Optimal. Leaf size=101 \[ -\frac {a^6 B}{6 x^6}-\frac {6 a^5 b B}{5 x^5}-\frac {15 a^4 b^2 B}{4 x^4}-\frac {20 a^3 b^3 B}{3 x^3}-\frac {15 a^2 b^4 B}{2 x^2}-\frac {A (a+b x)^7}{7 a x^7}-\frac {6 a b^5 B}{x}+b^6 B \log (x) \]
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Rubi [A] time = 0.04, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 78, 43} \begin {gather*} -\frac {15 a^2 b^4 B}{2 x^2}-\frac {20 a^3 b^3 B}{3 x^3}-\frac {15 a^4 b^2 B}{4 x^4}-\frac {6 a^5 b B}{5 x^5}-\frac {a^6 B}{6 x^6}-\frac {A (a+b x)^7}{7 a x^7}-\frac {6 a b^5 B}{x}+b^6 B \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rule 78
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^8} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{x^8} \, dx\\ &=-\frac {A (a+b x)^7}{7 a x^7}+B \int \frac {(a+b x)^6}{x^7} \, dx\\ &=-\frac {A (a+b x)^7}{7 a x^7}+B \int \left (\frac {a^6}{x^7}+\frac {6 a^5 b}{x^6}+\frac {15 a^4 b^2}{x^5}+\frac {20 a^3 b^3}{x^4}+\frac {15 a^2 b^4}{x^3}+\frac {6 a b^5}{x^2}+\frac {b^6}{x}\right ) \, dx\\ &=-\frac {a^6 B}{6 x^6}-\frac {6 a^5 b B}{5 x^5}-\frac {15 a^4 b^2 B}{4 x^4}-\frac {20 a^3 b^3 B}{3 x^3}-\frac {15 a^2 b^4 B}{2 x^2}-\frac {6 a b^5 B}{x}-\frac {A (a+b x)^7}{7 a x^7}+b^6 B \log (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 132, normalized size = 1.31 \begin {gather*} -\frac {a^6 (6 A+7 B x)}{42 x^7}-\frac {a^5 b (5 A+6 B x)}{5 x^6}-\frac {3 a^4 b^2 (4 A+5 B x)}{4 x^5}-\frac {5 a^3 b^3 (3 A+4 B x)}{3 x^4}-\frac {5 a^2 b^4 (2 A+3 B x)}{2 x^3}-\frac {3 a b^5 (A+2 B x)}{x^2}-\frac {A b^6}{x}+b^6 B \log (x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 149, normalized size = 1.48 \begin {gather*} \frac {420 \, B b^{6} x^{7} \log \relax (x) - 60 \, A a^{6} - 420 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} - 630 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} - 700 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 70 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{420 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 147, normalized size = 1.46 \begin {gather*} B b^{6} \log \left ({\left | x \right |}\right ) - \frac {60 \, A a^{6} + 420 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 630 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 700 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 70 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{420 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 148, normalized size = 1.47 \begin {gather*} B \,b^{6} \ln \relax (x )-\frac {A \,b^{6}}{x}-\frac {6 B a \,b^{5}}{x}-\frac {3 A a \,b^{5}}{x^{2}}-\frac {15 B \,a^{2} b^{4}}{2 x^{2}}-\frac {5 A \,a^{2} b^{4}}{x^{3}}-\frac {20 B \,a^{3} b^{3}}{3 x^{3}}-\frac {5 A \,a^{3} b^{3}}{x^{4}}-\frac {15 B \,a^{4} b^{2}}{4 x^{4}}-\frac {3 A \,a^{4} b^{2}}{x^{5}}-\frac {6 B \,a^{5} b}{5 x^{5}}-\frac {A \,a^{5} b}{x^{6}}-\frac {B \,a^{6}}{6 x^{6}}-\frac {A \,a^{6}}{7 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 146, normalized size = 1.45 \begin {gather*} B b^{6} \log \relax (x) - \frac {60 \, A a^{6} + 420 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 630 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 700 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 252 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 70 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x}{420 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 140, normalized size = 1.39 \begin {gather*} B\,b^6\,\ln \relax (x)-\frac {x\,\left (\frac {B\,a^6}{6}+A\,b\,a^5\right )+\frac {A\,a^6}{7}+x^2\,\left (\frac {6\,B\,a^5\,b}{5}+3\,A\,a^4\,b^2\right )+x^5\,\left (\frac {15\,B\,a^2\,b^4}{2}+3\,A\,a\,b^5\right )+x^6\,\left (A\,b^6+6\,B\,a\,b^5\right )+x^3\,\left (\frac {15\,B\,a^4\,b^2}{4}+5\,A\,a^3\,b^3\right )+x^4\,\left (\frac {20\,B\,a^3\,b^3}{3}+5\,A\,a^2\,b^4\right )}{x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.37, size = 156, normalized size = 1.54 \begin {gather*} B b^{6} \log {\relax (x )} + \frac {- 60 A a^{6} + x^{6} \left (- 420 A b^{6} - 2520 B a b^{5}\right ) + x^{5} \left (- 1260 A a b^{5} - 3150 B a^{2} b^{4}\right ) + x^{4} \left (- 2100 A a^{2} b^{4} - 2800 B a^{3} b^{3}\right ) + x^{3} \left (- 2100 A a^{3} b^{3} - 1575 B a^{4} b^{2}\right ) + x^{2} \left (- 1260 A a^{4} b^{2} - 504 B a^{5} b\right ) + x \left (- 420 A a^{5} b - 70 B a^{6}\right )}{420 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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