Integrand size = 24, antiderivative size = 116 \[ \int \frac {1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {1}{3 a^3 d e^4 (c+d x)^3}-\frac {b}{6 a^2 d e^4 \left (a+b (c+d x)^3\right )^2}-\frac {2 b}{3 a^3 d e^4 \left (a+b (c+d x)^3\right )}-\frac {3 b \log (c+d x)}{a^4 d e^4}+\frac {b \log \left (a+b (c+d x)^3\right )}{a^4 d e^4} \]
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Time = 0.07 (sec) , antiderivative size = 116, 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} \[ \int \frac {1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {3 b \log (c+d x)}{a^4 d e^4}+\frac {b \log \left (a+b (c+d x)^3\right )}{a^4 d e^4}-\frac {2 b}{3 a^3 d e^4 \left (a+b (c+d x)^3\right )}-\frac {1}{3 a^3 d e^4 (c+d x)^3}-\frac {b}{6 a^2 d e^4 \left (a+b (c+d x)^3\right )^2} \]
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Rule 46
Rule 272
Rule 379
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^4 \left (a+b x^3\right )^3} \, dx,x,c+d x\right )}{d e^4} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x^2 (a+b x)^3} \, dx,x,(c+d x)^3\right )}{3 d e^4} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{a^3 x^2}-\frac {3 b}{a^4 x}+\frac {b^2}{a^2 (a+b x)^3}+\frac {2 b^2}{a^3 (a+b x)^2}+\frac {3 b^2}{a^4 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d e^4} \\ & = -\frac {1}{3 a^3 d e^4 (c+d x)^3}-\frac {b}{6 a^2 d e^4 \left (a+b (c+d x)^3\right )^2}-\frac {2 b}{3 a^3 d e^4 \left (a+b (c+d x)^3\right )}-\frac {3 b \log (c+d x)}{a^4 d e^4}+\frac {b \log \left (a+b (c+d x)^3\right )}{a^4 d e^4} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {a \left (-\frac {2}{(c+d x)^3}-\frac {a b}{\left (a+b (c+d x)^3\right )^2}-\frac {4 b}{a+b (c+d x)^3}\right )-18 b \log (c+d x)+6 b \log \left (a+b (c+d x)^3\right )}{6 a^4 d e^4} \]
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Time = 4.05 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.42
method | result | size |
default | \(\frac {-\frac {1}{3 a^{3} d \left (d x +c \right )^{3}}-\frac {3 b \ln \left (d x +c \right )}{a^{4} d}+\frac {b^{2} \left (\frac {-\frac {2 a \,d^{2} x^{3}}{3}-2 a c d \,x^{2}-2 a \,c^{2} x -\frac {a \left (4 c^{3} b +5 a \right )}{6 d b}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{b d}\right )}{a^{4}}}{e^{4}}\) | \(165\) |
risch | \(\frac {-\frac {b^{2} d^{5} x^{6}}{a^{3}}-\frac {6 b^{2} c \,d^{4} x^{5}}{a^{3}}-\frac {15 b^{2} c^{2} d^{3} x^{4}}{a^{3}}-\frac {d^{2} \left (40 c^{3} b +3 a \right ) b \,x^{3}}{2 a^{3}}-\frac {3 b c d \left (10 c^{3} b +3 a \right ) x^{2}}{2 a^{3}}-\frac {3 \left (4 c^{3} b +3 a \right ) b \,c^{2} x}{2 a^{3}}-\frac {6 b^{2} c^{6}+9 a b \,c^{3}+2 a^{2}}{6 d \,a^{3}}}{e^{4} \left (d x +c \right )^{3} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}-\frac {3 b \ln \left (d x +c \right )}{a^{4} d \,e^{4}}+\frac {b \ln \left (-b \,d^{3} x^{3}-3 b c \,d^{2} x^{2}-3 b \,c^{2} d x -c^{3} b -a \right )}{a^{4} d \,e^{4}}\) | \(254\) |
norman | \(\frac {-\frac {b^{2} d^{5} x^{6}}{a^{3} e}+\frac {-6 b^{4} c^{6} d^{8}-9 a \,b^{3} c^{3} d^{8}-2 a^{2} b^{2} d^{8}}{6 a^{3} d^{9} e \,b^{2}}+\frac {\left (-40 b^{4} c^{3} d^{8}-3 a \,b^{3} d^{8}\right ) x^{3}}{2 a^{3} d^{6} e \,b^{2}}+\frac {3 c^{2} \left (-4 b^{4} c^{3} d^{8}-3 a \,b^{3} d^{8}\right ) x}{2 a^{3} d^{8} e \,b^{2}}+\frac {3 c \left (-10 b^{4} c^{3} d^{8}-3 a \,b^{3} d^{8}\right ) x^{2}}{2 a^{3} d^{7} e \,b^{2}}-\frac {6 c \,b^{2} d^{4} x^{5}}{e \,a^{3}}-\frac {15 c^{2} b^{2} d^{3} x^{4}}{e \,a^{3}}}{e^{3} \left (d x +c \right )^{3} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )^{2}}+\frac {b \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{a^{4} d \,e^{4}}-\frac {3 b \ln \left (d x +c \right )}{a^{4} d \,e^{4}}\) | \(332\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1628\) |
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Leaf count of result is larger than twice the leaf count of optimal. 919 vs. \(2 (110) = 220\).
