Optimal. Leaf size=17 \[ \frac {x^3}{3 a (a+b x)^3} \]
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Rubi [A] time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {37} \begin {gather*} \frac {x^3}{3 a (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin {align*} \int \frac {x^2}{(a+b x)^4} \, dx &=\frac {x^3}{3 a (a+b x)^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.82 \begin {gather*} -\frac {a^2+3 a b x+3 b^2 x^2}{3 b^3 (a+b x)^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)^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.04, size = 54, normalized size = 3.18 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 29, normalized size = 1.71 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b x + a\right )}^{3} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 41, normalized size = 2.41 \begin {gather*} -\frac {a^{2}}{3 \left (b x +a \right )^{3} b^{3}}+\frac {a}{\left (b x +a \right )^{2} b^{3}}-\frac {1}{\left (b x +a \right ) b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.40, size = 54, normalized size = 3.18 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 56, normalized size = 3.29 \begin {gather*} -\frac {a^2+3\,a\,b\,x+3\,b^2\,x^2}{3\,a^3\,b^3+9\,a^2\,b^4\,x+9\,a\,b^5\,x^2+3\,b^6\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.30, size = 56, normalized size = 3.29 \begin {gather*} \frac {- a^{2} - 3 a b x - 3 b^{2} x^{2}}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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