Optimal. Leaf size=121 \[ -\frac {a^4 (A b-a B)}{3 b^6 (a+b x)^3}+\frac {a^3 (4 A b-5 a B)}{2 b^6 (a+b x)^2}-\frac {2 a^2 (3 A b-5 a B)}{b^6 (a+b x)}-\frac {2 a (2 A b-5 a B) \log (a+b x)}{b^6}+\frac {x (A b-4 a B)}{b^5}+\frac {B x^2}{2 b^4} \]
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Rubi [A] time = 0.12, antiderivative size = 121, 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^4 (A b-a B)}{3 b^6 (a+b x)^3}+\frac {a^3 (4 A b-5 a B)}{2 b^6 (a+b x)^2}-\frac {2 a^2 (3 A b-5 a B)}{b^6 (a+b x)}+\frac {x (A b-4 a B)}{b^5}-\frac {2 a (2 A b-5 a B) \log (a+b x)}{b^6}+\frac {B x^2}{2 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {x^4 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {x^4 (A+B x)}{(a+b x)^4} \, dx\\ &=\int \left (\frac {A b-4 a B}{b^5}+\frac {B x}{b^4}-\frac {a^4 (-A b+a B)}{b^5 (a+b x)^4}+\frac {a^3 (-4 A b+5 a B)}{b^5 (a+b x)^3}-\frac {2 a^2 (-3 A b+5 a B)}{b^5 (a+b x)^2}+\frac {2 a (-2 A b+5 a B)}{b^5 (a+b x)}\right ) \, dx\\ &=\frac {(A b-4 a B) x}{b^5}+\frac {B x^2}{2 b^4}-\frac {a^4 (A b-a B)}{3 b^6 (a+b x)^3}+\frac {a^3 (4 A b-5 a B)}{2 b^6 (a+b x)^2}-\frac {2 a^2 (3 A b-5 a B)}{b^6 (a+b x)}-\frac {2 a (2 A b-5 a B) \log (a+b x)}{b^6}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 129, normalized size = 1.07 \begin {gather*} \frac {2 \left (5 a^2 B-2 a A b\right ) \log (a+b x)}{b^6}+\frac {a^5 B-a^4 A b}{3 b^6 (a+b x)^3}+\frac {4 a^3 A b-5 a^4 B}{2 b^6 (a+b x)^2}+\frac {2 \left (5 a^3 B-3 a^2 A b\right )}{b^6 (a+b x)}+\frac {x (A b-4 a B)}{b^5}+\frac {B x^2}{2 b^4} \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^4 (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 = 0.41, size = 231, normalized size = 1.91 \begin {gather*} \frac {3 \, B b^{5} x^{5} + 47 \, B a^{5} - 26 \, A a^{4} b - 3 \, {\left (5 \, B a b^{4} - 2 \, A b^{5}\right )} x^{4} - 9 \, {\left (7 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} - 9 \, {\left (B a^{3} b^{2} + 2 \, A a^{2} b^{3}\right )} x^{2} + 27 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x + 12 \, {\left (5 \, B a^{5} - 2 \, A a^{4} b + {\left (5 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{3} + 3 \, {\left (5 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 3 \, {\left (5 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \log \left (b x + a\right )}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 124, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac {B b^{4} x^{2} - 8 \, B a b^{3} x + 2 \, A b^{4} x}{2 \, b^{8}} + \frac {47 \, B a^{5} - 26 \, A a^{4} b + 12 \, {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 15 \, {\left (7 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{6 \, {\left (b x + a\right )}^{3} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 149, normalized size = 1.23 \begin {gather*} -\frac {A \,a^{4}}{3 \left (b x +a \right )^{3} b^{5}}+\frac {B \,a^{5}}{3 \left (b x +a \right )^{3} b^{6}}+\frac {2 A \,a^{3}}{\left (b x +a \right )^{2} b^{5}}-\frac {5 B \,a^{4}}{2 \left (b x +a \right )^{2} b^{6}}+\frac {B \,x^{2}}{2 b^{4}}-\frac {6 A \,a^{2}}{\left (b x +a \right ) b^{5}}-\frac {4 A a \ln \left (b x +a \right )}{b^{5}}+\frac {A x}{b^{4}}+\frac {10 B \,a^{3}}{\left (b x +a \right ) b^{6}}+\frac {10 B \,a^{2} \ln \left (b x +a \right )}{b^{6}}-\frac {4 B a x}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 143, normalized size = 1.18 \begin {gather*} \frac {47 \, B a^{5} - 26 \, A a^{4} b + 12 \, {\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{2} + 15 \, {\left (7 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x}{6 \, {\left (b^{9} x^{3} + 3 \, a b^{8} x^{2} + 3 \, a^{2} b^{7} x + a^{3} b^{6}\right )}} + \frac {B b x^{2} - 2 \, {\left (4 \, B a - A b\right )} x}{2 \, b^{5}} + \frac {2 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} \log \left (b x + a\right )}{b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 141, normalized size = 1.17 \begin {gather*} x\,\left (\frac {A}{b^4}-\frac {4\,B\,a}{b^5}\right )+\frac {x\,\left (\frac {35\,B\,a^4}{2}-10\,A\,a^3\,b\right )-x^2\,\left (6\,A\,a^2\,b^2-10\,B\,a^3\,b\right )+\frac {47\,B\,a^5-26\,A\,a^4\,b}{6\,b}}{a^3\,b^5+3\,a^2\,b^6\,x+3\,a\,b^7\,x^2+b^8\,x^3}+\frac {B\,x^2}{2\,b^4}+\frac {\ln \left (a+b\,x\right )\,\left (10\,B\,a^2-4\,A\,a\,b\right )}{b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.15, size = 144, normalized size = 1.19 \begin {gather*} \frac {B x^{2}}{2 b^{4}} + \frac {2 a \left (- 2 A b + 5 B a\right ) \log {\left (a + b x \right )}}{b^{6}} + x \left (\frac {A}{b^{4}} - \frac {4 B a}{b^{5}}\right ) + \frac {- 26 A a^{4} b + 47 B a^{5} + x^{2} \left (- 36 A a^{2} b^{3} + 60 B a^{3} b^{2}\right ) + x \left (- 60 A a^{3} b^{2} + 105 B a^{4} b\right )}{6 a^{3} b^{6} + 18 a^{2} b^{7} x + 18 a b^{8} x^{2} + 6 b^{9} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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