Optimal. Leaf size=15 \[ -\frac {d}{4 b^2 (c+d x)^4} \]
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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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32}
\begin {gather*} -\frac {d}{4 b^2 (c+d x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{\left (\frac {b c}{d}+b x\right )^2 (c+d x)^3} \, dx &=\frac {d^2 \int \frac {1}{(c+d x)^5} \, dx}{b^2}\\ &=-\frac {d}{4 b^2 (c+d x)^4}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {d}{4 b^2 (c+d x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 14, normalized size = 0.93
method | result | size |
gosper | \(-\frac {d}{4 b^{2} \left (d x +c \right )^{4}}\) | \(14\) |
default | \(-\frac {d}{4 b^{2} \left (d x +c \right )^{4}}\) | \(14\) |
norman | \(-\frac {d}{4 b^{2} \left (d x +c \right )^{4}}\) | \(14\) |
risch | \(-\frac {d}{4 b^{2} \left (d x +c \right )^{4}}\) | \(14\) |
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. 59 vs.
\(2 (13) = 26\).
time = 0.29, size = 59, normalized size = 3.93 \begin {gather*} -\frac {d}{4 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\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. 59 vs.
\(2 (13) = 26\).
time = 0.80, size = 59, normalized size = 3.93 \begin {gather*} -\frac {d}{4 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\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. 68 vs.
\(2 (14) = 28\).
time = 0.19, size = 68, normalized size = 4.53 \begin {gather*} - \frac {d^{2}}{4 b^{2} c^{4} d + 16 b^{2} c^{3} d^{2} x + 24 b^{2} c^{2} d^{3} x^{2} + 16 b^{2} c d^{4} x^{3} + 4 b^{2} d^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.21, size = 20, normalized size = 1.33 \begin {gather*} -\frac {b^{2}}{4 \, {\left (b x + \frac {b c}{d}\right )}^{4} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 61, normalized size = 4.07 \begin {gather*} -\frac {d}{4\,\left (b^2\,c^4+4\,b^2\,c^3\,d\,x+6\,b^2\,c^2\,d^2\,x^2+4\,b^2\,c\,d^3\,x^3+b^2\,d^4\,x^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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