Time = 0.30 (sec) , antiderivative size = 919, normalized size of antiderivative = 7.92 \[ \int \frac {1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {6 \, a b^{2} d^{6} x^{6} + 36 \, a b^{2} c d^{5} x^{5} + 90 \, a b^{2} c^{2} d^{4} x^{4} + 6 \, a b^{2} c^{6} + 3 \, {\left (40 \, a b^{2} c^{3} + 3 \, a^{2} b\right )} d^{3} x^{3} + 9 \, a^{2} b c^{3} + 9 \, {\left (10 \, a b^{2} c^{4} + 3 \, a^{2} b c\right )} d^{2} x^{2} + 2 \, a^{3} + 9 \, {\left (4 \, a b^{2} c^{5} + 3 \, a^{2} b c^{2}\right )} d x - 6 \, {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 2 \, {\left (42 \, b^{3} c^{3} + a b^{2}\right )} d^{6} x^{6} + b^{3} c^{9} + 6 \, {\left (21 \, b^{3} c^{4} + 2 \, a b^{2} c\right )} d^{5} x^{5} + 2 \, a b^{2} c^{6} + 6 \, {\left (21 \, b^{3} c^{5} + 5 \, a b^{2} c^{2}\right )} d^{4} x^{4} + {\left (84 \, b^{3} c^{6} + 40 \, a b^{2} c^{3} + a^{2} b\right )} d^{3} x^{3} + a^{2} b c^{3} + 3 \, {\left (12 \, b^{3} c^{7} + 10 \, a b^{2} c^{4} + a^{2} b c\right )} d^{2} x^{2} + 3 \, {\left (3 \, b^{3} c^{8} + 4 \, a b^{2} c^{5} + a^{2} b c^{2}\right )} d x\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 ) + 18 \, {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 2 \, {\left (42 \, b^{3} c^{3} + a b^{2}\right )} d^{6} x^{6} + b^{3} c^{9} + 6 \, {\left (21 \, b^{3} c^{4} + 2 \, a b^{2} c\right )} d^{5} x^{5} + 2 \, a b^{2} c^{6} + 6 \, {\left (21 \, b^{3} c^{5} + 5 \, a b^{2} c^{2}\right )} d^{4} x^{4} + {\left (84 \, b^{3} c^{6} + 40 \, a b^{2} c^{3} + a^{2} b\right )} d^{3} x^{3} + a^{2} b c^{3} + 3 \, {\left (12 \, b^{3} c^{7} + 10 \, a b^{2} c^{4} + a^{2} b c\right )} d^{2} x^{2} + 3 \, {\left (3 \, b^{3} c^{8} + 4 \, a b^{2} c^{5} + a^{2} b c^{2}\right )} d x\right )} \log \left (d x + c\right )}{6 \, {\left (a^{4} b^{2} d^{10} e^{4} x^{9} + 9 \, a^{4} b^{2} c d^{9} e^{4} x^{8} + 36 \, a^{4} b^{2} c^{2} d^{8} e^{4} x^{7} + 2 \, {\left (42 \, a^{4} b^{2} c^{3} + a^{5} b\right )} d^{7} e^{4} x^{6} + 6 \, {\left (21 \, a^{4} b^{2} c^{4} + 2 \, a^{5} b c\right )} d^{6} e^{4} x^{5} + 6 \, {\left (21 \, a^{4} b^{2} c^{5} + 5 \, a^{5} b c^{2}\right )} d^{5} e^{4} x^{4} + {\left (84 \, a^{4} b^{2} c^{6} + 40 \, a^{5} b c^{3} + a^{6}\right )} d^{4} e^{4} x^{3} + 3 \, {\left (12 \, a^{4} b^{2} c^{7} + 10 \, a^{5} b c^{4} + a^{6} c\right )} d^{3} e^{4} x^{2} + 3 \, {\left (3 \, a^{4} b^{2} c^{8} + 4 \, a^{5} b c^{5} + a^{6} c^{2}\right )} d^{2} e^{4} x + {\left (a^{4} b^{2} c^{9} + 2 \, a^{5} b c^{6} + a^{6} c^{3}\right )} d e^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (105) = 210\).
