Optimal. Leaf size=17 \[ -\frac {b^2}{4 d^3 (a+b x)^4} \]
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Rubi [A] time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32} \begin {gather*} -\frac {b^2}{4 d^3 (a+b x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^2 \left (\frac {a d}{b}+d x\right )^3} \, dx &=\frac {b^3 \int \frac {1}{(a+b x)^5} \, dx}{d^3}\\ &=-\frac {b^2}{4 d^3 (a+b x)^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} -\frac {b^2}{4 d^3 (a+b x)^4} \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 {1}{(a+b x)^2 \left (\frac {a d}{b}+d x\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.36, size = 61, normalized size = 3.59 \begin {gather*} -\frac {b^{2}}{4 \, {\left (b^{4} d^{3} x^{4} + 4 \, a b^{3} d^{3} x^{3} + 6 \, a^{2} b^{2} d^{3} x^{2} + 4 \, a^{3} b d^{3} x + a^{4} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 15, normalized size = 0.88 \begin {gather*} -\frac {b^{2}}{4 \, {\left (b x + a\right )}^{4} 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 = 0.94 \begin {gather*} -\frac {b^{2}}{4 \left (b x +a \right )^{4} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.35, size = 61, normalized size = 3.59 \begin {gather*} -\frac {b^{2}}{4 \, {\left (b^{4} d^{3} x^{4} + 4 \, a b^{3} d^{3} x^{3} + 6 \, a^{2} b^{2} d^{3} x^{2} + 4 \, a^{3} b d^{3} x + a^{4} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 63, normalized size = 3.71 \begin {gather*} -\frac {b^2}{4\,\left (a^4\,d^3+4\,a^3\,b\,d^3\,x+6\,a^2\,b^2\,d^3\,x^2+4\,a\,b^3\,d^3\,x^3+b^4\,d^3\,x^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.36, size = 68, normalized size = 4.00 \begin {gather*} - \frac {b^{3}}{4 a^{4} b d^{3} + 16 a^{3} b^{2} d^{3} x + 24 a^{2} b^{3} d^{3} x^{2} + 16 a b^{4} d^{3} x^{3} + 4 b^{5} d^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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