Optimal. Leaf size=94 \[ -\frac {\log \left (a+b (c+d x)^3\right )}{3 a^3 d e}+\frac {\log (c+d x)}{a^3 d e}+\frac {1}{3 a^2 d e \left (a+b (c+d x)^3\right )}+\frac {1}{6 a d e \left (a+b (c+d x)^3\right )^2} \]
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Rubi [A] time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {372, 266, 44} \[ \frac {1}{3 a^2 d e \left (a+b (c+d x)^3\right )}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a^3 d e}+\frac {\log (c+d x)}{a^3 d e}+\frac {1}{6 a d e \left (a+b (c+d x)^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rule 372
Rubi steps
\begin {align*} \int \frac {1}{(c e+d e x) \left (a+b (c+d x)^3\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (a+b x)^3} \, dx,x,(c+d x)^3\right )}{3 d e}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{a^3 x}-\frac {b}{a (a+b x)^3}-\frac {b}{a^2 (a+b x)^2}-\frac {b}{a^3 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d e}\\ &=\frac {1}{6 a d e \left (a+b (c+d x)^3\right )^2}+\frac {1}{3 a^2 d e \left (a+b (c+d x)^3\right )}+\frac {\log (c+d x)}{a^3 d e}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a^3 d e}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 0.70 \[ \frac {\frac {a \left (2 \left (a+b (c+d x)^3\right )+a\right )}{\left (a+b (c+d x)^3\right )^2}-2 \log \left (a+b (c+d x)^3\right )+6 \log (c+d x)}{6 a^3 d e} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.10, size = 474, normalized size = 5.04 \[ \frac {2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2} - 2 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \, {\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \, {\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \, {\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \, {\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (d x + c\right )}{6 \, {\left (a^{3} b^{2} d^{7} e x^{6} + 6 \, a^{3} b^{2} c d^{6} e x^{5} + 15 \, a^{3} b^{2} c^{2} d^{5} e x^{4} + 2 \, {\left (10 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{4} e x^{3} + 3 \, {\left (5 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{3} e x^{2} + 6 \, {\left (a^{3} b^{2} c^{5} + a^{4} b c^{2}\right )} d^{2} e x + {\left (a^{3} b^{2} c^{6} + 2 \, a^{4} b c^{3} + a^{5}\right )} d e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 150, normalized size = 1.60 \[ -\frac {e^{\left (-1\right )} \log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{3} d} + \frac {e^{\left (-1\right )} \log \left ({\left | d x + c \right |}\right )}{a^{3} d} + \frac {{\left (2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2}\right )} e^{\left (-1\right )}}{6 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 304, normalized size = 3.23 \[ \frac {b \,d^{2} x^{3}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2} e}+\frac {b c d \,x^{2}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2} e}+\frac {b \,c^{2} x}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2} e}+\frac {b \,c^{3}}{3 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a^{2} d e}+\frac {1}{2 \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )^{2} a d e}+\frac {\ln \left (d x +c \right )}{a^{3} d e}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a \right )}{3 a^{3} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 258, normalized size = 2.74 \[ \frac {2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + 3 \, a}{6 \, {\left (a^{2} b^{2} d^{7} e x^{6} + 6 \, a^{2} b^{2} c d^{6} e x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} e x^{4} + 2 \, {\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} e x^{3} + 3 \, {\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} e x^{2} + 6 \, {\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} e x + {\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d e\right )}} - \frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d e} + \frac {\log \left (d x + c\right )}{a^{3} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.91, size = 249, normalized size = 2.65 \[ \frac {\frac {2\,b\,c^3+3\,a}{6\,a^2\,d}+\frac {b\,d^2\,x^3}{3\,a^2}+\frac {b\,c^2\,x}{a^2}+\frac {b\,c\,d\,x^2}{a^2}}{x^2\,\left (15\,e\,b^2\,c^4\,d^2+6\,a\,e\,b\,c\,d^2\right )+a^2\,e+x\,\left (6\,d\,e\,b^2\,c^5+6\,a\,d\,e\,b\,c^2\right )+x^3\,\left (20\,e\,b^2\,c^3\,d^3+2\,a\,e\,b\,d^3\right )+b^2\,c^6\,e+b^2\,d^6\,e\,x^6+2\,a\,b\,c^3\,e+6\,b^2\,c\,d^5\,e\,x^5+15\,b^2\,c^2\,d^4\,e\,x^4}+\frac {\ln \left (c+d\,x\right )}{a^3\,d\,e}-\frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^3\,d\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.06, size = 292, normalized size = 3.11 \[ \frac {3 a + 2 b c^{3} + 6 b c^{2} d x + 6 b c d^{2} x^{2} + 2 b d^{3} x^{3}}{6 a^{4} d e + 12 a^{3} b c^{3} d e + 6 a^{2} b^{2} c^{6} d e + 90 a^{2} b^{2} c^{2} d^{5} e x^{4} + 36 a^{2} b^{2} c d^{6} e x^{5} + 6 a^{2} b^{2} d^{7} e x^{6} + x^{3} \left (12 a^{3} b d^{4} e + 120 a^{2} b^{2} c^{3} d^{4} e\right ) + x^{2} \left (36 a^{3} b c d^{3} e + 90 a^{2} b^{2} c^{4} d^{3} e\right ) + x \left (36 a^{3} b c^{2} d^{2} e + 36 a^{2} b^{2} c^{5} d^{2} e\right )} + \frac {\log {\left (\frac {c}{d} + x \right )}}{a^{3} d e} - \frac {\log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{3} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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