. Leaf size=72 \[ -\frac{b (a+b x)^7 (2 A b-9 a B)}{504 a^3 x^7}+\frac{(a+b x)^7 (2 A b-9 a B)}{72 a^2 x^8}-\frac{A (a+b x)^7}{9 a x^9} \]
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Rubi [A] time = 0.0219145, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {27, 78, 45, 37} \[ -\frac{b (a+b x)^7 (2 A b-9 a B)}{504 a^3 x^7}+\frac{(a+b x)^7 (2 A b-9 a B)}{72 a^2 x^8}-\frac{A (a+b x)^7}{9 a x^9} \]
Antiderivative was successfully verified.
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Rule 27
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{10}} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{x^{10}} \, dx\\ &=-\frac{A (a+b x)^7}{9 a x^9}+\frac{(-2 A b+9 a B) \int \frac{(a+b x)^6}{x^9} \, dx}{9 a}\\ &=-\frac{A (a+b x)^7}{9 a x^9}+\frac{(2 A b-9 a B) (a+b x)^7}{72 a^2 x^8}+\frac{(b (2 A b-9 a B)) \int \frac{(a+b x)^6}{x^8} \, dx}{72 a^2}\\ &=-\frac{A (a+b x)^7}{9 a x^9}+\frac{(2 A b-9 a B) (a+b x)^7}{72 a^2 x^8}-\frac{b (2 A b-9 a B) (a+b x)^7}{504 a^3 x^7}\\ \end{align*}
Mathematica [A] time = 0.0329323, size = 126, normalized size = 1.75 \[ -\frac{180 a^4 b^2 x^2 (6 A+7 B x)+336 a^3 b^3 x^3 (5 A+6 B x)+378 a^2 b^4 x^4 (4 A+5 B x)+54 a^5 b x (7 A+8 B x)+7 a^6 (8 A+9 B x)+252 a b^5 x^5 (3 A+4 B x)+84 b^6 x^6 (2 A+3 B x)}{504 x^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 128, normalized size = 1.8 \begin{align*} -{\frac{{a}^{2}{b}^{3} \left ( 3\,Ab+4\,aB \right ) }{{x}^{5}}}-{\frac{A{a}^{6}}{9\,{x}^{9}}}-{\frac{5\,{a}^{3}{b}^{2} \left ( 4\,Ab+3\,aB \right ) }{6\,{x}^{6}}}-{\frac{{b}^{5} \left ( Ab+6\,aB \right ) }{3\,{x}^{3}}}-{\frac{{a}^{5} \left ( 6\,Ab+aB \right ) }{8\,{x}^{8}}}-{\frac{B{b}^{6}}{2\,{x}^{2}}}-{\frac{3\,{a}^{4}b \left ( 5\,Ab+2\,aB \right ) }{7\,{x}^{7}}}-{\frac{3\,a{b}^{4} \left ( 2\,Ab+5\,aB \right ) }{4\,{x}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00113, size = 198, normalized size = 2.75 \begin{align*} -\frac{252 \, B b^{6} x^{7} + 56 \, A a^{6} + 168 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 378 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 504 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 420 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 216 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 63 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{504 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.31562, size = 329, normalized size = 4.57 \begin{align*} -\frac{252 \, B b^{6} x^{7} + 56 \, A a^{6} + 168 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 378 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 504 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 420 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 216 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 63 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x}{504 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 13.3428, size = 150, normalized size = 2.08 \begin{align*} - \frac{56 A a^{6} + 252 B b^{6} x^{7} + x^{6} \left (168 A b^{6} + 1008 B a b^{5}\right ) + x^{5} \left (756 A a b^{5} + 1890 B a^{2} b^{4}\right ) + x^{4} \left (1512 A a^{2} b^{4} + 2016 B a^{3} b^{3}\right ) + x^{3} \left (1680 A a^{3} b^{3} + 1260 B a^{4} b^{2}\right ) + x^{2} \left (1080 A a^{4} b^{2} + 432 B a^{5} b\right ) + x \left (378 A a^{5} b + 63 B a^{6}\right )}{504 x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18118, size = 198, normalized size = 2.75 \begin{align*} -\frac{252 \, B b^{6} x^{7} + 1008 \, B a b^{5} x^{6} + 168 \, A b^{6} x^{6} + 1890 \, B a^{2} b^{4} x^{5} + 756 \, A a b^{5} x^{5} + 2016 \, B a^{3} b^{3} x^{4} + 1512 \, A a^{2} b^{4} x^{4} + 1260 \, B a^{4} b^{2} x^{3} + 1680 \, A a^{3} b^{3} x^{3} + 432 \, B a^{5} b x^{2} + 1080 \, A a^{4} b^{2} x^{2} + 63 \, B a^{6} x + 378 \, A a^{5} b x + 56 \, A a^{6}}{504 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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