Optimal. Leaf size=488 \[ \frac{3^{2/3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\frac{f \left (b^2 h^2-b c g h+c^2 g^2\right )}{3 c^2 h^2}+\frac{b f x}{c}+f x^2\right )}{2 f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}-\frac{3\ 3^{2/3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\left (1-\frac{3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}\right )}{2 f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}+\frac{3 \sqrt [6]{3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left (1-\frac{3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}}{\sqrt{3} \sqrt [3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}}\right )}{f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}} \]
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Rubi [A] time = 0.360419, antiderivative size = 488, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 104, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.019, Rules used = {1041, 1040} \[ \frac{3^{2/3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\frac{f \left (b^2 h^2-b c g h+c^2 g^2\right )}{3 c^2 h^2}+\frac{b f x}{c}+f x^2\right )}{2 f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}-\frac{3\ 3^{2/3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\left (1-\frac{3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}\right )}{2 f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}+\frac{3 \sqrt [6]{3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left (1-\frac{3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}}{\sqrt{3} \sqrt [3]{\frac{3 h (b+2 c x)}{2 c g-b h}+1}}\right )}{f \sqrt [3]{-\frac{(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}} \]
Antiderivative was successfully verified.
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Rule 1041
Rule 1040
Rubi steps
\begin{align*} \int \frac{g+h x}{\sqrt [3]{\frac{-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac{f \left (b^2-\frac{-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac{b f x}{c}+f x^2\right )} \, dx &=\frac{\sqrt [3]{-\frac{c \left (\frac{-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2\right )}{b^2-\frac{4 \left (-c^2 g^2+b c g h+2 b^2 h^2\right )}{9 h^2}}} \int \frac{g+h x}{\left (\frac{f \left (b^2-\frac{-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac{b f x}{c}+f x^2\right ) \sqrt [3]{-\frac{-c^2 g^2+b c g h+2 b^2 h^2}{9 h^2 \left (b^2-\frac{4 \left (-c^2 g^2+b c g h+2 b^2 h^2\right )}{9 h^2}\right )}-\frac{b c x}{b^2-\frac{4 \left (-c^2 g^2+b c g h+2 b^2 h^2\right )}{9 h^2}}-\frac{c^2 x^2}{b^2-\frac{4 \left (-c^2 g^2+b c g h+2 b^2 h^2\right )}{9 h^2}}}} \, dx}{\sqrt [3]{\frac{-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2}}\\ &=\frac{3 \sqrt [6]{3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (c g+b h)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2^{2/3} \left (1-\frac{3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}}{\sqrt{3} \sqrt [3]{1+\frac{3 h (b+2 c x)}{2 c g-b h}}}\right )}{f \sqrt [3]{-\frac{(c g-2 b h) (c g+b h)}{c h^2}+9 b x+9 c x^2}}+\frac{3^{2/3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (c g+b h)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\frac{f \left (c^2 g^2-b c g h+b^2 h^2\right )}{3 c^2 h^2}+\frac{b f x}{c}+f x^2\right )}{2 f \sqrt [3]{-\frac{(c g-2 b h) (c g+b h)}{c h^2}+9 b x+9 c x^2}}-\frac{3\ 3^{2/3} h \sqrt [3]{\frac{c h^2 \left (\frac{(c g-2 b h) (c g+b h)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\left (1-\frac{3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\frac{3 h (b+2 c x)}{2 c g-b h}}\right )}{2 f \sqrt [3]{-\frac{(c g-2 b h) (c g+b h)}{c h^2}+9 b x+9 c x^2}}\\ \end{align*}
Mathematica [F] time = 0.549227, size = 0, normalized size = 0. \[ \int \frac{g+h x}{\sqrt [3]{\frac{-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac{f \left (b^2-\frac{-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac{b f x}{c}+f x^2\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 3.165, size = 0, normalized size = 0. \begin{align*} \int{(hx+g){\frac{1}{\sqrt [3]{{\frac{2\,{b}^{2}{h}^{2}+bcgh-{c}^{2}{g}^{2}}{9\,c{h}^{2}}}+bx+c{x}^{2}}}} \left ({\frac{f}{{c}^{2}} \left ({b}^{2}+{\frac{-2\,{b}^{2}{h}^{2}-bcgh+{c}^{2}{g}^{2}}{3\,{h}^{2}}} \right ) }+{\frac{bfx}{c}}+f{x}^{2} \right ) ^{-1}}\, 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*} 3 \, \int \frac{h x + g}{{\left (c x^{2} + b x - \frac{c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{9 \, c h^{2}}\right )}^{\frac{1}{3}}{\left (3 \, f x^{2} + \frac{3 \, b f x}{c} + \frac{{\left (3 \, b^{2} + \frac{c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{h^{2}}\right )} f}{c^{2}}\right )}}\,{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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{3 \,{\left (h x + g\right )}}{{\left (c x^{2} + b x - \frac{c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{9 \, c h^{2}}\right )}^{\frac{1}{3}}{\left (3 \, f x^{2} + \frac{3 \, b f x}{c} + \frac{{\left (3 \, b^{2} + \frac{c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{h^{2}}\right )} f}{c^{2}}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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