Optimal. Leaf size=178 \[ -\frac{\log (a+b x)}{2 b^{2/3} (b c-a d)^{2/3}}+\frac{3 \log \left (\frac{b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{a d+b c+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 b^{2/3} (c+d x)^{2/3}}{\sqrt{3} \sqrt [3]{b c-a d} \sqrt [3]{a d+b c+2 b d x}}+\frac{1}{\sqrt{3}}\right )}{2 b^{2/3} (b c-a d)^{2/3}} \]
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Rubi [A] time = 0.0715279, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {123} \[ -\frac{\log (a+b x)}{2 b^{2/3} (b c-a d)^{2/3}}+\frac{3 \log \left (\frac{b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{a d+b c+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 b^{2/3} (c+d x)^{2/3}}{\sqrt{3} \sqrt [3]{b c-a d} \sqrt [3]{a d+b c+2 b d x}}+\frac{1}{\sqrt{3}}\right )}{2 b^{2/3} (b c-a d)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 123
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 b^{2/3} (c+d x)^{2/3}}{\sqrt{3} \sqrt [3]{b c-a d} \sqrt [3]{b c+a d+2 b d x}}\right )}{2 b^{2/3} (b c-a d)^{2/3}}-\frac{\log (a+b x)}{2 b^{2/3} (b c-a d)^{2/3}}+\frac{3 \log \left (\frac{b^{2/3} (c+d x)^{2/3}}{\sqrt [3]{b c-a d}}-\sqrt [3]{b c+a d+2 b d x}\right )}{4 b^{2/3} (b c-a d)^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.148075, size = 140, normalized size = 0.79 \[ -\frac{3 \sqrt [3]{\frac{b c-a d}{2 d (a+b x)}+1} \sqrt [3]{\frac{b c-a d}{d (a+b x)}+1} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{b c-a d}{d (a+b x)},-\frac{b c-a d}{2 d (a+b x)}\right )}{2 b \sqrt [3]{c+d x} \sqrt [3]{a d+b (c+2 d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.055, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{bx+a}{\frac{1}{\sqrt [3]{dx+c}}}{\frac{1}{\sqrt [3]{2\,bdx+ad+bc}}}}\, 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{1}{{\left (2 \, b d x + b c + a d\right )}^{\frac{1}{3}}{\left (b x + a\right )}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{\left (a + b x\right ) \sqrt [3]{c + d x} \sqrt [3]{a d + b c + 2 b d 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{1}{{\left (2 \, b d x + b c + a d\right )}^{\frac{1}{3}}{\left (b x + a\right )}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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