Optimal. Leaf size=172 \[ -\frac{(b c-a d) \log (c+d x)}{6 b^{4/3} d^{2/3}}-\frac{(b c-a d) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{2/3}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} d^{2/3}}+\frac{(a+b x)^{2/3} \sqrt [3]{c+d x}}{b} \]
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Rubi [A] time = 0.0455278, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {50, 59} \[ -\frac{(b c-a d) \log (c+d x)}{6 b^{4/3} d^{2/3}}-\frac{(b c-a d) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{2/3}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} b^{4/3} d^{2/3}}+\frac{(a+b x)^{2/3} \sqrt [3]{c+d x}}{b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 59
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{c+d x}}{\sqrt [3]{a+b x}} \, dx &=\frac{(a+b x)^{2/3} \sqrt [3]{c+d x}}{b}+\frac{(b c-a d) \int \frac{1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{3 b}\\ &=\frac{(a+b x)^{2/3} \sqrt [3]{c+d x}}{b}-\frac{(b c-a d) \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt{3} b^{4/3} d^{2/3}}-\frac{(b c-a d) \log (c+d x)}{6 b^{4/3} d^{2/3}}-\frac{(b c-a d) \log \left (-1+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 b^{4/3} d^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0235564, size = 73, normalized size = 0.42 \[ \frac{3 (a+b x)^{2/3} \sqrt [3]{c+d x} \, _2F_1\left (-\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{d (a+b x)}{a d-b c}\right )}{2 b \sqrt [3]{\frac{b (c+d x)}{b c-a d}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.015, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [3]{dx+c}{\frac{1}{\sqrt [3]{bx+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.00917, size = 1547, normalized size = 8.99 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [3]{c + d x}}{\sqrt [3]{a + b x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{\frac{1}{3}}}{{\left (b x + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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