Optimal. Leaf size=104 \[ \frac{\left (-4 a c C+8 A c^2+3 b^2 C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2}}-\frac{3 b C \sqrt{a+b x+c x^2}}{4 c^2}+\frac{C x \sqrt{a+b x+c x^2}}{2 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0795335, antiderivative size = 104, 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 = {1661, 640, 621, 206} \[ \frac{\left (-4 a c C+8 A c^2+3 b^2 C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2}}-\frac{3 b C \sqrt{a+b x+c x^2}}{4 c^2}+\frac{C x \sqrt{a+b x+c x^2}}{2 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1661
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{A+C x^2}{\sqrt{a+b x+c x^2}} \, dx &=\frac{C x \sqrt{a+b x+c x^2}}{2 c}+\frac{\int \frac{2 A c-a C-\frac{3 b C x}{2}}{\sqrt{a+b x+c x^2}} \, dx}{2 c}\\ &=-\frac{3 b C \sqrt{a+b x+c x^2}}{4 c^2}+\frac{C x \sqrt{a+b x+c x^2}}{2 c}+\frac{\left (\frac{3 b^2 C}{2}+2 c (2 A c-a C)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{4 c^2}\\ &=-\frac{3 b C \sqrt{a+b x+c x^2}}{4 c^2}+\frac{C x \sqrt{a+b x+c x^2}}{2 c}+\frac{\left (\frac{3 b^2 C}{2}+2 c (2 A c-a C)\right ) \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 )}{2 c^2}\\ &=-\frac{3 b C \sqrt{a+b x+c x^2}}{4 c^2}+\frac{C x \sqrt{a+b x+c x^2}}{2 c}+\frac{\left (8 A c^2+3 b^2 C-4 a c C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.145834, size = 86, normalized size = 0.83 \[ \frac{\left (-4 a c C+8 A c^2+3 b^2 C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{8 c^{5/2}}+\frac{C (2 c x-3 b) \sqrt{a+x (b+c x)}}{4 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.051, size = 136, normalized size = 1.3 \begin{align*}{\frac{Cx}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,bC}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,C{b}^{2}}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{aC}{2}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{A\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}} \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 [A] time = 1.98015, size = 494, normalized size = 4.75 \begin{align*} \left [\frac{{\left (3 \, C b^{2} - 4 \, C a c + 8 \, A c^{2}\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 (2 \, C c^{2} x - 3 \, C b c\right )} \sqrt{c x^{2} + b x + a}}{16 \, c^{3}}, -\frac{{\left (3 \, C b^{2} - 4 \, C a c + 8 \, A c^{2}\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 (2 \, C c^{2} x - 3 \, C b c\right )} \sqrt{c x^{2} + b x + a}}{8 \, c^{3}}\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}}{\sqrt{a + b x + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.34466, size = 113, normalized size = 1.09 \begin{align*} \frac{1}{4} \, \sqrt{c x^{2} + b x + a}{\left (\frac{2 \, C x}{c} - \frac{3 \, C b}{c^{2}}\right )} - \frac{{\left (3 \, C b^{2} - 4 \, C a c + 8 \, A c^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]