Optimal. Leaf size=59 \[ -\frac {A b-2 a B}{2 b^3 (a+b x)^2}+\frac {a (A b-a B)}{3 b^3 (a+b x)^3}-\frac {B}{b^3 (a+b x)} \]
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Rubi [A] time = 0.04, antiderivative size = 59, 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} \[ -\frac {A b-2 a B}{2 b^3 (a+b x)^2}+\frac {a (A b-a B)}{3 b^3 (a+b x)^3}-\frac {B}{b^3 (a+b x)} \]
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 )^2} \, dx &=\int \frac {x (A+B x)}{(a+b x)^4} \, dx\\ &=\int \left (\frac {a (-A b+a B)}{b^2 (a+b x)^4}+\frac {A b-2 a B}{b^2 (a+b x)^3}+\frac {B}{b^2 (a+b x)^2}\right ) \, dx\\ &=\frac {a (A b-a B)}{3 b^3 (a+b x)^3}-\frac {A b-2 a B}{2 b^3 (a+b x)^2}-\frac {B}{b^3 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.71 \[ -\frac {2 a^2 B+a b (A+6 B x)+3 b^2 x (A+2 B x)}{6 b^3 (a+b x)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 71, normalized size = 1.20 \[ -\frac {6 \, B b^{2} x^{2} + 2 \, B a^{2} + A a b + 3 \, {\left (2 \, B a b + A b^{2}\right )} x}{6 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 45, normalized size = 0.76 \[ -\frac {6 \, B b^{2} x^{2} + 6 \, B a b x + 3 \, A b^{2} x + 2 \, B a^{2} + A a b}{6 \, {\left (b x + a\right )}^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 56, normalized size = 0.95 \[ -\frac {B}{\left (b x +a \right ) b^{3}}+\frac {\left (A b -B a \right ) a}{3 \left (b x +a \right )^{3} b^{3}}-\frac {A b -2 B a}{2 \left (b x +a \right )^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 71, normalized size = 1.20 \[ -\frac {6 \, B b^{2} x^{2} + 2 \, B a^{2} + A a b + 3 \, {\left (2 \, B a b + A b^{2}\right )} x}{6 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 68, normalized size = 1.15 \[ -\frac {\frac {B\,x^2}{b}+\frac {a\,\left (A\,b+2\,B\,a\right )}{6\,b^3}+\frac {x\,\left (A\,b+2\,B\,a\right )}{2\,b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 75, normalized size = 1.27 \[ \frac {- A a b - 2 B a^{2} - 6 B b^{2} x^{2} + x \left (- 3 A b^{2} - 6 B a b\right )}{6 a^{3} b^{3} + 18 a^{2} b^{4} x + 18 a b^{5} x^{2} + 6 b^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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