Optimal. Leaf size=98 \[ \frac{C \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}}-\frac{2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac{a C}{c}+A\right )\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0770029, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1660, 12, 621, 206} \[ \frac{C \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}}-\frac{2 \left (x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac{a C}{c}+A\right )\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1660
Rule 12
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{A+C x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 \left (b c \left (A+\frac{a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\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 ) C}{2 c \sqrt{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac{2 \left (b c \left (A+\frac{a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{C \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{c}\\ &=-\frac{2 \left (b c \left (A+\frac{a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\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 )}{c}\\ &=-\frac{2 \left (b c \left (A+\frac{a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{C \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 = 0.763087, size = 104, normalized size = 1.06 \[ \frac{\frac{2 \sqrt{c} \left (a C (b-2 c x)+A c (b+2 c x)+b^2 C x\right )}{\sqrt{a+x (b+c x)}}-C \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{c^{3/2} \left (4 a c-b^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.052, size = 169, normalized size = 1.7 \begin{align*} -{\frac{Cx}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{Cb}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{C{b}^{2}x}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{C{b}^{3}}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{C\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+2\,{\frac{A \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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 3.14125, size = 892, normalized size = 9.1 \begin{align*} \left [\frac{{\left (C a b^{2} - 4 \, C a^{2} c +{\left (C b^{2} c - 4 \, C a c^{2}\right )} x^{2} +{\left (C b^{3} - 4 \, C a b c\right )} x\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 (C a b c + A b c^{2} +{\left (C b^{2} c - 2 \, C a c^{2} + 2 \, A c^{3}\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 (C a b^{2} - 4 \, C a^{2} c +{\left (C b^{2} c - 4 \, C a c^{2}\right )} x^{2} +{\left (C b^{3} - 4 \, C a b c\right )} x\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 (C a b c + A b c^{2} +{\left (C b^{2} c - 2 \, C a c^{2} + 2 \, A c^{3}\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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + C 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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.33972, size = 149, normalized size = 1.52 \begin{align*} -\frac{2 \,{\left (\frac{{\left (C b^{2} - 2 \, C a c + 2 \, A c^{2}\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac{C a b + A b c}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt{c x^{2} + b x + a}} - \frac{C \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.
[In]
[Out]