Integrand size = 24, antiderivative size = 94 \[ \int \frac {1}{(c e+d e x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {1}{6 a d e \left (a+b (c+d x)^3\right )^2}+\frac {1}{3 a^2 d e \left (a+b (c+d x)^3\right )}+\frac {\log (c+d x)}{a^3 d e}-\frac {\log \left (a+b (c+d x)^3\right )}{3 a^3 d e} \]
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Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {379, 272, 46} \[ \int \frac {1}{(c e+d e x) \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {\log \left (a+b (c+d x)^3\right )}{3 a^3 d e}+\frac {\log (c+d x)}{a^3 d e}+\frac {1}{3 a^2 d e \left (a+b (c+d x)^3\right )}+\frac {1}{6 a d e \left (a+b (c+d x)^3\right )^2} \]
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Rule 46
Rule 272
Rule 379
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d e} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^3} \, dx,x,(c+d x)^3\right )}{3 d e} \\ & = \frac {\text {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*}
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(c e+d e x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {\frac {a \left (a+2 \left (a+b (c+d x)^3\right )\right )}{\left (a+b (c+d x)^3\right )^2}+6 \log (c+d x)-2 \log \left (a+b (c+d x)^3\right )}{6 a^3 d e} \]
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Time = 4.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.57
method | result | size |
default | \(\frac {\frac {\ln \left (d x +c \right )}{a^{3} d}-\frac {b \left (\frac {-\frac {a \,d^{2} x^{3}}{3}-a c d \,x^{2}-a \,c^{2} x -\frac {a \left (2 c^{3} b +3 a \right )}{6 d b}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\right )}{a^{3}}}{e}\) | \(148\) |
risch | \(\frac {\frac {b \,d^{2} x^{3}}{3 a^{2}}+\frac {b c d \,x^{2}}{a^{2}}+\frac {b x \,c^{2}}{a^{2}}+\frac {2 c^{3} b +3 a}{6 a^{2} d}}{e \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {\ln \left (d x +c \right )}{a^{3} d e}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 a^{3} d e}\) | \(152\) |
norman | \(\frac {\frac {c b d \,x^{2}}{a^{2} e}+\frac {c^{2} b x}{a^{2} e}+\frac {2 b^{3} c^{3} d^{5}+3 a \,b^{2} d^{5}}{6 a^{2} d^{6} e \,b^{2}}+\frac {b \,d^{2} x^{3}}{3 a^{2} e}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {\ln \left (d x +c \right )}{a^{3} d e}-\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 a^{3} d e}\) | \(175\) |
parallelrisch | \(\frac {6 \ln \left (d x +c \right ) a^{2} b^{2} d^{5}-12 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{2} a \,b^{3} c \,d^{7}+6 x^{2} a \,b^{3} c \,d^{7}+6 x a \,b^{3} c^{2} d^{6}+12 \ln \left (d x +c \right ) a \,b^{3} c^{3} d^{5}-4 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) a \,b^{3} c^{3} d^{5}+36 \ln \left (d x +c \right ) x^{5} b^{4} c \,d^{10}-12 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{5} b^{4} c \,d^{10}+90 \ln \left (d x +c \right ) x^{4} b^{4} c^{2} d^{9}-30 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{4} b^{4} c^{2} d^{9}+120 \ln \left (d x +c \right ) x^{3} b^{4} c^{3} d^{8}-40 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{3} b^{4} c^{3} d^{8}+90 \ln \left (d x +c \right ) x^{2} b^{4} c^{4} d^{7}-30 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{2} b^{4} c^{4} d^{7}-12 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x a \,b^{3} c^{2} d^{6}+36 \ln \left (d x +c \right ) x^{2} a \,b^{3} c \,d^{7}+36 \ln \left (d x +c \right ) x a \,b^{3} c^{2} d^{6}-2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{6} b^{4} d^{11}+3 a^{2} b^{2} d^{5}+2 a \,b^{3} c^{3} d^{5}+36 \ln \left (d x +c \right ) x \,b^{4} c^{5} d^{6}-12 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x \,b^{4} c^{5} d^{6}+12 \ln \left (d x +c \right ) x^{3} a \,b^{3} d^{8}-4 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{3} a \,b^{3} d^{8}-2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) a^{2} b^{2} d^{5}+6 \ln \left (d x +c \right ) x^{6} b^{4} d^{11}+2 x^{3} a \,b^{3} d^{8}+6 \ln \left (d x +c \right ) b^{4} c^{6} d^{5}-2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) b^{4} c^{6} d^{5}}{6 a^{3} b^{2} d^{6} e \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}\) | \(890\) |
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Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (88) = 176\).