Time = 3.11 (sec) , antiderivative size = 578, normalized size of antiderivative = 4.98 \[ \int \frac {1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {- 2 a^{2} - 9 a b c^{3} - 6 b^{2} c^{6} - 90 b^{2} c^{2} d^{4} x^{4} - 36 b^{2} c d^{5} x^{5} - 6 b^{2} d^{6} x^{6} + x^{3} \left (- 9 a b d^{3} - 120 b^{2} c^{3} d^{3}\right ) + x^{2} \left (- 27 a b c d^{2} - 90 b^{2} c^{4} d^{2}\right ) + x \left (- 27 a b c^{2} d - 36 b^{2} c^{5} d\right )}{6 a^{5} c^{3} d e^{4} + 12 a^{4} b c^{6} d e^{4} + 6 a^{3} b^{2} c^{9} d e^{4} + 216 a^{3} b^{2} c^{2} d^{8} e^{4} x^{7} + 54 a^{3} b^{2} c d^{9} e^{4} x^{8} + 6 a^{3} b^{2} d^{10} e^{4} x^{9} + x^{6} \cdot \left (12 a^{4} b d^{7} e^{4} + 504 a^{3} b^{2} c^{3} d^{7} e^{4}\right ) + x^{5} \cdot \left (72 a^{4} b c d^{6} e^{4} + 756 a^{3} b^{2} c^{4} d^{6} e^{4}\right ) + x^{4} \cdot \left (180 a^{4} b c^{2} d^{5} e^{4} + 756 a^{3} b^{2} c^{5} d^{5} e^{4}\right ) + x^{3} \cdot \left (6 a^{5} d^{4} e^{4} + 240 a^{4} b c^{3} d^{4} e^{4} + 504 a^{3} b^{2} c^{6} d^{4} e^{4}\right ) + x^{2} \cdot \left (18 a^{5} c d^{3} e^{4} + 180 a^{4} b c^{4} d^{3} e^{4} + 216 a^{3} b^{2} c^{7} d^{3} e^{4}\right ) + x \left (18 a^{5} c^{2} d^{2} e^{4} + 72 a^{4} b c^{5} d^{2} e^{4} + 54 a^{3} b^{2} c^{8} d^{2} e^{4}\right )} - \frac {3 b \log {\left (\frac {c}{d} + x \right )}}{a^{4} d e^{4}} + \frac {b \log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{a^{4} d e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (110) = 220\).
