Integrand size = 13, antiderivative size = 130 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=-\frac {(a+b x)^{2/3}}{2 a x^2}+\frac {2 b (a+b x)^{2/3}}{3 a^2 x}+\frac {2 b^2 \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}-\frac {b^2 \log (x)}{9 a^{7/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{7/3}} \]
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Time = 0.03 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {44, 57, 631, 210, 31} \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\frac {2 b^2 \arctan \left (\frac {2 \sqrt [3]{a+b x}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}-\frac {b^2 \log (x)}{9 a^{7/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{7/3}}+\frac {2 b (a+b x)^{2/3}}{3 a^2 x}-\frac {(a+b x)^{2/3}}{2 a x^2} \]
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Rule 31
Rule 44
Rule 57
Rule 210
Rule 631
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{2/3}}{2 a x^2}-\frac {(2 b) \int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx}{3 a} \\ & = -\frac {(a+b x)^{2/3}}{2 a x^2}+\frac {2 b (a+b x)^{2/3}}{3 a^2 x}+\frac {\left (2 b^2\right ) \int \frac {1}{x \sqrt [3]{a+b x}} \, dx}{9 a^2} \\ & = -\frac {(a+b x)^{2/3}}{2 a x^2}+\frac {2 b (a+b x)^{2/3}}{3 a^2 x}-\frac {b^2 \log (x)}{9 a^{7/3}}-\frac {b^2 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x}\right )}{3 a^{7/3}}+\frac {b^2 \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x}\right )}{3 a^2} \\ & = -\frac {(a+b x)^{2/3}}{2 a x^2}+\frac {2 b (a+b x)^{2/3}}{3 a^2 x}-\frac {b^2 \log (x)}{9 a^{7/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{7/3}}-\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}\right )}{3 a^{7/3}} \\ & = -\frac {(a+b x)^{2/3}}{2 a x^2}+\frac {2 b (a+b x)^{2/3}}{3 a^2 x}+\frac {2 b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{7/3}}-\frac {b^2 \log (x)}{9 a^{7/3}}+\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{3 a^{7/3}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=-\frac {(a+b x)^{2/3} (7 a-4 (a+b x))}{6 a^2 x^2}+\frac {2 b^2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{7/3}}+\frac {2 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x}\right )}{9 a^{7/3}}-\frac {b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x}+(a+b x)^{2/3}\right )}{9 a^{7/3}} \]
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Time = 0.10 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.84
method | result | size |
risch | \(-\frac {\left (b x +a \right )^{\frac {2}{3}} \left (-4 b x +3 a \right )}{6 a^{2} x^{2}}+\frac {2 b^{2} \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{9 a^{\frac {7}{3}}}-\frac {b^{2} \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{9 a^{\frac {7}{3}}}+\frac {2 b^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{9 a^{\frac {7}{3}}}\) | \(109\) |
pseudoelliptic | \(\frac {4 b^{2} \arctan \left (\frac {\left (a^{\frac {1}{3}}+2 \left (b x +a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a^{\frac {1}{3}}}\right ) \sqrt {3}\, x^{2}+4 b^{2} \ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right ) x^{2}-2 b^{2} \ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right ) x^{2}+12 b x \left (b x +a \right )^{\frac {2}{3}} a^{\frac {1}{3}}-9 \left (b x +a \right )^{\frac {2}{3}} a^{\frac {4}{3}}}{18 a^{\frac {7}{3}} x^{2}}\) | \(122\) |
derivativedivides | \(3 b^{2} \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{6 a \,b^{2} x^{2}}-\frac {2 \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 a b x}+\frac {-\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{3 a}\right )}{3 a}\right )\) | \(130\) |
default | \(3 b^{2} \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{6 a \,b^{2} x^{2}}-\frac {2 \left (-\frac {\left (b x +a \right )^{\frac {2}{3}}}{3 a b x}+\frac {-\frac {\ln \left (\left (b x +a \right )^{\frac {1}{3}}-a^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x +a \right )^{\frac {2}{3}}+a^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x +a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{3 a}\right )}{3 a}\right )\) | \(130\) |
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Time = 0.24 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.28 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\left [\frac {6 \, \sqrt {\frac {1}{3}} a b^{2} x^{2} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x}\right ) - 2 \, a^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 4 \, a^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b x - 3 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, a^{3} x^{2}}, \frac {12 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} b^{2} x^{2} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right ) - 2 \, a^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 4 \, a^{\frac {2}{3}} b^{2} x^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (4 \, a b x - 3 \, a^{2}\right )} {\left (b x + a\right )}^{\frac {2}{3}}}{18 \, a^{3} x^{2}}\right ] \]
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Result contains complex when optimal does not.
Time = 2.94 (sec) , antiderivative size = 2730, normalized size of antiderivative = 21.00 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\text {Too large to display} \]
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Time = 0.30 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{9 \, a^{\frac {7}{3}}} - \frac {b^{2} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{9 \, a^{\frac {7}{3}}} + \frac {2 \, b^{2} \log \left ({\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{9 \, a^{\frac {7}{3}}} + \frac {4 \, {\left (b x + a\right )}^{\frac {5}{3}} b^{2} - 7 \, {\left (b x + a\right )}^{\frac {2}{3}} a b^{2}}{6 \, {\left ({\left (b x + a\right )}^{2} a^{2} - 2 \, {\left (b x + a\right )} a^{3} + a^{4}\right )}} \]
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Time = 0.58 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\frac {\frac {4 \, \sqrt {3} b^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {7}{3}}} - \frac {2 \, b^{3} \log \left ({\left (b x + a\right )}^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {7}{3}}} + \frac {4 \, b^{3} \log \left ({\left | {\left (b x + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {7}{3}}} + \frac {3 \, {\left (4 \, {\left (b x + a\right )}^{\frac {5}{3}} b^{3} - 7 \, {\left (b x + a\right )}^{\frac {2}{3}} a b^{3}\right )}}{a^{2} b^{2} x^{2}}}{18 \, b} \]
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Time = 0.42 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x}} \, dx=\frac {2\,b^2\,\ln \left ({\left (a+b\,x\right )}^{1/3}-a^{1/3}\right )}{9\,a^{7/3}}-\frac {\frac {7\,b^2\,{\left (a+b\,x\right )}^{2/3}}{6\,a}-\frac {2\,b^2\,{\left (a+b\,x\right )}^{5/3}}{3\,a^2}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}-\frac {\ln \left (\frac {4\,b^4\,{\left (a+b\,x\right )}^{1/3}}{9\,a^4}-\frac {{\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )}^2}{9\,a^{11/3}}\right )\,\left (b^2+\sqrt {3}\,b^2\,1{}\mathrm {i}\right )}{9\,a^{7/3}}+\frac {b^2\,\ln \left (\frac {4\,b^4\,{\left (a+b\,x\right )}^{1/3}}{9\,a^4}-\frac {9\,b^4\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2}{a^{11/3}}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}{a^{7/3}} \]
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