Optimal. Leaf size=17 \[ -\frac {b^2}{3 d^3 (a+b x)^3} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 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}{3 d^3 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{(a+b x) \left (\frac {a d}{b}+d x\right )^3} \, dx &=\frac {b^3 \int \frac {1}{(a+b x)^4} \, dx}{d^3}\\ &=-\frac {b^2}{3 d^3 (a+b x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {b^2}{3 d^3 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 16, normalized size = 0.94
method | result | size |
gosper | \(-\frac {b^{2}}{3 d^{3} \left (b x +a \right )^{3}}\) | \(16\) |
default | \(-\frac {b^{2}}{3 d^{3} \left (b x +a \right )^{3}}\) | \(16\) |
norman | \(-\frac {b^{2}}{3 d^{3} \left (b x +a \right )^{3}}\) | \(16\) |
risch | \(-\frac {b^{2}}{3 d^{3} \left (b x +a \right )^{3}}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs.
\(2 (15) = 30\).
time = 0.31, size = 47, normalized size = 2.76 \begin {gather*} -\frac {b^{2}}{3 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs.
\(2 (15) = 30\).
time = 0.44, size = 47, normalized size = 2.76 \begin {gather*} -\frac {b^{2}}{3 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs.
\(2 (15) = 30\).
time = 0.12, size = 53, normalized size = 3.12 \begin {gather*} - \frac {b^{3}}{3 a^{3} b d^{3} + 9 a^{2} b^{2} d^{3} x + 9 a b^{3} d^{3} x^{2} + 3 b^{4} d^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.32, size = 15, normalized size = 0.88 \begin {gather*} -\frac {b^{2}}{3 \, {\left (b x + a\right )}^{3} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 49, normalized size = 2.88 \begin {gather*} -\frac {b^2}{3\,\left (a^3\,d^3+3\,a^2\,b\,d^3\,x+3\,a\,b^2\,d^3\,x^2+b^3\,d^3\,x^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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