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 = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 = {1608, 1482,
642} \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Rule 642
Rule 1482
Rule 1608
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 \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*}
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Mathematica [A]
time = 0.05, size = 47, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 36, normalized size = 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 | \(-\frac {3 \ln \left (c^{\frac {2}{3}} d^{\frac {2}{3}} x^{\frac {1}{3}}-c^{\frac {1}{3}} d \,x^{\frac {2}{3}}-c \,d^{\frac {1}{3}}\right )}{d^{\frac {2}{3}} c^{\frac {1}{3}}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 34, normalized size = 0.72 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 33, normalized size = 0.70 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (44) = 88\).
time = 2.15, size = 119, normalized size = 2.53 \begin {gather*} - \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.46, size = 31, normalized size = 0.66 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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