Time = 0.24 (sec) , antiderivative size = 474, normalized size of antiderivative = 4.09 \[ \int \frac {1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {6 \, b^{2} d^{6} x^{6} + 36 \, b^{2} c d^{5} x^{5} + 90 \, b^{2} c^{2} d^{4} x^{4} + 6 \, b^{2} c^{6} + 3 \, {\left (40 \, b^{2} c^{3} + 3 \, a b\right )} d^{3} x^{3} + 9 \, a b c^{3} + 9 \, {\left (10 \, b^{2} c^{4} + 3 \, a b c\right )} d^{2} x^{2} + 9 \, {\left (4 \, b^{2} c^{5} + 3 \, a b c^{2}\right )} d x + 2 \, a^{2}}{6 \, {\left (a^{3} b^{2} d^{10} e^{4} x^{9} + 9 \, a^{3} b^{2} c d^{9} e^{4} x^{8} + 36 \, a^{3} b^{2} c^{2} d^{8} e^{4} x^{7} + 2 \, {\left (42 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{7} e^{4} x^{6} + 6 \, {\left (21 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{6} e^{4} x^{5} + 6 \, {\left (21 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{5} e^{4} x^{4} + {\left (84 \, a^{3} b^{2} c^{6} + 40 \, a^{4} b c^{3} + a^{5}\right )} d^{4} e^{4} x^{3} + 3 \, {\left (12 \, a^{3} b^{2} c^{7} + 10 \, a^{4} b c^{4} + a^{5} c\right )} d^{3} e^{4} x^{2} + 3 \, {\left (3 \, a^{3} b^{2} c^{8} + 4 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d^{2} e^{4} x + {\left (a^{3} b^{2} c^{9} + 2 \, a^{4} b c^{6} + a^{5} c^{3}\right )} d e^{4}\right )}} + \frac {b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{a^{4} d e^{4}} - \frac {3 \, b \log \left (d x + c\right )}{a^{4} d e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (110) = 220\).
Time = 0.31 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.26 \[ \int \frac {1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {b \log \left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{a^{4} d e^{4}} - \frac {3 \, b \log \left ({\left | d x + c \right |}\right )}{a^{4} d e^{4}} - \frac {6 \, a b^{2} d^{6} x^{6} + 36 \, a b^{2} c d^{5} x^{5} + 90 \, a b^{2} c^{2} d^{4} x^{4} + 6 \, a b^{2} c^{6} + 9 \, a^{2} b c^{3} + 3 \, {\left (40 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b d^{3}\right )} x^{3} + 2 \, a^{3} + 9 \, {\left (10 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c d^{2}\right )} x^{2} + 9 \, {\left (4 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{2} d\right )} x}{6 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} {\left (d x + c\right )}^{3} a^{4} d e^{4}} \]
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Time = 8.16 (sec) , antiderivative size = 507, normalized size of antiderivative = 4.37 \[ \int \frac {1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {b\,\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{a^4\,d\,e^4}-\frac {\frac {2\,a^2+9\,a\,b\,c^3+6\,b^2\,c^6}{6\,a^3\,d}+\frac {3\,x^2\,\left (10\,d\,b^2\,c^4+3\,a\,d\,b\,c\right )}{2\,a^3}+\frac {3\,x\,\left (4\,b^2\,c^5+3\,a\,b\,c^2\right )}{2\,a^3}+\frac {x^3\,\left (40\,b^2\,c^3\,d^2+3\,a\,b\,d^2\right )}{2\,a^3}+\frac {b^2\,d^5\,x^6}{a^3}+\frac {15\,b^2\,c^2\,d^3\,x^4}{a^3}+\frac {6\,b^2\,c\,d^4\,x^5}{a^3}}{x^5\,\left (126\,b^2\,c^4\,d^5\,e^4+12\,a\,b\,c\,d^5\,e^4\right )+x^3\,\left (a^2\,d^3\,e^4+40\,a\,b\,c^3\,d^3\,e^4+84\,b^2\,c^6\,d^3\,e^4\right )+x\,\left (3\,d\,a^2\,c^2\,e^4+12\,d\,a\,b\,c^5\,e^4+9\,d\,b^2\,c^8\,e^4\right )+x^4\,\left (126\,b^2\,c^5\,d^4\,e^4+30\,a\,b\,c^2\,d^4\,e^4\right )+x^2\,\left (3\,a^2\,c\,d^2\,e^4+30\,a\,b\,c^4\,d^2\,e^4+36\,b^2\,c^7\,d^2\,e^4\right )+x^6\,\left (84\,b^2\,c^3\,d^6\,e^4+2\,a\,b\,d^6\,e^4\right )+a^2\,c^3\,e^4+b^2\,c^9\,e^4+b^2\,d^9\,e^4\,x^9+2\,a\,b\,c^6\,e^4+36\,b^2\,c^2\,d^7\,e^4\,x^7+9\,b^2\,c\,d^8\,e^4\,x^8}-\frac {3\,b\,\ln \left (c+d\,x\right )}{a^4\,d\,e^4} \]
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