Optimal. Leaf size=59 \[ -\frac {\log \left (a+b (c+d x)^3\right )}{3 a^2 d}+\frac {\log (c+d x)}{a^2 d}+\frac {1}{3 a d \left (a+b (c+d x)^3\right )} \]
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Rubi [A] time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {372, 266, 44} \[ -\frac {\log \left (a+b (c+d x)^3\right )}{3 a^2 d}+\frac {\log (c+d x)}{a^2 d}+\frac {1}{3 a d \left (a+b (c+d x)^3\right )} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 372
Rubi steps
\begin {align*} \int \frac {1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,(c+d x)^3\right )}{3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d}\\ &=\frac {1}{3 a d \left (a+b (c+d x)^3\right )}+\frac {\log (c+d x)}{a^2 d}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 0.81 \[ \frac {\frac {a}{a+b (c+d x)^3}-\log \left (a+b (c+d x)^3\right )+3 \log (c+d x)}{3 a^2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 169, normalized size = 2.86 \[ -\frac {{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (d x + c\right ) - a}{3 \, {\left (a^{2} b d^{4} x^{3} + 3 \, a^{2} b c d^{3} x^{2} + 3 \, a^{2} b c^{2} d^{2} x + {\left (a^{2} b c^{3} + a^{3}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 101, normalized size = 1.71 \[ -\frac {\log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{2} d} + \frac {\log \left ({\left | d x + c \right |}\right )}{a^{2} d} + \frac {1}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 100, normalized size = 1.69 \[ \frac {1}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right ) a d}+\frac {\ln \left (d x +c \right )}{a^{2} d}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{3 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 104, normalized size = 1.76 \[ \frac {1}{3 \, {\left (a b d^{4} x^{3} + 3 \, a b c d^{3} x^{2} + 3 \, a b c^{2} d^{2} x + {\left (a b c^{3} + a^{2}\right )} d\right )}} - \frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{2} d} + \frac {\log \left (d x + c\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 105, normalized size = 1.78 \[ \frac {1}{3\,\left (a^2\,d+b\,a\,c^3\,d+3\,b\,a\,c^2\,d^2\,x+3\,b\,a\,c\,d^3\,x^2+b\,a\,d^4\,x^3\right )}-\frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^2\,d}+\frac {\ln \left (c+d\,x\right )}{a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.48, size = 110, normalized size = 1.86 \[ \frac {1}{3 a^{2} d + 3 a b c^{3} d + 9 a b c^{2} d^{2} x + 9 a b c d^{3} x^{2} + 3 a b d^{4} x^{3}} + \frac {\log {\left (\frac {c}{d} + x \right )}}{a^{2} d} - \frac {\log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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