Time = 0.27 (sec) , antiderivative size = 474, normalized size of antiderivative = 5.04 \[ \int \frac {1}{(c e+d e x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2} - 2 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \, {\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \, {\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \, {\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \, {\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (d x + c\right )}{6 \, {\left (a^{3} b^{2} d^{7} e x^{6} + 6 \, a^{3} b^{2} c d^{6} e x^{5} + 15 \, a^{3} b^{2} c^{2} d^{5} e x^{4} + 2 \, {\left (10 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{4} e x^{3} + 3 \, {\left (5 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{3} e x^{2} + 6 \, {\left (a^{3} b^{2} c^{5} + a^{4} b c^{2}\right )} d^{2} e x + {\left (a^{3} b^{2} c^{6} + 2 \, a^{4} b c^{3} + a^{5}\right )} d e\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (73) = 146\).
Time = 1.83 (sec) , antiderivative size = 292, normalized size of antiderivative = 3.11 \[ \int \frac {1}{(c e+d e x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {3 a + 2 b c^{3} + 6 b c^{2} d x + 6 b c d^{2} x^{2} + 2 b d^{3} x^{3}}{6 a^{4} d e + 12 a^{3} b c^{3} d e + 6 a^{2} b^{2} c^{6} d e + 90 a^{2} b^{2} c^{2} d^{5} e x^{4} + 36 a^{2} b^{2} c d^{6} e x^{5} + 6 a^{2} b^{2} d^{7} e x^{6} + x^{3} \cdot \left (12 a^{3} b d^{4} e + 120 a^{2} b^{2} c^{3} d^{4} e\right ) + x^{2} \cdot \left (36 a^{3} b c d^{3} e + 90 a^{2} b^{2} c^{4} d^{3} e\right ) + x \left (36 a^{3} b c^{2} d^{2} e + 36 a^{2} b^{2} c^{5} d^{2} e\right )} + \frac {\log {\left (\frac {c}{d} + x \right )}}{a^{3} d e} - \frac {\log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{3} d e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (88) = 176\).
Time = 0.23 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.74 \[ \int \frac {1}{(c e+d e x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + 3 \, a}{6 \, {\left (a^{2} b^{2} d^{7} e x^{6} + 6 \, a^{2} b^{2} c d^{6} e x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} e x^{4} + 2 \, {\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} e x^{3} + 3 \, {\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} e x^{2} + 6 \, {\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} e x + {\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d e\right )}} - \frac {\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d e} + \frac {\log \left (d x + c\right )}{a^{3} d e} \]
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Time = 0.31 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(c e+d e x) \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {\log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{3} d e} + \frac {\log \left ({\left | d x + c \right |}\right )}{a^{3} d e} + \frac {2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2}}{6 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a^{3} d e} \]
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Time = 6.49 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.65 \[ \int \frac {1}{(c e+d e x) \left (a+b (c+d x)^3\right )^3} \, dx=\frac {\frac {2\,b\,c^3+3\,a}{6\,a^2\,d}+\frac {b\,d^2\,x^3}{3\,a^2}+\frac {b\,c^2\,x}{a^2}+\frac {b\,c\,d\,x^2}{a^2}}{x^2\,\left (15\,e\,b^2\,c^4\,d^2+6\,a\,e\,b\,c\,d^2\right )+a^2\,e+x\,\left (6\,d\,e\,b^2\,c^5+6\,a\,d\,e\,b\,c^2\right )+x^3\,\left (20\,e\,b^2\,c^3\,d^3+2\,a\,e\,b\,d^3\right )+b^2\,c^6\,e+b^2\,d^6\,e\,x^6+2\,a\,b\,c^3\,e+6\,b^2\,c\,d^5\,e\,x^5+15\,b^2\,c^2\,d^4\,e\,x^4}+\frac {\ln \left (c+d\,x\right )}{a^3\,d\,e}-\frac {\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^3\,d\,e} \]
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