Optimal. Leaf size=61 \[ -\frac {A b-2 a B}{4 b^3 (a+b x)^4}+\frac {a (A b-a B)}{5 b^3 (a+b x)^5}-\frac {B}{3 b^3 (a+b x)^3} \]
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Rubi [A] time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {27, 77} \begin {gather*} -\frac {A b-2 a B}{4 b^3 (a+b x)^4}+\frac {a (A b-a B)}{5 b^3 (a+b x)^5}-\frac {B}{3 b^3 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 77
Rubi steps
\begin {align*} \int \frac {x (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {x (A+B x)}{(a+b x)^6} \, dx\\ &=\int \left (\frac {a (-A b+a B)}{b^2 (a+b x)^6}+\frac {A b-2 a B}{b^2 (a+b x)^5}+\frac {B}{b^2 (a+b x)^4}\right ) \, dx\\ &=\frac {a (A b-a B)}{5 b^3 (a+b x)^5}-\frac {A b-2 a B}{4 b^3 (a+b x)^4}-\frac {B}{3 b^3 (a+b x)^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.75 \begin {gather*} -\frac {2 a^2 B+a b (3 A+10 B x)+5 b^2 x (3 A+4 B x)}{60 b^3 (a+b x)^5} \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 (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 [A] time = 0.40, size = 95, normalized size = 1.56 \begin {gather*} -\frac {20 \, B b^{2} x^{2} + 2 \, B a^{2} + 3 \, A a b + 5 \, {\left (2 \, B a b + 3 \, A b^{2}\right )} x}{60 \, {\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 46, normalized size = 0.75 \begin {gather*} -\frac {20 \, B b^{2} x^{2} + 10 \, B a b x + 15 \, A b^{2} x + 2 \, B a^{2} + 3 \, A a b}{60 \, {\left (b x + a\right )}^{5} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 56, normalized size = 0.92 \begin {gather*} -\frac {B}{3 \left (b x +a \right )^{3} b^{3}}+\frac {\left (A b -B a \right ) a}{5 \left (b x +a \right )^{5} b^{3}}-\frac {A b -2 B a}{4 \left (b x +a \right )^{4} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 95, normalized size = 1.56 \begin {gather*} -\frac {20 \, B b^{2} x^{2} + 2 \, B a^{2} + 3 \, A a b + 5 \, {\left (2 \, B a b + 3 \, A b^{2}\right )} x}{60 \, {\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 93, normalized size = 1.52 \begin {gather*} -\frac {\frac {B\,x^2}{3\,b}+\frac {a\,\left (3\,A\,b+2\,B\,a\right )}{60\,b^3}+\frac {x\,\left (3\,A\,b+2\,B\,a\right )}{12\,b^2}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.68, size = 100, normalized size = 1.64 \begin {gather*} \frac {- 3 A a b - 2 B a^{2} - 20 B b^{2} x^{2} + x \left (- 15 A b^{2} - 10 B a b\right )}{60 a^{5} b^{3} + 300 a^{4} b^{4} x + 600 a^{3} b^{5} x^{2} + 600 a^{2} b^{6} x^{3} + 300 a b^{7} x^{4} + 60 b^{8} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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