Optimal. Leaf size=143 \[ \frac {a^5 (A b-a B)}{3 b^7 (a+b x)^3}-\frac {a^4 (5 A b-6 a B)}{2 b^7 (a+b x)^2}+\frac {5 a^3 (2 A b-3 a B)}{b^7 (a+b x)}+\frac {10 a^2 (A b-2 a B) \log (a+b x)}{b^7}-\frac {2 a x (2 A b-5 a B)}{b^6}+\frac {x^2 (A b-4 a B)}{2 b^5}+\frac {B x^3}{3 b^4} \]
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Rubi [A] time = 0.16, antiderivative size = 143, 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)}{3 b^7 (a+b x)^3}-\frac {a^4 (5 A b-6 a B)}{2 b^7 (a+b x)^2}+\frac {5 a^3 (2 A b-3 a B)}{b^7 (a+b x)}+\frac {10 a^2 (A b-2 a B) \log (a+b x)}{b^7}+\frac {x^2 (A b-4 a B)}{2 b^5}-\frac {2 a x (2 A b-5 a B)}{b^6}+\frac {B x^3}{3 b^4} \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 )^2} \, dx &=\int \frac {x^5 (A+B x)}{(a+b x)^4} \, dx\\ &=\int \left (\frac {2 a (-2 A b+5 a B)}{b^6}+\frac {(A b-4 a B) x}{b^5}+\frac {B x^2}{b^4}+\frac {a^5 (-A b+a B)}{b^6 (a+b x)^4}-\frac {a^4 (-5 A b+6 a B)}{b^6 (a+b x)^3}+\frac {5 a^3 (-2 A b+3 a B)}{b^6 (a+b x)^2}-\frac {10 a^2 (-A b+2 a B)}{b^6 (a+b x)}\right ) \, dx\\ &=-\frac {2 a (2 A b-5 a B) x}{b^6}+\frac {(A b-4 a B) x^2}{2 b^5}+\frac {B x^3}{3 b^4}+\frac {a^5 (A b-a B)}{3 b^7 (a+b x)^3}-\frac {a^4 (5 A b-6 a B)}{2 b^7 (a+b x)^2}+\frac {5 a^3 (2 A b-3 a B)}{b^7 (a+b x)}+\frac {10 a^2 (A b-2 a B) \log (a+b x)}{b^7}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 147, normalized size = 1.03 \begin {gather*} \frac {-74 a^6 B+a^5 b (47 A-102 B x)+3 a^4 b^2 x (27 A+26 B x)+a^3 b^3 x^2 (146 B x-9 A)+3 a^2 b^4 x^3 (10 B x-21 A)-60 a^2 (a+b x)^3 (2 a B-A b) \log (a+b x)-3 a b^5 x^4 (5 A+2 B x)+b^6 x^5 (3 A+2 B x)}{6 b^7 (a+b x)^3} \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 )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.40, size = 257, normalized size = 1.80 \begin {gather*} \frac {2 \, B b^{6} x^{6} - 74 \, B a^{6} + 47 \, A a^{5} b - 3 \, {\left (2 \, B a b^{5} - A b^{6}\right )} x^{5} + 15 \, {\left (2 \, B a^{2} b^{4} - A a b^{5}\right )} x^{4} + {\left (146 \, B a^{3} b^{3} - 63 \, A a^{2} b^{4}\right )} x^{3} + 3 \, {\left (26 \, B a^{4} b^{2} - 3 \, A a^{3} b^{3}\right )} x^{2} - 3 \, {\left (34 \, B a^{5} b - 27 \, A a^{4} b^{2}\right )} x - 60 \, {\left (2 \, B a^{6} - A a^{5} b + {\left (2 \, B a^{3} b^{3} - A a^{2} b^{4}\right )} x^{3} + 3 \, {\left (2 \, B a^{4} b^{2} - A a^{3} b^{3}\right )} x^{2} + 3 \, {\left (2 \, B a^{5} b - A a^{4} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} 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 = 149, normalized size = 1.04 \begin {gather*} -\frac {10 \, {\left (2 \, B a^{3} - A a^{2} b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {74 \, B a^{6} - 47 \, A a^{5} b + 30 \, {\left (3 \, B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{2} + 