Optimal. Leaf size=41 \[ -\frac {a^2}{2 b^3 (a+b x)^2}+\frac {2 a}{b^3 (a+b x)}+\frac {\log (a+b x)}{b^3} \]
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Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \begin {gather*} -\frac {a^2}{2 b^3 (a+b x)^2}+\frac {2 a}{b^3 (a+b x)}+\frac {\log (a+b x)}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^2}{(a+b x)^3} \, dx &=\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\\ &=-\frac {a^2}{2 b^3 (a+b x)^2}+\frac {2 a}{b^3 (a+b x)}+\frac {\log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.80 \begin {gather*} \frac {\frac {a (3 a+4 b x)}{(a+b x)^2}+2 \log (a+b x)}{2 b^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^2}{(a+b x)^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.65, size = 61, normalized size = 1.49 \begin {gather*} \frac {4 \, a b x + 3 \, a^{2} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 37, normalized size = 0.90 \begin {gather*} \frac {\log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac {4 \, a x + \frac {3 \, a^{2}}{b}}{2 \, {\left (b x + a\right )}^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.98 \begin {gather*} -\frac {a^{2}}{2 \left (b x +a \right )^{2} b^{3}}+\frac {2 a}{\left (b x +a \right ) b^{3}}+\frac {\ln \left (b x +a \right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 48, normalized size = 1.17 \begin {gather*} \frac {4 \, a b x + 3 \, a^{2}}{2 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac {\log \left (b x + a\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 46, normalized size = 1.12 \begin {gather*} \frac {\ln \left (a+b\,x\right )}{b^3}+\frac {\frac {3\,a^2}{2\,b^3}+\frac {2\,a\,x}{b^2}}{a^2+2\,a\,b\,x+b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 46, normalized size = 1.12 \begin {gather*} \frac {3 a^{2} + 4 a b x}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac {\log {\left (a + b x \right )}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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