Optimal. Leaf size=111 \[ \frac{2 \left (c \left (2 a e-b \left (\frac{a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}} \]
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Rubi [A] time = 0.0797038, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {1660, 12, 621, 206} \[ \frac{2 \left (c \left (2 a e-b \left (\frac{a f}{c}+d\right )\right )-x \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=\frac{2 \left (c \left (2 a e-b \left (d+\frac{a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}-\frac{2 \int -\frac{\left (b^2-4 a c\right ) f}{2 c \sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=\frac{2 \left (c \left (2 a e-b \left (d+\frac{a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{c}\\ &=\frac{2 \left (c \left (2 a e-b \left (d+\frac{a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{(2 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 )}{c}\\ &=\frac{2 \left (c \left (2 a e-b \left (d+\frac{a f}{c}\right )\right )-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{f \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.328344, size = 113, normalized size = 1.02 \[ \frac{\frac{2 \sqrt{c} \left (a b f-2 a c (e+f x)+b^2 f x+b c (d-e x)+2 c^2 d x\right )}{\sqrt{a+x (b+c x)}}-f \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{3/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 249, normalized size = 2.2 \begin{align*} -{\frac{fx}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{bf}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{{b}^{2}fx}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{{b}^{3}f}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{f\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}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}-2\,{\frac{bxe}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-{\frac{{b}^{2}e}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+2\,{\frac{d \left ( 2\,cx+b \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}} \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 [B] time = 4.44962, size = 941, normalized size = 8.48 \begin{align*} \left [\frac{{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} f x^{2} +{\left (b^{3} - 4 \, a b c\right )} f x +{\left (a b^{2} - 4 \, a^{2} c\right )} f\right )} \sqrt{c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) - 4 \,{\left (b c^{2} d - 2 \, a c^{2} e + a b c f +{\left (2 \, c^{3} d - b c^{2} e +{\left (b^{2} c - 2 \, a c^{2}\right )} f\right )} x\right )} \sqrt{c x^{2} + b x + a}}{2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )}}, -\frac{{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} f x^{2} +{\left (b^{3} - 4 \, a b c\right )} f x +{\left (a b^{2} - 4 \, a^{2} c\right )} f\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \,{\left (b c^{2} d - 2 \, a c^{2} e + a b c f +{\left (2 \, c^{3} d - b c^{2} e +{\left (b^{2} c - 2 \, a c^{2}\right )} f\right )} x\right )} \sqrt{c x^{2} + b x + a}}{a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x}\right ] \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 (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50197, size = 165, normalized size = 1.49 \begin{align*} -\frac{2 \,{\left (\frac{{\left (2 \, c^{2} d + b^{2} f - 2 \, a c f - b c e\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac{b c d + a b f - 2 \, a c e}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt{c x^{2} + b x + a}} - \frac{f \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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