Optimal. Leaf size=22 \[ \frac {\log \left (a+b (c+d x)^3\right )}{3 b d} \]
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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 = {372, 260} \[ \frac {\log \left (a+b (c+d x)^3\right )}{3 b d} \]
Antiderivative was successfully verified.
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Rule 260
Rule 372
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{a+b (c+d x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{a+b x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\log \left (a+b (c+d x)^3\right )}{3 b d}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \frac {\log \left (a+b (c+d x)^3\right )}{3 b d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 42, normalized size = 1.91 \[ \frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 21, normalized size = 0.95 \[ \frac {\log \left ({\left | {\left (d x + c\right )}^{3} b + a \right |}\right )}{3 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 43, normalized size = 1.95 \[ \frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{3 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 20, normalized size = 0.91 \[ \frac {\log \left ({\left (d x + c\right )}^{3} b + a\right )}{3 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 42, normalized size = 1.91 \[ \frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,b\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.27, size = 42, normalized size = 1.91 \[ \frac {\log {\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}}{3 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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