Optimal. Leaf size=15 \[ -\frac {b^2}{d^3 (a+b x)} \]
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Rubi [A] time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 32} \begin {gather*} -\frac {b^2}{d^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin {align*} \int \frac {a+b x}{\left (\frac {a d}{b}+d x\right )^3} \, dx &=\frac {b^3 \int \frac {1}{(a+b x)^2} \, dx}{d^3}\\ &=-\frac {b^2}{d^3 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {b^2}{d^3 (a+b x)} \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 {a+b x}{\left (\frac {a d}{b}+d x\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.34, size = 19, normalized size = 1.27 \begin {gather*} -\frac {b^{2}}{b d^{3} x + a d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 15, normalized size = 1.00 \begin {gather*} -\frac {b^{2}}{{\left (b x + a\right )} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 1.07 \begin {gather*} -\frac {b^{2}}{\left (b x +a \right ) d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 19, normalized size = 1.27 \begin {gather*} -\frac {b^{2}}{b d^{3} x + a d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 15, normalized size = 1.00 \begin {gather*} -\frac {b^2}{d^3\,\left (a+b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 19, normalized size = 1.27 \begin {gather*} - \frac {b^{3}}{a b d^{3} + b^{2} d^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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