Integrand size = 24, antiderivative size = 204 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {5}{6 a^2 d e^3 (c+d x)^2}+\frac {1}{3 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac {5 b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3} d e^3}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{8/3} d e^3}+\frac {5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{8/3} d e^3} \]
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Time = 0.12 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {379, 296, 331, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^2} \, dx=\frac {5 b^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3} d e^3}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{8/3} d e^3}+\frac {5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{8/3} d e^3}-\frac {5}{6 a^2 d e^3 (c+d x)^2}+\frac {1}{3 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )} \]
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Rule 31
Rule 206
Rule 210
Rule 296
Rule 331
Rule 379
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d e^3} \\ & = \frac {1}{3 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac {5 \text {Subst}\left (\int \frac {1}{x^3 \left (a+b x^3\right )} \, dx,x,c+d x\right )}{3 a d e^3} \\ & = -\frac {5}{6 a^2 d e^3 (c+d x)^2}+\frac {1}{3 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {(5 b) \text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,c+d x\right )}{3 a^2 d e^3} \\ & = -\frac {5}{6 a^2 d e^3 (c+d x)^2}+\frac {1}{3 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {(5 b) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,c+d x\right )}{9 a^{8/3} d e^3}-\frac {(5 b) \text {Subst}\left (\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{9 a^{8/3} d e^3} \\ & = -\frac {5}{6 a^2 d e^3 (c+d x)^2}+\frac {1}{3 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{8/3} d e^3}+\frac {\left (5 b^{2/3}\right ) \text {Subst}\left (\int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{18 a^{8/3} d e^3}-\frac {(5 b) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,c+d x\right )}{6 a^{7/3} d e^3} \\ & = -\frac {5}{6 a^2 d e^3 (c+d x)^2}+\frac {1}{3 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{8/3} d e^3}+\frac {5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{8/3} d e^3}-\frac {\left (5 b^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{3 a^{8/3} d e^3} \\ & = -\frac {5}{6 a^2 d e^3 (c+d x)^2}+\frac {1}{3 a d e^3 (c+d x)^2 \left (a+b (c+d x)^3\right )}+\frac {5 b^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{8/3} d e^3}-\frac {5 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{8/3} d e^3}+\frac {5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{8/3} d e^3} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^2} \, dx=\frac {-\frac {9 a^{2/3}}{(c+d x)^2}-\frac {6 a^{2/3} b (c+d x)}{a+b (c+d x)^3}-10 \sqrt {3} b^{2/3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )-10 b^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )+5 b^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{8/3} d e^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.04 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {-\frac {1}{2 a^{2} d \left (d x +c \right )^{2}}-\frac {b \left (\frac {\frac {x}{3}+\frac {c}{3 d}}{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b \,d^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{9 b d}\right )}{a^{2}}}{e^{3}}\) | \(144\) |
risch | \(\frac {-\frac {5 b \,d^{2} x^{3}}{6 a^{2}}-\frac {5 b c d \,x^{2}}{2 a^{2}}-\frac {5 b x \,c^{2}}{2 a^{2}}-\frac {5 c^{3} b +3 a}{6 d \,a^{2}}}{e^{3} \left (d x +c \right )^{2} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{8} d^{3} e^{9} \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 a^{8} d^{4} e^{9} \textit {\_R}^{3}-3 b^{2} d \right ) x -4 a^{8} c \,d^{3} e^{9} \textit {\_R}^{3}-a^{3} b d \,e^{3} \textit {\_R} -3 b^{2} c \right )\right )}{9}\) | \(182\) |
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Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (163) = 326\).
