Integrand size = 104, antiderivative size = 488 \[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\frac {3 \sqrt [6]{3} h \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (c g+b h)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \left (1-\frac {3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}}{\sqrt {3} \sqrt [3]{1+\frac {3 h (b+2 c x)}{2 c g-b h}}}\right )}{f \sqrt [3]{-\frac {(c g-2 b h) (c g+b h)}{c h^2}+9 b x+9 c x^2}}+\frac {3^{2/3} h \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (c g+b h)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\frac {f \left (c^2 g^2-b c g h+b^2 h^2\right )}{3 c^2 h^2}+\frac {b f x}{c}+f x^2\right )}{2 f \sqrt [3]{-\frac {(c g-2 b h) (c g+b h)}{c h^2}+9 b x+9 c x^2}}-\frac {3\ 3^{2/3} h \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (c g+b h)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\left (1-\frac {3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{1+\frac {3 h (b+2 c x)}{2 c g-b h}}\right )}{2 f \sqrt [3]{-\frac {(c g-2 b h) (c g+b h)}{c h^2}+9 b x+9 c x^2}} \]
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Time = 0.23 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {1055, 1054} \[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\frac {3 \sqrt [6]{3} h \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \left (1-\frac {3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}}{\sqrt {3} \sqrt [3]{\frac {3 h (b+2 c x)}{2 c g-b h}+1}}\right )}{f \sqrt [3]{-\frac {(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}+\frac {3^{2/3} h \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\frac {f \left (b^2 h^2-b c g h+c^2 g^2\right )}{3 c^2 h^2}+\frac {b f x}{c}+f x^2\right )}{2 f \sqrt [3]{-\frac {(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}}-\frac {3\ 3^{2/3} h \sqrt [3]{\frac {c h^2 \left (\frac {(c g-2 b h) (b h+c g)}{c h^2}-9 b x-9 c x^2\right )}{(2 c g-b h)^2}} \log \left (\left (1-\frac {3 h (b+2 c x)}{2 c g-b h}\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{\frac {3 h (b+2 c x)}{2 c g-b h}+1}\right )}{2 f \sqrt [3]{-\frac {(c g-2 b h) (b h+c g)}{c h^2}+9 b x+9 c x^2}} \]
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Rule 1054
Rule 1055
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 2.28 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.22 \[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\frac {3^{2/3} \sqrt [3]{c} h^{5/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{c} h^{2/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}}{-4 b h+2 c (g-3 h x)+\sqrt [3]{c} h^{2/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}}\right )-2 \log \left (\sqrt {h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}\right )+\log \left (h \left (\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2\right )^{2/3}\right )+2 \log \left (\sqrt {c} \left (c g-2 b h-3 c h x-\sqrt [3]{c} h^{2/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}\right )\right )-\log \left (c \left (4 b^2 h^2-4 b c h (g-3 h x)+c^2 (g-3 h x)^2-2 b \sqrt [3]{c} h^{5/3} \sqrt [3]{2 c g-b h} \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}+c^{4/3} h^{2/3} \sqrt [3]{2 c g-b h} (g-3 h x) \sqrt [3]{\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2}+c^{2/3} h^{4/3} (2 c g-b h)^{2/3} \left (\frac {2 b^2}{c}-\frac {c g^2}{h^2}+\frac {b g}{h}+9 b x+9 c x^2\right )^{2/3}\right )\right )\right )}{2 f (2 c g-b h)^{2/3}} \]
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\[\int \frac {h x +g}{\left (\frac {2 b^{2} h^{2}+b c g h -c^{2} g^{2}}{9 c \,h^{2}}+b x +c \,x^{2}\right )^{\frac {1}{3}} \left (\frac {f \left (b^{2}+\frac {-2 b^{2} h^{2}-b c g h +c^{2} g^{2}}{3 h^{2}}\right )}{c^{2}}+\frac {b f x}{c}+f \,x^{2}\right )}d x\]
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Timed out. \[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\frac {3 \cdot 3^{\frac {2}{3}} c^{2} h^{2} \left (\int \frac {g}{b^{2} h^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} - b c g h \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + 3 b c h^{2} x \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + c^{2} g^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + 3 c^{2} h^{2} x^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}}}\, dx + \int \frac {h x}{b^{2} h^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} - b c g h \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + 3 b c h^{2} x \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + c^{2} g^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}} + 3 c^{2} h^{2} x^{2} \sqrt [3]{\frac {2 b^{2}}{c} + \frac {b g}{h} + 9 b x - \frac {c g^{2}}{h^{2}} + 9 c x^{2}}}\, dx\right )}{f} \]
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\[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\int { \frac {3 \, {\left (h x + g\right )}}{{\left (c x^{2} + b x - \frac {c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{9 \, c h^{2}}\right )}^{\frac {1}{3}} {\left (3 \, f x^{2} + \frac {3 \, b f x}{c} + \frac {{\left (3 \, b^{2} + \frac {c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{h^{2}}\right )} f}{c^{2}}\right )}} \,d x } \]
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\[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\int { \frac {3 \, {\left (h x + g\right )}}{{\left (c x^{2} + b x - \frac {c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{9 \, c h^{2}}\right )}^{\frac {1}{3}} {\left (3 \, f x^{2} + \frac {3 \, b f x}{c} + \frac {{\left (3 \, b^{2} + \frac {c^{2} g^{2} - b c g h - 2 \, b^{2} h^{2}}{h^{2}}\right )} f}{c^{2}}\right )}} \,d x } \]
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Timed out. \[ \int \frac {g+h x}{\sqrt [3]{\frac {-c^2 g^2+b c g h+2 b^2 h^2}{9 c h^2}+b x+c x^2} \left (\frac {f \left (b^2-\frac {-c^2 g^2+b c g h+2 b^2 h^2}{3 h^2}\right )}{c^2}+\frac {b f x}{c}+f x^2\right )} \, dx=\int \frac {g+h\,x}{{\left (b\,x+c\,x^2+\frac {\frac {2\,b^2\,h^2}{9}+\frac {b\,c\,g\,h}{9}-\frac {c^2\,g^2}{9}}{c\,h^2}\right )}^{1/3}\,\left (f\,x^2-\frac {f\,\left (\frac {\frac {2\,b^2\,h^2}{3}+\frac {b\,c\,g\,h}{3}-\frac {c^2\,g^2}{3}}{h^2}-b^2\right )}{c^2}+\frac {b\,f\,x}{c}\right )} \,d x \]
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