Optimal. Leaf size=70 \[ -\frac {b (a+b x)^6 (A b-4 a B)}{168 a^3 x^6}+\frac {(a+b x)^6 (A b-4 a B)}{28 a^2 x^7}-\frac {A (a+b x)^6}{8 a x^8} \]
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Rubi [A] time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {78, 45, 37} \[ -\frac {b (a+b x)^6 (A b-4 a B)}{168 a^3 x^6}+\frac {(a+b x)^6 (A b-4 a B)}{28 a^2 x^7}-\frac {A (a+b x)^6}{8 a x^8} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx &=-\frac {A (a+b x)^6}{8 a x^8}+\frac {(-2 A b+8 a B) \int \frac {(a+b x)^5}{x^8} \, dx}{8 a}\\ &=-\frac {A (a+b x)^6}{8 a x^8}+\frac {(A b-4 a B) (a+b x)^6}{28 a^2 x^7}+\frac {(b (A b-4 a B)) \int \frac {(a+b x)^5}{x^7} \, dx}{28 a^2}\\ &=-\frac {A (a+b x)^6}{8 a x^8}+\frac {(A b-4 a B) (a+b x)^6}{28 a^2 x^7}-\frac {b (A b-4 a B) (a+b x)^6}{168 a^3 x^6}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 107, normalized size = 1.53 \[ -\frac {3 a^5 (7 A+8 B x)+20 a^4 b x (6 A+7 B x)+56 a^3 b^2 x^2 (5 A+6 B x)+84 a^2 b^3 x^3 (4 A+5 B x)+70 a b^4 x^4 (3 A+4 B x)+28 b^5 x^5 (2 A+3 B x)}{168 x^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 119, normalized size = 1.70 \[ -\frac {84 \, B b^{5} x^{6} + 21 \, A a^{5} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 210 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 24 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{168 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.31, size = 123, normalized size = 1.76 \[ -\frac {84 \, B b^{5} x^{6} + 280 \, B a b^{4} x^{5} + 56 \, A b^{5} x^{5} + 420 \, B a^{2} b^{3} x^{4} + 210 \, A a b^{4} x^{4} + 336 \, B a^{3} b^{2} x^{3} + 336 \, A a^{2} b^{3} x^{3} + 140 \, B a^{4} b x^{2} + 280 \, A a^{3} b^{2} x^{2} + 24 \, B a^{5} x + 120 \, A a^{4} b x + 21 \, A a^{5}}{168 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 104, normalized size = 1.49 \[ -\frac {B \,b^{5}}{2 x^{2}}-\frac {\left (A b +5 B a \right ) b^{4}}{3 x^{3}}-\frac {5 \left (A b +2 B a \right ) a \,b^{3}}{4 x^{4}}-\frac {2 \left (A b +B a \right ) a^{2} b^{2}}{x^{5}}-\frac {A \,a^{5}}{8 x^{8}}-\frac {5 \left (2 A b +B a \right ) a^{3} b}{6 x^{6}}-\frac {\left (5 A b +B a \right ) a^{4}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 119, normalized size = 1.70 \[ -\frac {84 \, B b^{5} x^{6} + 21 \, A a^{5} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 210 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 24 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{168 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 120, normalized size = 1.71 \[ -\frac {x\,\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )+\frac {A\,a^5}{8}+x^4\,\left (\frac {5\,B\,a^2\,b^3}{2}+\frac {5\,A\,a\,b^4}{4}\right )+x^2\,\left (\frac {5\,B\,a^4\,b}{6}+\frac {5\,A\,a^3\,b^2}{3}\right )+x^5\,\left (\frac {A\,b^5}{3}+\frac {5\,B\,a\,b^4}{3}\right )+x^3\,\left (2\,B\,a^3\,b^2+2\,A\,a^2\,b^3\right )+\frac {B\,b^5\,x^6}{2}}{x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.57, size = 133, normalized size = 1.90 \[ \frac {- 21 A a^{5} - 84 B b^{5} x^{6} + x^{5} \left (- 56 A b^{5} - 280 B a b^{4}\right ) + x^{4} \left (- 210 A a b^{4} - 420 B a^{2} b^{3}\right ) + x^{3} \left (- 336 A a^{2} b^{3} - 336 B a^{3} b^{2}\right ) + x^{2} \left (- 280 A a^{3} b^{2} - 140 B a^{4} b\right ) + x \left (- 120 A a^{4} b - 24 B a^{5}\right )}{168 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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