Optimal. Leaf size=736 \[ -\frac{f \left (a \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b d \left (\sqrt{e^2-4 d f}+e\right )+2 c d^2\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (e-\sqrt{e^2-4 d f}\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{f \left (a \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b d \left (e-\sqrt{e^2-4 d f}\right )+2 c d^2\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\sqrt{a} e \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{d^2}-\frac{b e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} d^2}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} d^2}-\frac{\sqrt{a+b x+c x^2}}{d x}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a} d}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{d} \]
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Rubi [A] time = 3.47554, antiderivative size = 736, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6728, 732, 843, 621, 206, 724, 734, 1019, 1076, 1032} \[ -\frac{f \left (a \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b d \left (\sqrt{e^2-4 d f}+e\right )+2 c d^2\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (e-\sqrt{e^2-4 d f}\right )\right )-b \left (e-\sqrt{e^2-4 d f}\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2-\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (e-\sqrt{e^2-4 d f}\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{f \left (a \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )-b d \left (e-\sqrt{e^2-4 d f}\right )+2 c d^2\right ) \tanh ^{-1}\left (\frac{4 a f+2 x \left (b f-c \left (\sqrt{e^2-4 d f}+e\right )\right )-b \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+b x+c x^2} \sqrt{2 a f^2+\sqrt{e^2-4 d f} (c e-b f)-b e f-2 c d f+c e^2}}\right )}{\sqrt{2} d^2 \sqrt{e^2-4 d f} \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}+\frac{\sqrt{a} e \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{d^2}-\frac{b e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} d^2}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} d^2}-\frac{\sqrt{a+b x+c x^2}}{d x}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a} d}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 6728
Rule 732
Rule 843
Rule 621
Rule 206
Rule 724
Rule 734
Rule 1019
Rule 1076
Rule 1032
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x+c x^2}}{x^2 \left (d+e x+f x^2\right )} \, dx &=\int \left (\frac{\sqrt{a+b x+c x^2}}{d x^2}-\frac{e \sqrt{a+b x+c x^2}}{d^2 x}+\frac{\left (e^2-d f+e f x\right ) \sqrt{a+b x+c x^2}}{d^2 \left (d+e x+f x^2\right )}\right ) \, dx\\ &=\frac{\int \frac{\left (e^2-d f+e f x\right ) \sqrt{a+b x+c x^2}}{d+e x+f x^2} \, dx}{d^2}+\frac{\int \frac{\sqrt{a+b x+c x^2}}{x^2} \, dx}{d}-\frac{e \int \frac{\sqrt{a+b x+c x^2}}{x} \, dx}{d^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{d x}+\frac{\int \frac{b+2 c x}{x \sqrt{a+b x+c x^2}} \, dx}{2 d}+\frac{e \int \frac{-2 a-b x}{x \sqrt{a+b x+c x^2}} \, dx}{2 d^2}-\frac{\int \frac{\frac{1}{2} f \left (b d e-2 a e^2+2 a d f\right )+\frac{1}{2} f \left (2 c d e-b e^2+2 b d f-2 a e f\right ) x+\frac{1}{2} (2 c d-b e) f^2 x^2}{\sqrt{a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{d^2 f}\\ &=-\frac{\sqrt{a+b x+c x^2}}{d x}+\frac{b \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{2 d}+\frac{c \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{d}-\frac{(a e) \int \frac{1}{x \sqrt{a+b x+c x^2}} \, dx}{d^2}-\frac{(b e) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 d^2}-\frac{(2 c d-b e) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 d^2}-\frac{\int \frac{-\frac{1}{2} d (2 c d-b e) f^2+\frac{1}{2} f^2 \left (b d e-2 a e^2+2 a d f\right )+\left (-\frac{1}{2} e (2 c d-b e) f^2+\frac{1}{2} f^2 \left (2 c d e-b e^2+2 b d f-2 a e f\right )\right ) x}{\sqrt{a+b x+c x^2} \left (d+e x+f x^2\right )} \, dx}{d^2 