3 \, {\left (54 \, B a^{5} b - 35 \, A a^{4} b^{2}\right )} x}{6 \, {\left (b x + a\right )}^{3} b^{7}} + \frac {2 \, B b^{8} x^{3} - 12 \, B a b^{7} x^{2} + 3 \, A b^{8} x^{2} + 60 \, B a^{2} b^{6} x - 24 \, A a b^{7} x}{6 \, b^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 174, normalized size = 1.22 \begin {gather*} \frac {A \,a^{5}}{3 \left (b x +a \right )^{3} b^{6}}-\frac {B \,a^{6}}{3 \left (b x +a \right )^{3} b^{7}}+\frac {B \,x^{3}}{3 b^{4}}-\frac {5 A \,a^{4}}{2 \left (b x +a \right )^{2} b^{6}}+\frac {A \,x^{2}}{2 b^{4}}+\frac {3 B \,a^{5}}{\left (b x +a \right )^{2} b^{7}}-\frac {2 B a \,x^{2}}{b^{5}}+\frac {10 A \,a^{3}}{\left (b x +a \right ) b^{6}}+\frac {10 A \,a^{2} \ln \left (b x +a \right )}{b^{6}}-\frac {4 A a x}{b^{5}}-\frac {15 B \,a^{4}}{\left (b x +a \right ) b^{7}}-\frac {20 B \,a^{3} \ln \left (b x +a \right )}{b^{7}}+\frac {10 B \,a^{2} x}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 168, normalized size = 1.17 \begin {gather*} -\frac {74 \, B a^{6} - 47 \, A a^{5} b + 30 \, {\left (3 \, B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{2} + 3 \, {\left (54 \, B a^{5} b - 35 \, A a^{4} b^{2}\right )} x}{6 \, {\left (b^{10} x^{3} + 3 \, a b^{9} x^{2} + 3 \, a^{2} b^{8} x + a^{3} b^{7}\right )}} + \frac {2 \, B b^{2} x^{3} - 3 \, {\left (4 \, B a b - A b^{2}\right )} x^{2} + 12 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} x}{6 \, b^{6}} - \frac {10 \, {\left (2 \, B a^{3} - A a^{2} 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 = 1.11, size = 180, normalized size = 1.26 \begin {gather*} x^2\,\left (\frac {A}{2\,b^4}-\frac {2\,B\,a}{b^5}\right )-\frac {x\,\left (27\,B\,a^5-\frac {35\,A\,a^4\,b}{2}\right )-x^2\,\left (10\,A\,a^3\,b^2-15\,B\,a^4\,b\right )+\frac {74\,B\,a^6-47\,A\,a^5\,b}{6\,b}}{a^3\,b^6+3\,a^2\,b^7\,x+3\,a\,b^8\,x^2+b^9\,x^3}-x\,\left (\frac {4\,a\,\left (\frac {A}{b^4}-\frac {4\,B\,a}{b^5}\right )}{b}+\frac {6\,B\,a^2}{b^6}\right )-\frac {\ln \left (a+b\,x\right )\,\left (20\,B\,a^3-10\,A\,a^2\,b\right )}{b^7}+\frac {B\,x^3}{3\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.25, size = 168, normalized size = 1.17 \begin {gather*} \frac {B x^{3}}{3 b^{4}} - \frac {10 a^{2} \left (- A b + 2 B a\right ) \log {\left (a + b x \right )}}{b^{7}} + x^{2} \left (\frac {A}{2 b^{4}} - \frac {2 B a}{b^{5}}\right ) + x \left (- \frac {4 A a}{b^{5}} + \frac {10 B a^{2}}{b^{6}}\right ) + \frac {47 A a^{5} b - 74 B a^{6} + x^{2} \left (60 A a^{3} b^{3} - 90 B a^{4} b^{2}\right ) + x \left (105 A a^{4} b^{2} - 162 B a^{5} b\right )}{6 a^{3} b^{7} + 18 a^{2} b^{8} x + 18 a b^{9} x^{2} + 6 b^{10} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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