Optimal. Leaf size=146 \[ \frac {a^5 (A b-a B)}{5 b^7 (a+b x)^5}-\frac {a^4 (5 A b-6 a B)}{4 b^7 (a+b x)^4}+\frac {5 a^3 (2 A b-3 a B)}{3 b^7 (a+b x)^3}-\frac {5 a^2 (A b-2 a B)}{b^7 (a+b x)^2}+\frac {5 a (A b-3 a B)}{b^7 (a+b x)}+\frac {(A b-6 a B) \log (a+b x)}{b^7}+\frac {B x}{b^6} \]
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Rubi [A] time = 0.16, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 77} \begin {gather*} \frac {a^5 (A b-a B)}{5 b^7 (a+b x)^5}-\frac {a^4 (5 A b-6 a B)}{4 b^7 (a+b x)^4}+\frac {5 a^3 (2 A b-3 a B)}{3 b^7 (a+b x)^3}-\frac {5 a^2 (A b-2 a B)}{b^7 (a+b x)^2}+\frac {5 a (A b-3 a B)}{b^7 (a+b x)}+\frac {(A b-6 a B) \log (a+b x)}{b^7}+\frac {B x}{b^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {x^5 (A+B x)}{(a+b x)^6} \, dx\\ &=\int \left (\frac {B}{b^6}+\frac {a^5 (-A b+a B)}{b^6 (a+b x)^6}-\frac {a^4 (-5 A b+6 a B)}{b^6 (a+b x)^5}+\frac {5 a^3 (-2 A b+3 a B)}{b^6 (a+b x)^4}-\frac {10 a^2 (-A b+2 a B)}{b^6 (a+b x)^3}+\frac {5 a (-A b+3 a B)}{b^6 (a+b x)^2}+\frac {A b-6 a B}{b^6 (a+b x)}\right ) \, dx\\ &=\frac {B x}{b^6}+\frac {a^5 (A b-a B)}{5 b^7 (a+b x)^5}-\frac {a^4 (5 A b-6 a B)}{4 b^7 (a+b x)^4}+\frac {5 a^3 (2 A b-3 a B)}{3 b^7 (a+b x)^3}-\frac {5 a^2 (A b-2 a B)}{b^7 (a+b x)^2}+\frac {5 a (A b-3 a B)}{b^7 (a+b x)}+\frac {(A b-6 a B) \log (a+b x)}{b^7}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 130, normalized size = 0.89 \begin {gather*} \frac {\frac {12 a^5 (A b-a B)}{(a+b x)^5}+\frac {15 a^4 (6 a B-5 A b)}{(a+b x)^4}+\frac {100 a^3 (2 A b-3 a B)}{(a+b x)^3}+\frac {300 a^2 (2 a B-A b)}{(a+b x)^2}+\frac {300 a (A b-3 a B)}{a+b x}+60 (A b-6 a B) \log (a+b x)+60 b B x}{60 b^7} \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 {x^5 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, size = 311, normalized size = 2.13 \begin {gather*} \frac {60 \, B b^{6} x^{6} + 300 \, B a b^{5} x^{5} - 522 \, B a^{6} + 137 \, A a^{5} b - 300 \, {\left (B a^{2} b^{4} - A a b^{5}\right )} x^{4} - 300 \, {\left (8 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} - 100 \, {\left (36 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} - 125 \, {\left (18 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x - 60 \, {\left (6 \, B a^{6} - A a^{5} b + {\left (6 \, B a b^{5} - A b^{6}\right )} x^{5} + 5 \, {\left (6 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 10 \, {\left (6 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 10 \, {\left (6 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (6 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \log \left (b x + a\right )}{60 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 144, normalized size = 0.99 \begin {gather*} \frac {B x}{b^{6}} - \frac {{\left (6 \, B a - A b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {522 \, B a^{6} - 137 \, A a^{5} b + 300 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 300 \, {\left (10 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 100 \, {\left (39 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (462 \, B a^{5} b - 125 \, A a^{4} b^{2}\right )} x}{60 \, {\left (b x + a\right )}^{5} b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 190, normalized size = 1.30 \begin {gather*} \frac {A \,a^{5}}{5 \left (b x +a \right )^{5} b^{6}}-\frac {B \,a^{6}}{5 \left (b x +a \right )^{5} b^{7}}-\frac {5 A \,a^{4}}{4 \left (b x +a \right )^{4} b^{6}}+\frac {3 B \,a^{5}}{2 \left (b x +a \right )^{4} b^{7}}+\frac {10 A \,a^{3}}{3 \left (b x +a \right )^{3} b^{6}}-\frac {5 B \,a^{4}}{\left (b x +a \right )^{3} b^{7}}-\frac {5 A \,a^{2}}{\left (b x +a \right )^{2} b^{6}}+\frac {10 B \,a^{3}}{\left (b x +a \right )^{2} b^{7}}+\frac {5 A a}{\left (b x +a \right ) b^{6}}+\frac {A \ln \left (b x +a \right )}{b^{6}}-\frac {15 B \,a^{2}}{\left (b x +a \right ) b^{7}}-\frac {6 B a \ln \left (b x +a \right )}{b^{7}}+\frac {B x}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 190, normalized size = 1.30 \begin {gather*} -\frac {522 \, B a^{6} - 137 \, A a^{5} b + 300 \, {\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + 300 \, {\left (10 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} x^{3} + 100 \, {\left (39 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{2} + 5 \, {\left (462 \, B a^{5} b - 125 \, A a^{4} b^{2}\right )} x}{60 \, {\left (b^{12} x^{5} + 5 \, a b^{11} x^{4} + 10 \, a^{2} b^{10} x^{3} + 10 \, a^{3} b^{9} x^{2} + 5 \, a^{4} b^{8} x + a^{5} b^{7}\right )}} + \frac {B x}{b^{6}} - \frac {{\left (6 \, B a - A b\right )} \log \left (b x + a\right )}{b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 187, normalized size = 1.28 \begin {gather*} \frac {B\,x}{b^6}-\frac {x\,\left (\frac {77\,B\,a^5}{2}-\frac {125\,A\,a^4\,b}{12}\right )+x^4\,\left (15\,B\,a^2\,b^3-5\,A\,a\,b^4\right )-x^2\,\left (\frac {55\,A\,a^3\,b^2}{3}-65\,B\,a^4\,b\right )+\frac {522\,B\,a^6-137\,A\,a^5\,b}{60\,b}-x^3\,\left (15\,A\,a^2\,b^3-50\,B\,a^3\,b^2\right )}{a^5\,b^6+5\,a^4\,b^7\,x+10\,a^3\,b^8\,x^2+10\,a^2\,b^9\,x^3+5\,a\,b^{10}\,x^4+b^{11}\,x^5}+\frac {\ln \left (a+b\,x\right )\,\left (A\,b-6\,B\,a\right )}{b^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.48, size = 190, normalized size = 1.30 \begin {gather*} \frac {B x}{b^{6}} + \frac {137 A a^{5} b - 522 B a^{6} + x^{4} \left (300 A a b^{5} - 900 B a^{2} b^{4}\right ) + x^{3} \left (900 A a^{2} b^{4} - 3000 B a^{3} b^{3}\right ) + x^{2} \left (1100 A a^{3} b^{3} - 3900 B a^{4} b^{2}\right ) + x \left (625 A a^{4} b^{2} - 2310 B a^{5} b\right )}{60 a^{5} b^{7} + 300 a^{4} b^{8} x + 600 a^{3} b^{9} x^{2} + 600 a^{2} b^{10} x^{3} + 300 a b^{11} x^{4} + 60 b^{12} x^{5}} - \frac {\left (- A b + 6 B a\right ) \log {\left (a + b x \right )}}{b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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