Optimal. Leaf size=140 \[ -\frac{\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac{3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1}{\sqrt{3}}\right )}{\sqrt [3]{b} (b c-a d)^{2/3}} \]
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Rubi [A] time = 0.0744044, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {57, 617, 204, 31} \[ -\frac{\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac{3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1}{\sqrt{3}}\right )}{\sqrt [3]{b} (b c-a d)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) (c+d x)^{2/3}} \, dx &=-\frac{\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{b c-a d}}{\sqrt [3]{b}}-x} \, dx,x,\sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{(b c-a d)^{2/3}}{b^{2/3}}+\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{b}}+x^2} \, dx,x,\sqrt [3]{c+d x}\right )}{2 b^{2/3} \sqrt [3]{b c-a d}}\\ &=-\frac{\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac{3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}\right )}{\sqrt [3]{b} (b c-a d)^{2/3}}\\ &=-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt [3]{b} (b c-a d)^{2/3}}-\frac{\log (a+b x)}{2 \sqrt [3]{b} (b c-a d)^{2/3}}+\frac{3 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0702584, size = 154, normalized size = 1.1 \[ -\frac{\log \left (\sqrt [3]{b} \sqrt [3]{c+d x} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+b^{2/3} (c+d x)^{2/3}\right )-2 \log \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{b c-a d}}+1}{\sqrt{3}}\right )}{2 \sqrt [3]{b} (b c-a d)^{2/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 160, normalized size = 1.1 \begin{align*}{\frac{1}{b}\ln \left ( \sqrt [3]{dx+c}+\sqrt [3]{{\frac{ad-bc}{b}}} \right ) \left ({\frac{ad-bc}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{2\,b}\ln \left ( \left ( dx+c \right ) ^{{\frac{2}{3}}}-\sqrt [3]{{\frac{ad-bc}{b}}}\sqrt [3]{dx+c}+ \left ({\frac{ad-bc}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{ad-bc}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sqrt [3]{dx+c}{\frac{1}{\sqrt [3]{{\frac{ad-bc}{b}}}}}}-1 \right ) } \right ) \left ({\frac{ad-bc}{b}} \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.4266, size = 1995, normalized size = 14.25 \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{1}{\left (a + b x\right ) \left (c + d x\right )^{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12254, size = 279, normalized size = 1.99 \begin{align*} -\frac{3 \,{\left (b^{3} c - a b^{2} d\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (d x + c\right )}^{\frac{1}{3}} + \left (\frac{b c - a d}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (\frac{b c - a d}{b}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{2} c - \sqrt{3} a b d} - \frac{{\left (b^{3} c - a b^{2} d\right )}^{\frac{1}{3}} \log \left ({\left (d x + c\right )}^{\frac{2}{3}} +{\left (d x + c\right )}^{\frac{1}{3}} \left (\frac{b c - a d}{b}\right )^{\frac{1}{3}} + \left (\frac{b c - a d}{b}\right )^{\frac{2}{3}}\right )}{2 \,{\left (b^{2} c - a b d\right )}} + \frac{\left (\frac{b c - a d}{b}\right )^{\frac{1}{3}} \log \left ({\left |{\left (d x + c\right )}^{\frac{1}{3}} - \left (\frac{b c - a d}{b}\right )^{\frac{1}{3}} \right |}\right )}{b c - a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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