f^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{d x}-\frac{b \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x}{\sqrt{a+b x+c x^2}}\right )}{d}+\frac{(2 c) \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 )}{d}+\frac{(2 a e) \operatorname{Subst}\left (\int \frac{1}{4 a-x^2} \, dx,x,\frac{2 a+b x}{\sqrt{a+b x+c x^2}}\right )}{d^2}-\frac{(b e) \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 )}{d^2}-\frac{(2 c d-b e) \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 )}{d^2}-\frac{\left (f \left (2 c d^2-b d \left (e-\sqrt{e^2-4 d f}\right )+a \left (e^2-2 d f-e \sqrt{e^2-4 d f}\right )\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}{d^2 \sqrt{e^2-4 d f}}+\frac{\left (f \left (2 c d^2-b d \left (e+\sqrt{e^2-4 d f}\right )+a \left (e^2-2 d f+e \sqrt{e^2-4 d f}\right )\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}{d^2 \sqrt{e^2-4 d f}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{d x}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a} d}+\frac{\sqrt{a} e \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{d^2}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{d}-\frac{b e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} d^2}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} d^2}+\frac{\left (2 f \left (2 c d^2-b d \left (e-\sqrt{e^2-4 d f}\right )+a \left (e^2-2 d f-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 )}{d^2 \sqrt{e^2-4 d f}}-\frac{\left (2 f \left (2 c d^2-b d \left (e+\sqrt{e^2-4 d f}\right )+a \left (e^2-2 d f+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 )}{d^2 \sqrt{e^2-4 d f}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{d x}-\frac{b \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{a} d}+\frac{\sqrt{a} e \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{d^2}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{d}-\frac{b e \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} d^2}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 \sqrt{c} d^2}-\frac{f \left (2 c d^2-b d \left (e+\sqrt{e^2-4 d f}\right )+a \left (e^2-2 d f+e \sqrt{e^2-4 d f}\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} d^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{f \left (2 c d^2-b d \left (e-\sqrt{e^2-4 d f}\right )+a \left (e^2-2 d f-e \sqrt{e^2-4 d f}\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} d^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 = 1.83877, size = 520, normalized size = 0.71 \[ \frac{(2 a e-b d) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )}{2 \sqrt{a} d^2}-\frac{4 d f \sqrt{e^2-4 d f} \sqrt{a+x (b+c x)}+\sqrt{2} x \left (e \sqrt{e^2-4 d f}+2 d f-e^2\right ) \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{4 a f-b \left (\sqrt{e^2-4 d f}+e-2 f x\right )-2 c x \left (\sqrt{e^2-4 d f}+e\right )}{2 \sqrt{2} \sqrt{a+x (b+c x)} \sqrt{f \left (2 a f-b \left (\sqrt{e^2-4 d f}+e\right )\right )+c \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )+\sqrt{2} x \left (e \sqrt{e^2-4 d f}-2 d f+e^2\right ) \sqrt{f \left (2 a f+b \left (\sqrt{e^2-4 d f}-e\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )} \tanh ^{-1}\left (\frac{4 a f+b \left (\sqrt{e^2-4 d f}-e+2 f x\right )+2 c x \left (\sqrt{e^2-4 d f}-e\right )}{2 \sqrt{2} \sqrt{a+x (b+c x)} \sqrt{f \left (2 a f+b \left (\sqrt{e^2-4 d f}-e\right )\right )+c \left (-e \sqrt{e^2-4 d f}-2 d f+e^2\right )}}\right )}{4 d^2 f x \sqrt{e^2-4 d f}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.313, size = 6765, normalized size = 9.2 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + b x + a}}{{\left (f x^{2} + e x + d\right )} x^{2}}\,{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{\sqrt{a + b x + c x^{2}}}{x^{2} \left (d + e x + f x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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