Integrand size = 15, antiderivative size = 115 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=\frac {b \sqrt [3]{(a+b x)^2}}{a}-\frac {\sqrt [3]{(a+b x)^5}}{a x}+\frac {b \left (\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a+b x}}{2 \sqrt [3]{a}+\sqrt [3]{a+b x}}\right )+\frac {3}{2} \log \left (\frac {-\sqrt [3]{a}+\sqrt [3]{a+b x}}{\sqrt [3]{x}}\right )\right )}{\sqrt [3]{a^2}} \]
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Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.32, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1973, 43, 57, 632, 210, 31} \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=\frac {2 b \sqrt [3]{(a+b x)^2} \arctan \left (\frac {2 \sqrt [3]{\frac {b x}{a}+1}+1}{\sqrt {3}}\right )}{\sqrt {3} a \left (\frac {b x}{a}+1\right )^{2/3}}-\frac {\sqrt [3]{(a+b x)^2}}{x}-\frac {b \log (x) \sqrt [3]{(a+b x)^2}}{3 a \left (\frac {b x}{a}+1\right )^{2/3}}+\frac {b \sqrt [3]{(a+b x)^2} \log \left (1-\sqrt [3]{\frac {b x}{a}+1}\right )}{a \left (\frac {b x}{a}+1\right )^{2/3}} \]
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Rule 31
Rule 43
Rule 57
Rule 210
Rule 632
Rule 1973
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{(a+b x)^2} \int \frac {\left (1+\frac {b x}{a}\right )^{2/3}}{x^2} \, dx}{\left (1+\frac {b x}{a}\right )^{2/3}} \\ & = -\frac {\sqrt [3]{(a+b x)^2}}{x}+\frac {\left (2 b \sqrt [3]{(a+b x)^2}\right ) \int \frac {1}{x \sqrt [3]{1+\frac {b x}{a}}} \, dx}{3 a \left (1+\frac {b x}{a}\right )^{2/3}} \\ & = -\frac {\sqrt [3]{(a+b x)^2}}{x}-\frac {b \sqrt [3]{(a+b x)^2} \log (x)}{3 a \left (1+\frac {b x}{a}\right )^{2/3}}-\frac {\left (b \sqrt [3]{(a+b x)^2}\right ) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+\frac {b x}{a}}\right )}{a \left (1+\frac {b x}{a}\right )^{2/3}}+\frac {\left (b \sqrt [3]{(a+b x)^2}\right ) \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+\frac {b x}{a}}\right )}{a \left (1+\frac {b x}{a}\right )^{2/3}} \\ & = -\frac {\sqrt [3]{(a+b x)^2}}{x}-\frac {b \sqrt [3]{(a+b x)^2} \log (x)}{3 a \left (1+\frac {b x}{a}\right )^{2/3}}+\frac {b \sqrt [3]{(a+b x)^2} \log \left (1-\sqrt [3]{1+\frac {b x}{a}}\right )}{a \left (1+\frac {b x}{a}\right )^{2/3}}-\frac {\left (2 b \sqrt [3]{(a+b x)^2}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+\frac {b x}{a}}\right )}{a \left (1+\frac {b x}{a}\right )^{2/3}} \\ & = -\frac {\sqrt [3]{(a+b x)^2}}{x}+\frac {2 b \sqrt [3]{(a+b x)^2} \arctan \left (\frac {1+2 \sqrt [3]{1+\frac {b x}{a}}}{\sqrt {3}}\right )}{\sqrt {3} a \left (1+\frac {b x}{a}\right )^{2/3}}-\frac {b \sqrt [3]{(a+b x)^2} \log (x)}{3 a \left (1+\frac {b x}{a}\right )^{2/3}}+\frac {b \sqrt [3]{(a+b x)^2} \log \left (1-\sqrt [3]{1+\frac {b x}{a}}\right )}{a \left (1+\frac {b x}{a}\right )^{2/3}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=-\frac {\sqrt [3]{(a+b x)^2} \left (3 \sqrt [3]{a} (a+b x)^{2/3}-2 \sqrt {3} b x \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b x \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )+b x \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )\right )}{3 \sqrt [3]{a} x (a+b x)^{2/3}} \]
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Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95
method | result | size |
risch | \(-\frac {\left (\left (b x +a \right )^{2}\right )^{\frac {1}{3}}}{x}+\frac {2 b \left (\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{a^{\frac {1}{3}}}-\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+\left (b x +a \right )^{\frac {1}{3}} a^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{2 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{a^{\frac {1}{3}}}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {1}{3}}}{3 \left (b x +a \right )^{\frac {2}{3}}}\) | \(109\) |
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Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (93) = 186\).
Time = 0.26 (sec) , antiderivative size = 560, normalized size of antiderivative = 4.87 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b x \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (-\frac {b^{2} x^{2} + 4 \, a b x + 3 \, a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left ({\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} a^{\frac {2}{3}} - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {2}{3}} a + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} a^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} a^{\frac {1}{3}}}{b x^{2} + a x}\right ) - a^{\frac {2}{3}} b x \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} a^{\frac {2}{3}} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {2}{3}} a + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} a^{\frac {1}{3}}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + 2 \, a^{\frac {2}{3}} b x \log \left (-\frac {{\left (b x + a\right )} a^{\frac {2}{3}} - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a}{b x + a}\right ) - 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a}{3 \, a x}, -\frac {6 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} b x \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left ({\left (b x + a\right )} a^{\frac {1}{3}} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a^{\frac {2}{3}}\right )}}{{\left (b x + a\right )} a^{\frac {1}{3}}}\right ) + a^{\frac {2}{3}} b x \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} {\left (b x + a\right )} a^{\frac {2}{3}} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {2}{3}} a + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} a^{\frac {1}{3}}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - 2 \, a^{\frac {2}{3}} b x \log \left (-\frac {{\left (b x + a\right )} a^{\frac {2}{3}} - {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a}{b x + a}\right ) + 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {1}{3}} a}{3 \, a x}\right ] \]
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\[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=\int \frac {\sqrt [3]{\left (a + b x\right )^{2}}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=\int { \frac {{\left ({\left (b x + a\right )}^{2}\right )}^{\frac {1}{3}}}{x^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (93) = 186\).
Time = 3.04 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=\frac {\frac {2 \, \sqrt {3} \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} + \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}}\right )}}{3 \, \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}}}\right )}{a} - \frac {\left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} b^{2} \log \left ({\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {2}{3}} + {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} + \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}}\right )}{a} + \frac {2 \, \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {2}{3}} b^{2} \log \left ({\left | {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {1}{3}} - \left (a \mathrm {sgn}\left (b x + a\right )\right )^{\frac {1}{3}} \right |}\right )}{a} - \frac {3 \, {\left (b x \mathrm {sgn}\left (b x + a\right ) + a \mathrm {sgn}\left (b x + a\right )\right )}^{\frac {2}{3}} b}{x}}{3 \, b} \]
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Timed out. \[ \int \frac {\sqrt [3]{(a+b x)^2}}{x^2} \, dx=\int \frac {{\left ({\left (a+b\,x\right )}^2\right )}^{1/3}}{x^2} \,d x \]
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