Optimal. Leaf size=33 \[ -\frac {a^2}{b^3 (a+b x)}-\frac {2 a \log (a+b x)}{b^3}+\frac {x}{b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 33, 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}{b^3 (a+b x)}-\frac {2 a \log (a+b x)}{b^3}+\frac {x}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {x^2}{(a+b x)^2} \, dx &=\int \left (\frac {1}{b^2}+\frac {a^2}{b^2 (a+b x)^2}-\frac {2 a}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {x}{b^2}-\frac {a^2}{b^3 (a+b x)}-\frac {2 a \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.88 \begin {gather*} \frac {-\frac {a^2}{a+b x}-2 a \log (a+b x)+b x}{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)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.28, size = 47, normalized size = 1.42 \begin {gather*} \frac {b^{2} x^{2} + a b x - a^{2} - 2 \, {\left (a b x + a^{2}\right )} \log \left (b x + a\right )}{b^{4} x + a b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 50, normalized size = 1.52 \begin {gather*} \frac {2 \, a \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3}} + \frac {b x + a}{b^{3}} - \frac {a^{2}}{{\left (b x + a\right )} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 34, normalized size = 1.03 \begin {gather*} -\frac {a^{2}}{\left (b x +a \right ) b^{3}}-\frac {2 a \ln \left (b x +a \right )}{b^{3}}+\frac {x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 36, normalized size = 1.09 \begin {gather*} -\frac {a^{2}}{b^{4} x + a b^{3}} + \frac {x}{b^{2}} - \frac {2 \, a \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 = 0.08, size = 36, normalized size = 1.09 \begin {gather*} \frac {x}{b^2}-\frac {a^2}{x\,b^4+a\,b^3}-\frac {2\,a\,\ln \left (a+b\,x\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 31, normalized size = 0.94 \begin {gather*} - \frac {a^{2}}{a b^{3} + b^{4} x} - \frac {2 a \log {\left (a + b x \right )}}{b^{3}} + \frac {x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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