Optimal. Leaf size=179 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (b f h-2 c e h+2 c f g)}{2 c^{3/2} h^2}+\frac{\left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right )}{h^2 \sqrt{a h^2-b g h+c g^2}}+\frac{f \sqrt{a+b x+c x^2}}{c h} \]
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Rubi [A] time = 0.293372, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1653, 843, 621, 206, 724} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (b f h-2 c e h+2 c f g)}{2 c^{3/2} h^2}+\frac{\left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\left (\frac{-2 a h+x (2 c g-b h)+b g}{2 \sqrt{a+b x+c x^2} \sqrt{a h^2-b g h+c g^2}}\right )}{h^2 \sqrt{a h^2-b g h+c g^2}}+\frac{f \sqrt{a+b x+c x^2}}{c h} \]
Antiderivative was successfully verified.
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Rule 1653
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2}{(g+h x) \sqrt{a+b x+c x^2}} \, dx &=\frac{f \sqrt{a+b x+c x^2}}{c h}+\frac{\int \frac{-\frac{1}{2} h (b f g-2 c d h)-\frac{1}{2} h (2 c f g-2 c e h+b f h) x}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{c h^2}\\ &=\frac{f \sqrt{a+b x+c x^2}}{c h}-\frac{(2 c f g-2 c e h+b f h) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{2 c h^2}+\frac{\left (f g^2-e g h+d h^2\right ) \int \frac{1}{(g+h x) \sqrt{a+b x+c x^2}} \, dx}{h^2}\\ &=\frac{f \sqrt{a+b x+c x^2}}{c h}-\frac{(2 c f g-2 c e h+b f h) \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 )}{c h^2}-\frac{\left (2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 c g^2-4 b g h+4 a h^2-x^2} \, dx,x,\frac{-b g+2 a h-(2 c g-b h) x}{\sqrt{a+b x+c x^2}}\right )}{h^2}\\ &=\frac{f \sqrt{a+b x+c x^2}}{c h}-\frac{(2 c f g-2 c e h+b f h) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 c^{3/2} h^2}+\frac{\left (f g^2-h (e g-d h)\right ) \tanh ^{-1}\left (\frac{b g-2 a h+(2 c g-b h) x}{2 \sqrt{c g^2-b g h+a h^2} \sqrt{a+b x+c x^2}}\right )}{h^2 \sqrt{c g^2-b g h+a h^2}}\\ \end{align*}
Mathematica [A] time = 0.285503, size = 172, normalized size = 0.96 \[ -\frac{\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) (b f h-2 c e h+2 c f g)}{c^{3/2}}+\frac{2 \left (h (d h-e g)+f g^2\right ) \tanh ^{-1}\left (\frac{2 a h-b g+b h x-2 c g x}{2 \sqrt{a+x (b+c x)} \sqrt{h (a h-b g)+c g^2}}\right )}{\sqrt{h (a h-b g)+c g^2}}-\frac{2 f h \sqrt{a+x (b+c x)}}{c}}{2 h^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.317, size = 599, normalized size = 3.4 \begin{align*}{\frac{f}{ch}\sqrt{c{x}^{2}+bx+a}}-{\frac{bf}{2\,h}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{e}{h}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{fg}{{h}^{2}}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{d}{h}\ln \left ({ \left ( 2\,{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}+{\frac{bh-2\,cg}{h} \left ( x+{\frac{g}{h}} \right ) }+2\,\sqrt{{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}}\sqrt{ \left ( x+{\frac{g}{h}} \right ) ^{2}c+{\frac{bh-2\,cg}{h} \left ( x+{\frac{g}{h}} \right ) }+{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}} \right ) \left ( x+{\frac{g}{h}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}}}}}+{\frac{eg}{{h}^{2}}\ln \left ({ \left ( 2\,{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}+{\frac{bh-2\,cg}{h} \left ( x+{\frac{g}{h}} \right ) }+2\,\sqrt{{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}}\sqrt{ \left ( x+{\frac{g}{h}} \right ) ^{2}c+{\frac{bh-2\,cg}{h} \left ( x+{\frac{g}{h}} \right ) }+{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}} \right ) \left ( x+{\frac{g}{h}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}}}}}-{\frac{f{g}^{2}}{{h}^{3}}\ln \left ({ \left ( 2\,{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}+{\frac{bh-2\,cg}{h} \left ( x+{\frac{g}{h}} \right ) }+2\,\sqrt{{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}}\sqrt{ \left ( x+{\frac{g}{h}} \right ) ^{2}c+{\frac{bh-2\,cg}{h} \left ( x+{\frac{g}{h}} \right ) }+{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}} \right ) \left ( x+{\frac{g}{h}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{h}^{2}-bgh+c{g}^{2}}{{h}^{2}}}}}}} \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{d + e x + f x^{2}}{\left (g + h x\right ) \sqrt{a + b x + c 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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