Optimal. Leaf size=23 \[ -\frac {1}{3 b d \left (a+b (c+d x)^3\right )} \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {379, 267}
\begin {gather*} -\frac {1}{3 b d \left (a+b (c+d x)^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 379
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{\left (a+b (c+d x)^3\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^2}{\left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d}\\ &=-\frac {1}{3 b d \left (a+b (c+d x)^3\right )}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} -\frac {1}{3 b d \left (a+b (c+d x)^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(43\) vs.
\(2(21)=42\).
time = 0.22, size = 44, normalized size = 1.91
method | result | size |
derivativedivides | \(-\frac {1}{3 b d \left (a +b \left (d x +c \right )^{3}\right )}\) | \(22\) |
gosper | \(-\frac {1}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) b d}\) | \(44\) |
default | \(-\frac {1}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) b d}\) | \(44\) |
norman | \(-\frac {1}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) b d}\) | \(44\) |
risch | \(-\frac {1}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) b d}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 21, normalized size = 0.91 \begin {gather*} -\frac {1}{3 \, {\left ({\left (d x + c\right )}^{3} b + a\right )} b d} \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. 52 vs.
\(2 (21) = 42\).
time = 0.37, size = 52, normalized size = 2.26 \begin {gather*} -\frac {1}{3 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + {\left (b^{2} c^{3} + a b\right )} d\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. 58 vs.
\(2 (17) = 34\).
time = 0.59, size = 58, normalized size = 2.52 \begin {gather*} - \frac {1}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.52, size = 21, normalized size = 0.91 \begin {gather*} -\frac {1}{3 \, {\left ({\left (d x + c\right )}^{3} b + a\right )} b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 43, normalized size = 1.87 \begin {gather*} -\frac {1}{3\,b\,d\,\left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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