Optimal. Leaf size=22 \[ \frac {\log \left (a+b (c+d x)^4\right )}{4 b d} \]
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Rubi [A]
time = 0.02, antiderivative size = 22, 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, 266}
\begin {gather*} \frac {\log \left (a+b (c+d x)^4\right )}{4 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 379
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{a+b (c+d x)^4} \, dx &=\frac {\text {Subst}\left (\int \frac {x^3}{a+b x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\log \left (a+b (c+d x)^4\right )}{4 b d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log \left (a+b (c+d x)^4\right )}{4 b d} \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. \(54\) vs.
\(2(20)=40\).
time = 0.22, size = 55, normalized size = 2.50
method | result | size |
derivativedivides | \(\frac {\ln \left (a +b \left (d x +c \right )^{4}\right )}{4 b d}\) | \(21\) |
default | \(\frac {\ln \left (b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a \right )}{4 b d}\) | \(55\) |
norman | \(\frac {\ln \left (b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a \right )}{4 b d}\) | \(55\) |
risch | \(\frac {\ln \left (b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a \right )}{4 b d}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 20, normalized size = 0.91 \begin {gather*} \frac {\log \left ({\left (d x + c\right )}^{4} b + a\right )}{4 \, 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. 54 vs.
\(2 (20) = 40\).
time = 0.39, size = 54, normalized size = 2.45 \begin {gather*} \frac {\log \left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}{4 \, b d} \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. 56 vs.
\(2 (15) = 30\).
time = 0.16, size = 56, normalized size = 2.55 \begin {gather*} \frac {\log {\left (a + b c^{4} + 4 b c^{3} d x + 6 b c^{2} d^{2} x^{2} + 4 b c d^{3} x^{3} + b d^{4} x^{4} \right )}}{4 b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 20, normalized size = 0.91 \begin {gather*} \frac {\log \left ({\left (d x + c\right )}^{4} b + a\right )}{4 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.26, size = 54, normalized size = 2.45 \begin {gather*} \frac {\ln \left (b\,c^4+4\,b\,c^3\,d\,x+6\,b\,c^2\,d^2\,x^2+4\,b\,c\,d^3\,x^3+b\,d^4\,x^4+a\right )}{4\,b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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