Optimal. Leaf size=72 \[ -\frac {a^2 B}{2 b^4 (a+b x)^2}+\frac {x^3 (A b-a B)}{3 a b (a+b x)^3}+\frac {2 a B}{b^4 (a+b x)}+\frac {B \log (a+b x)}{b^4} \]
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Rubi [A] time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 78, 43} \[ -\frac {a^2 B}{2 b^4 (a+b x)^2}+\frac {x^3 (A b-a B)}{3 a b (a+b x)^3}+\frac {2 a B}{b^4 (a+b x)}+\frac {B \log (a+b x)}{b^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rule 78
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac {x^2 (A+B x)}{(a+b x)^4} \, dx\\ &=\frac {(A b-a B) x^3}{3 a b (a+b x)^3}+\frac {B \int \frac {x^2}{(a+b x)^3} \, dx}{b}\\ &=\frac {(A b-a B) x^3}{3 a b (a+b x)^3}+\frac {B \int \left (\frac {a^2}{b^2 (a+b x)^3}-\frac {2 a}{b^2 (a+b x)^2}+\frac {1}{b^2 (a+b x)}\right ) \, dx}{b}\\ &=\frac {(A b-a B) x^3}{3 a b (a+b x)^3}-\frac {a^2 B}{2 b^4 (a+b x)^2}+\frac {2 a B}{b^4 (a+b x)}+\frac {B \log (a+b x)}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 1.01 \[ \frac {11 a^3 B+a^2 (27 b B x-2 A b)-6 a b^2 x (A-3 B x)+6 B (a+b x)^3 \log (a+b x)-6 A b^3 x^2}{6 b^4 (a+b x)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 128, normalized size = 1.78 \[ \frac {11 \, B a^{3} - 2 \, A a^{2} b + 6 \, {\left (3 \, B a b^{2} - A b^{3}\right )} x^{2} + 3 \, {\left (9 \, B a^{2} b - 2 \, A a b^{2}\right )} x + 6 \, {\left (B b^{3} x^{3} + 3 \, B a b^{2} x^{2} + 3 \, B a^{2} b x + B a^{3}\right )} \log \left (b x + a\right )}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 76, normalized size = 1.06 \[ \frac {B \log \left ({\left | b x + a \right |}\right )}{b^{4}} + \frac {6 \, {\left (3 \, B a b - A b^{2}\right )} x^{2} + 3 \, {\left (9 \, B a^{2} - 2 \, A a b\right )} x + \frac {11 \, B a^{3} - 2 \, A a^{2} b}{b}}{6 \, {\left (b x + a\right )}^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 101, normalized size = 1.40 \[ -\frac {A \,a^{2}}{3 \left (b x +a \right )^{3} b^{3}}+\frac {B \,a^{3}}{3 \left (b x +a \right )^{3} b^{4}}+\frac {A a}{\left (b x +a \right )^{2} b^{3}}-\frac {3 B \,a^{2}}{2 \left (b x +a \right )^{2} b^{4}}-\frac {A}{\left (b x +a \right ) b^{3}}+\frac {3 B a}{\left (b x +a \right ) b^{4}}+\frac {B \ln \left (b x +a \right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 100, normalized size = 1.39 \[ \frac {11 \, B a^{3} - 2 \, A a^{2} b + 6 \, {\left (3 \, B a b^{2} - A b^{3}\right )} x^{2} + 3 \, {\left (9 \, B a^{2} b - 2 \, A a b^{2}\right )} x}{6 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}} + \frac {B \log \left (b x + a\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 96, normalized size = 1.33 \[ \frac {\frac {11\,B\,a^3-2\,A\,a^2\,b}{6\,b^4}-\frac {x^2\,\left (A\,b-3\,B\,a\right )}{b^2}+\frac {x\,\left (9\,B\,a^2-2\,A\,a\,b\right )}{2\,b^3}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3}+\frac {B\,\ln \left (a+b\,x\right )}{b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.66, size = 100, normalized size = 1.39 \[ \frac {B \log {\left (a + b x \right )}}{b^{4}} + \frac {- 2 A a^{2} b + 11 B a^{3} + x^{2} \left (- 6 A b^{3} + 18 B a b^{2}\right ) + x \left (- 6 A a b^{2} + 27 B a^{2} b\right )}{6 a^{3} b^{4} + 18 a^{2} b^{5} x + 18 a b^{6} x^{2} + 6 b^{7} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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