Integrand size = 17, antiderivative size = 235 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx=\frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}-\frac {5 x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{54 a^{8/3} \sqrt [3]{b}} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {260, 205, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx=-\frac {5 x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b}}-\frac {5 x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac {5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2} \]
[In]
[Out]
Rule 31
Rule 205
Rule 206
Rule 210
Rule 260
Rule 631
Rule 642
Rule 648
Rubi steps \begin{align*} \text {integral}& = \left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{\left (a+b x^3\right )^3} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right ) \\ & = \frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{\left (a+b x^3\right )^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{6 a} \\ & = \frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{a+b x^3} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{9 a^2} \\ & = \frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3}}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \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,\left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3}} \\ & = \frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\left (c x^n\right )^{\frac {1}{n}}\right )}{18 a^{7/3}}-\frac {\left (5 x \left (c x^n\right )^{-1/n}\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,\left (c x^n\right )^{\frac {1}{n}}\right )}{54 a^{8/3} \sqrt [3]{b}} \\ & = \frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}+\frac {5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{54 a^{8/3} \sqrt [3]{b}}+\frac {\left (5 x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}\right )}{9 a^{8/3} \sqrt [3]{b}} \\ & = \frac {x}{6 a \left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {5 x}{18 a^2 \left (a+b \left (c x^n\right )^{3/n}\right )}-\frac {5 x \left (c x^n\right )^{-1/n} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {5 x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{27 a^{8/3} \sqrt [3]{b}}-\frac {5 x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{54 a^{8/3} \sqrt [3]{b}} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx=\frac {x \left (c x^n\right )^{-1/n} \left (\frac {9 a^{5/3} \left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{3/n}\right )^2}+\frac {15 a^{2/3} \left (c x^n\right )^{\frac {1}{n}}}{a+b \left (c x^n\right )^{3/n}}-\frac {10 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{\sqrt [3]{b}}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{\sqrt [3]{b}}\right )}{54 a^{8/3}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.29 (sec) , antiderivative size = 981, normalized size of antiderivative = 4.17
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (194) = 388\).
Time = 0.30 (sec) , antiderivative size = 885, normalized size of antiderivative = 3.77 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx=\left [\frac {15 \, a^{2} b^{2} c^{\frac {6}{n}} x^{4} + 24 \, a^{3} b c^{\frac {3}{n}} x + 15 \, \sqrt {\frac {1}{3}} {\left (a b^{3} c^{\frac {9}{n}} x^{6} + 2 \, a^{2} b^{2} c^{\frac {6}{n}} x^{3} + a^{3} b c^{\frac {3}{n}}\right )} \sqrt {-\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}} \log \left (\frac {2 \, a b c^{\frac {3}{n}} x^{3} - 3 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b c^{\frac {3}{n}} x^{2} + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}}}{b c^{\frac {3}{n}} x^{3} + a}\right ) - 5 \, {\left (b^{2} c^{\frac {6}{n}} x^{6} + 2 \, a b c^{\frac {3}{n}} x^{3} + a^{2}\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x^{2} - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right ) + 10 \, {\left (b^{2} c^{\frac {6}{n}} x^{6} + 2 \, a b c^{\frac {3}{n}} x^{3} + a^{2}\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{4} b^{3} c^{\frac {9}{n}} x^{6} + 2 \, a^{5} b^{2} c^{\frac {6}{n}} x^{3} + a^{6} b c^{\frac {3}{n}}\right )}}, \frac {15 \, a^{2} b^{2} c^{\frac {6}{n}} x^{4} + 24 \, a^{3} b c^{\frac {3}{n}} x + 30 \, \sqrt {\frac {1}{3}} {\left (a b^{3} c^{\frac {9}{n}} x^{6} + 2 \, a^{2} b^{2} c^{\frac {6}{n}} x^{3} + a^{3} b c^{\frac {3}{n}}\right )} \sqrt {\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}}}{a^{2}}\right ) - 5 \, {\left (b^{2} c^{\frac {6}{n}} x^{6} + 2 \, a b c^{\frac {3}{n}} x^{3} + a^{2}\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x^{2} - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right ) + 10 \, {\left (b^{2} c^{\frac {6}{n}} x^{6} + 2 \, a b c^{\frac {3}{n}} x^{3} + a^{2}\right )} \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{4} b^{3} c^{\frac {9}{n}} x^{6} + 2 \, a^{5} b^{2} c^{\frac {6}{n}} x^{3} + a^{6} b c^{\frac {3}{n}}\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx=\int \frac {1}{\left (a + b \left (c x^{n}\right )^{\frac {3}{n}}\right )^{3}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {3}{n}} b + a\right )}^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {3}{n}} b + a\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{3/n}\right )^3} \, dx=\int \frac {1}{{\left (a+b\,{\left (c\,x^n\right )}^{3/n}\right )}^3} \,d x \]
[In]
[Out]