Optimal. Leaf size=549 \[ -\frac{\left (\left (e-\sqrt{e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right )+2 d f (c e-b f)\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\left (\left (\sqrt{e^2-4 d f}+e\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right )+2 d f (c e-b f)\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{(2 c e-b f) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} f^2}+\frac{\sqrt{a+b x+c x^2}}{f} \]
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Rubi [A] time = 7.02783, antiderivative size = 549, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {1019, 1076, 621, 206, 1032, 724} \[ -\frac{\left (\left (e-\sqrt{e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right )+2 d f (c e-b f)\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}+\frac{\left (\left (\sqrt{e^2-4 d f}+e\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right )+2 d f (c e-b f)\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}-\frac{(2 c e-b f) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} f^2}+\frac{\sqrt{a+b x+c x^2}}{f} \]
Antiderivative was successfully verified.
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Rule 1019
Rule 1076
Rule 621
Rule 206
Rule 1032
Rule 724
Rubi steps
\begin{align*} \int \frac{x \sqrt{a+b x+c x^2}}{d+e x+f x^2} \, dx &=\frac{\sqrt{a+b x+c x^2}}{f}-\frac{\int \frac{\frac{b d}{2}+\frac{1}{2} (2 c d+b e-2 a f) x+\frac{1}{2} (2 c e-b f) x^2}{\sqrt{a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{f}\\ &=\frac{\sqrt{a+b x+c x^2}}{f}-\frac{\int \frac{\frac{b d f}{2}-\frac{1}{2} d (2 c e-b f)+\left (\frac{1}{2} f (2 c d+b e-2 a f)-\frac{1}{2} e (2 c e-b f)\right ) x}{\sqrt{a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{f^2}-\frac{(2 c e-b f) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 f^2}\\ &=\frac{\sqrt{a+b x+c x^2}}{f}-\frac{(2 c e-b f) \operatorname{Subst}\left (\int \frac{1}{4 c-x^2} \, dx,x,\frac{b+2 c x}{\sqrt{a+b x+c x^2}}\right )}{f^2}-\frac{\left (2 f \left (\frac{b d f}{2}-\frac{1}{2} d (2 c e-b f)\right )-\left (\frac{1}{2} f (2 c d+b e-2 a f)-\frac{1}{2} e (2 c e-b f)\right ) \left (e-\sqrt{e^2-4 d f}\right )\right ) \int \frac{1}{\left (e-\sqrt{e^2-4 d f}+2 f x\right ) \sqrt{a+b x+c x^2}} \, dx}{f^2 \sqrt{e^2-4 d f}}+\frac{\left (2 f \left (\frac{b d f}{2}-\frac{1}{2} d (2 c e-b f)\right )-\left (\frac{1}{2} f (2 c d+b e-2 a f)-\frac{1}{2} e (2 c e-b f)\right ) \left (e+\sqrt{e^2-4 d f}\right )\right ) \int \frac{1}{\left (e+\sqrt{e^2-4 d f}+2 f x\right ) \sqrt{a+b x+c x^2}} \, dx}{f^2 \sqrt{e^2-4 d f}}\\ &=\frac{\sqrt{a+b x+c x^2}}{f}-\frac{(2 c e-b f) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} f^2}+\frac{\left (2 \left (2 f \left (\frac{b d f}{2}-\frac{1}{2} d (2 c e-b f)\right )-\left (\frac{1}{2} f (2 c d+b e-2 a f)-\frac{1}{2} e (2 c e-b f)\right ) \left (e-\sqrt{e^2-4 d f}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{16 a f^2-8 b f \left (e-\sqrt{e^2-4 d f}\right )+4 c \left (e-\sqrt{e^2-4 d f}\right )^2-x^2} \, dx,x,\frac{4 a f-b \left (e-\sqrt{e^2-4 d f}\right )-\left (-2 b f+2 c \left (e-\sqrt{e^2-4 d f}\right )\right ) x}{\sqrt{a+b x+c x^2}}\right )}{f^2 \sqrt{e^2-4 d f}}-\frac{\left (2 \left (2 f \left (\frac{b d f}{2}-\frac{1}{2} d (2 c e-b f)\right )-\left (\frac{1}{2} f (2 c d+b e-2 a f)-\frac{1}{2} e (2 c e-b f)\right ) \left (e+\sqrt{e^2-4 d f}\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{16 a f^2-8 b f \left (e+\sqrt{e^2-4 d f}\right )+4 c \left (e+\sqrt{e^2-4 d f}\right )^2-x^2} \, dx,x,\frac{4 a f-b \left (e+\sqrt{e^2-4 d f}\right )-\left (-2 b f+2 c \left (e+\sqrt{e^2-4 d f}\right )\right ) x}{\sqrt{a+b x+c x^2}}\right )}{f^2 \sqrt{e^2-4 d f}}\\ &=\frac{\sqrt{a+b x+c x^2}}{f}-\frac{(2 c e-b f) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} f^2}-\frac{\left (2 f (c d e-b d f)+\left (e-\sqrt{e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{4 a f-b \left (e-\sqrt{e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right ) x}{2 \sqrt{2} \sqrt{c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt{e^2-4 d f}} \sqrt{a+b x+c x^2}}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt{e^2-4 d f}}}+\frac{\left (2 f (c d e-b d f)+\left (e+\sqrt{e^2-4 d f}\right ) \left (f (b e-a f)-c \left (e^2-d f\right )\right )\right ) \tanh ^{-1}\left (\frac{4 a f-b \left (e+\sqrt{e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt{e^2-4 d f}\right )\right ) x}{2 \sqrt{2} \sqrt{c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt{e^2-4 d f}} \sqrt{a+b x+c x^2}}\right )}{\sqrt{2} f^2 \sqrt{e^2-4 d f} \sqrt{c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt{e^2-4 d f}}}\\ \end{align*}
Mathematica [A] time = 1.94204, size = 496, normalized size = 0.9 \[ \frac{4 f \sqrt{e^2-4 d f} \sqrt{a+x (b+c x)}-\sqrt{2} \left (\sqrt{e^2-4 d f}+e\right ) \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{4 a f-b \left (\sqrt{e^2-4 d f}+e-2 f x\right )-2 c x \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+x (b+c x)} \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )-\sqrt{2} \left (\sqrt{e^2-4 d f}-e\right ) \sqrt{f \left (2 a f+b \sqrt{e^2-4 d f}+b (-e)\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{4 a f+b \left (\sqrt{e^2-4 d f}-e+2 f x\right )+2 c x \left (\sqrt{e^2-4 d f}-e\right )}{2 \sqrt{2} \sqrt{a+x (b+c x)} \sqrt{f \left (2 a f+b \sqrt{e^2-4 d f}+b (-e)\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{4 f^2 \sqrt{e^2-4 d f}}-\frac{(2 c e-b f) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{2 \sqrt{c} f^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.321, size = 10138, normalized size = 18.5 \begin{align*} \text{output too large to display} \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 [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{x \sqrt{a + b x + c x^{2}}}{d + e x + f x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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