Time = 0.28 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.58 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {15 \, b d^{3} x^{3} + 45 \, b c d^{2} x^{2} + 45 \, b c^{2} d x + 15 \, b c^{3} - 10 \, \sqrt {3} {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + b c^{5} + {\left (10 \, b c^{3} + a\right )} d^{2} x^{2} + a c^{2} + {\left (5 \, b c^{4} + 2 \, a c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (a d x + a c\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 5 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + b c^{5} + {\left (10 \, b c^{3} + a\right )} d^{2} x^{2} + a c^{2} + {\left (5 \, b c^{4} + 2 \, a c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} + {\left (a b d x + a b c\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) - 10 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + b c^{5} + {\left (10 \, b c^{3} + a\right )} d^{2} x^{2} + a c^{2} + {\left (5 \, b c^{4} + 2 \, a c\right )} d x\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b d x + b c - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 9 \, a}{18 \, {\left (a^{2} b d^{6} e^{3} x^{5} + 5 \, a^{2} b c d^{5} e^{3} x^{4} + 10 \, a^{2} b c^{2} d^{4} e^{3} x^{3} + {\left (10 \, a^{2} b c^{3} + a^{3}\right )} d^{3} e^{3} x^{2} + {\left (5 \, a^{2} b c^{4} + 2 \, a^{3} c\right )} d^{2} e^{3} x + {\left (a^{2} b c^{5} + a^{3} c^{2}\right )} d e^{3}\right )}} \]
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Time = 1.36 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^2} \, dx=\frac {- 3 a - 5 b c^{3} - 15 b c^{2} d x - 15 b c d^{2} x^{2} - 5 b d^{3} x^{3}}{6 a^{3} c^{2} d e^{3} + 6 a^{2} b c^{5} d e^{3} + 60 a^{2} b c^{2} d^{4} e^{3} x^{3} + 30 a^{2} b c d^{5} e^{3} x^{4} + 6 a^{2} b d^{6} e^{3} x^{5} + x^{2} \cdot \left (6 a^{3} d^{3} e^{3} + 60 a^{2} b c^{3} d^{3} e^{3}\right ) + x \left (12 a^{3} c d^{2} e^{3} + 30 a^{2} b c^{4} d^{2} e^{3}\right )} + \frac {\operatorname {RootSum} {\left (729 t^{3} a^{8} + 125 b^{2}, \left ( t \mapsto t \log {\left (x + \frac {- 9 t a^{3} + 5 b c}{5 b d} \right )} \right )\right )}}{d e^{3}} \]
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\[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^2} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{2} {\left (d e x + c e\right )}^{3}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^2} \, dx=\frac {5 \, {\left (2 \, \sqrt {3} \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}}\right ) - \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c + \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (-\frac {b^{2}}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | b d x + b c - \left (-a b^{2}\right )^{\frac {1}{3}} \right |}\right )\right )}}{18 \, a^{2} e^{3}} - \frac {b d x + b c}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} a^{2} d e^{3}} - \frac {1}{2 \, {\left (d x + c\right )}^{2} a^{2} d e^{3}} \]
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Time = 6.66 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.43 \[ \int \frac {1}{(c e+d e x)^3 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {\frac {5\,b\,c^3+3\,a}{6\,a^2\,d}+\frac {5\,b\,d^2\,x^3}{6\,a^2}+\frac {5\,b\,c^2\,x}{2\,a^2}+\frac {5\,b\,c\,d\,x^2}{2\,a^2}}{x\,\left (5\,b\,d\,c^4\,e^3+2\,a\,d\,c\,e^3\right )+x^2\,\left (10\,b\,c^3\,d^2\,e^3+a\,d^2\,e^3\right )+a\,c^2\,e^3+b\,c^5\,e^3+b\,d^5\,e^3\,x^5+5\,b\,c\,d^4\,e^3\,x^4+10\,b\,c^2\,d^3\,e^3\,x^3}-\frac {5\,b^{2/3}\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{9\,a^{8/3}\,d\,e^3}+\frac {5\,b^{2/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x-\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{8/3}\,d\,e^3}-\frac {5\,b^{2/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{9\,a^{8/3}\,d\,e^3} \]
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