Optimal. Leaf size=47 \[ -\frac{3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}} \]
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Rubi [A] time = 0.062168, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 59, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {1594, 1468, 628} \[ -\frac{3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}} \]
Antiderivative was successfully verified.
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Rule 1594
Rule 1468
Rule 628
Rubi steps
\begin{align*} \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 &=\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 \operatorname{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*}
Mathematica [A] time = 0.0198506, size = 47, normalized size = 1. \[ -\frac{3 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{x}+d^{2/3} x^{2/3}\right )}{\sqrt [3]{c} d^{2/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 36, normalized size = 0.8 \begin{align*} -3\,{\frac{\ln \left ({c}^{2/3}{d}^{2/3}\sqrt [3]{x}-\sqrt [3]{c}{x}^{2/3}d-c\sqrt [3]{d} \right ) }{{d}^{2/3}\sqrt [3]{c}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06346, size = 46, normalized size = 0.98 \begin{align*} -\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38533, size = 109, normalized size = 2.32 \begin{align*} -\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 7.14819, size = 126, normalized size = 2.68 \begin{align*} - \frac{3 \log{\left (- \frac{\sqrt [3]{c}}{2 \sqrt [3]{d}} + \sqrt [3]{x} - \frac{\sqrt{3} i \sqrt{c^{\frac{4}{3}}} \sqrt{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} i \sqrt{c^{\frac{4}{3}}} \sqrt{d^{\frac{4}{3}}}}{2 \sqrt [3]{c} d} \right )}}{\sqrt [3]{c} d^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16123, size = 46, normalized size = 0.98 \begin{align*} -\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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