Integrand size = 59, antiderivative size = 47 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}} \]
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Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {1608, 1482, 642} \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}} \]
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Rule 642
Rule 1482
Rule 1608
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{\left (c \sqrt [3]{d}-c^{2/3} d^{2/3} \sqrt [3]{x}+\sqrt [3]{c} d x^{2/3}\right ) x^{2/3}} \, dx \\ & = 3 \text {Subst}\left (\int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{c \sqrt [3]{d}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^2} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}} \]
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Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(-\frac {3 \ln \left (c^{\frac {2}{3}} d^{\frac {2}{3}} x^{\frac {1}{3}}-c^{\frac {1}{3}} d \,x^{\frac {2}{3}}-c \,d^{\frac {1}{3}}\right )}{d^{\frac {2}{3}} c^{\frac {1}{3}}}\) | \(36\) |
default | \(-\frac {3 \ln \left (c^{\frac {2}{3}} d^{\frac {2}{3}} x^{\frac {1}{3}}-c^{\frac {1}{3}} d \,x^{\frac {2}{3}}-c \,d^{\frac {1}{3}}\right )}{d^{\frac {2}{3}} c^{\frac {1}{3}}}\) | \(36\) |
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none
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3 \, \log \left (d x^{\frac {2}{3}} - c^{\frac {1}{3}} d^{\frac {2}{3}} x^{\frac {1}{3}} + c^{\frac {2}{3}} d^{\frac {1}{3}}\right )}{c^{\frac {1}{3}} d^{\frac {2}{3}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (44) = 88\).
Time = 2.61 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.53 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=- \frac {3 \log {\left (- \frac {\sqrt [3]{c}}{2 \sqrt [3]{d}} + \sqrt [3]{x} - \frac {\sqrt {3} \sqrt {- c^{\frac {4}{3}} d^{\frac {4}{3}}}}{2 \sqrt [3]{c} d} \right )}}{\sqrt [3]{c} d^{\frac {2}{3}}} - \frac {3 \log {\left (- \frac {\sqrt [3]{c}}{2 \sqrt [3]{d}} + \sqrt [3]{x} + \frac {\sqrt {3} \sqrt {- c^{\frac {4}{3}} d^{\frac {4}{3}}}}{2 \sqrt [3]{c} d} \right )}}{\sqrt [3]{c} d^{\frac {2}{3}}} \]
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none
Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3 \, \log \left (c^{\frac {1}{3}} d x^{\frac {2}{3}} - c^{\frac {2}{3}} d^{\frac {2}{3}} x^{\frac {1}{3}} + c d^{\frac {1}{3}}\right )}{c^{\frac {1}{3}} d^{\frac {2}{3}}} \]
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Exception generated. \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=\text {Exception raised: TypeError} \]
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Time = 8.82 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} \sqrt [3]{x}}{c \sqrt [3]{d} x^{2/3}-c^{2/3} d^{2/3} x+\sqrt [3]{c} d x^{4/3}} \, dx=-\frac {3\,\ln \left (x^{2/3}+\frac {c^{2/3}}{d^{2/3}}-\frac {c^{1/3}\,x^{1/3}}{d^{1/3}}\right )}{c^{1/3}\,d^{2/3}} \]
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