Integrand size = 15, antiderivative size = 80 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^2} \, dx=-\frac {3 b^3}{a^4 \left (a+b \sqrt [3]{x}\right )}-\frac {1}{a^2 x}+\frac {3 b}{a^3 x^{2/3}}-\frac {9 b^2}{a^4 \sqrt [3]{x}}+\frac {12 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^5}-\frac {4 b^3 \log (x)}{a^5} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 46} \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^2} \, dx=\frac {12 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^5}-\frac {4 b^3 \log (x)}{a^5}-\frac {3 b^3}{a^4 \left (a+b \sqrt [3]{x}\right )}-\frac {9 b^2}{a^4 \sqrt [3]{x}}+\frac {3 b}{a^3 x^{2/3}}-\frac {1}{a^2 x} \]
[In]
[Out]
Rule 46
Rule 272
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {1}{x^4 (a+b x)^2} \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {1}{a^2 x^4}-\frac {2 b}{a^3 x^3}+\frac {3 b^2}{a^4 x^2}-\frac {4 b^3}{a^5 x}+\frac {b^4}{a^4 (a+b x)^2}+\frac {4 b^4}{a^5 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {3 b^3}{a^4 \left (a+b \sqrt [3]{x}\right )}-\frac {1}{a^2 x}+\frac {3 b}{a^3 x^{2/3}}-\frac {9 b^2}{a^4 \sqrt [3]{x}}+\frac {12 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^5}-\frac {4 b^3 \log (x)}{a^5} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^2} \, dx=\frac {-\frac {a^4-2 a^3 b \sqrt [3]{x}+6 a^2 b^2 x^{2/3}+12 a b^3 x}{a x+b x^{4/3}}+12 b^3 \log \left (a+b \sqrt [3]{x}\right )-4 b^3 \log (x)}{a^5} \]
[In]
[Out]
Time = 3.75 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(-\frac {3 b^{3}}{a^{4} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{a^{2} x}+\frac {3 b}{a^{3} x^{\frac {2}{3}}}-\frac {9 b^{2}}{a^{4} x^{\frac {1}{3}}}+\frac {12 b^{3} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{5}}-\frac {4 b^{3} \ln \left (x \right )}{a^{5}}\) | \(73\) |
default | \(-\frac {3 b^{3}}{a^{4} \left (a +b \,x^{\frac {1}{3}}\right )}-\frac {1}{a^{2} x}+\frac {3 b}{a^{3} x^{\frac {2}{3}}}-\frac {9 b^{2}}{a^{4} x^{\frac {1}{3}}}+\frac {12 b^{3} \ln \left (a +b \,x^{\frac {1}{3}}\right )}{a^{5}}-\frac {4 b^{3} \ln \left (x \right )}{a^{5}}\) | \(73\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^2} \, dx=-\frac {4 \, a^{3} b^{3} x + a^{6} - 12 \, {\left (b^{6} x^{2} + a^{3} b^{3} x\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + 12 \, {\left (b^{6} x^{2} + a^{3} b^{3} x\right )} \log \left (x^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b^{5} x + 3 \, a^{4} b^{2}\right )} x^{\frac {2}{3}} - 3 \, {\left (2 \, a^{2} b^{4} x + a^{5} b\right )} x^{\frac {1}{3}}}{a^{5} b^{3} x^{2} + a^{8} x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (78) = 156\).
Time = 0.99 (sec) , antiderivative size = 272, normalized size of antiderivative = 3.40 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{3}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {1}{a^{2} x} & \text {for}\: b = 0 \\- \frac {3}{5 b^{2} x^{\frac {5}{3}}} & \text {for}\: a = 0 \\- \frac {a^{4} x^{\frac {2}{3}}}{a^{6} x^{\frac {5}{3}} + a^{5} b x^{2}} + \frac {2 a^{3} b x}{a^{6} x^{\frac {5}{3}} + a^{5} b x^{2}} - \frac {6 a^{2} b^{2} x^{\frac {4}{3}}}{a^{6} x^{\frac {5}{3}} + a^{5} b x^{2}} - \frac {4 a b^{3} x^{\frac {5}{3}} \log {\left (x \right )}}{a^{6} x^{\frac {5}{3}} + a^{5} b x^{2}} + \frac {12 a b^{3} x^{\frac {5}{3}} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{a^{6} x^{\frac {5}{3}} + a^{5} b x^{2}} - \frac {12 a b^{3} x^{\frac {5}{3}}}{a^{6} x^{\frac {5}{3}} + a^{5} b x^{2}} - \frac {4 b^{4} x^{2} \log {\left (x \right )}}{a^{6} x^{\frac {5}{3}} + a^{5} b x^{2}} + \frac {12 b^{4} x^{2} \log {\left (\frac {a}{b} + \sqrt [3]{x} \right )}}{a^{6} x^{\frac {5}{3}} + a^{5} b x^{2}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^2} \, dx=-\frac {12 \, b^{3} x + 6 \, a b^{2} x^{\frac {2}{3}} - 2 \, a^{2} b x^{\frac {1}{3}} + a^{3}}{a^{4} b x^{\frac {4}{3}} + a^{5} x} + \frac {12 \, b^{3} \log \left (b x^{\frac {1}{3}} + a\right )}{a^{5}} - \frac {4 \, b^{3} \log \left (x\right )}{a^{5}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^2} \, dx=\frac {12 \, b^{3} \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{a^{5}} - \frac {4 \, b^{3} \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {12 \, a b^{3} x + 6 \, a^{2} b^{2} x^{\frac {2}{3}} - 2 \, a^{3} b x^{\frac {1}{3}} + a^{4}}{{\left (b x^{\frac {1}{3}} + a\right )} a^{5} x} \]
[In]
[Out]
Time = 5.94 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a+b \sqrt [3]{x}\right )^2 x^2} \, dx=\frac {24\,b^3\,\mathrm {atanh}\left (\frac {2\,b\,x^{1/3}}{a}+1\right )}{a^5}-\frac {\frac {1}{a}-\frac {2\,b\,x^{1/3}}{a^2}+\frac {12\,b^3\,x}{a^4}+\frac {6\,b^2\,x^{2/3}}{a^3}}{a\,x+b\,x^{4/3}} \]
[In]
[Out]