Optimal. Leaf size=157 \[ \frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a c C+16 A c^2+5 b^2 C\right )}{64 c^3}-\frac{\left (b^2-4 a c\right ) \left (-4 a c C+16 A c^2+5 b^2 C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2}}-\frac{5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac{C x \left (a+b x+c x^2\right )^{3/2}}{4 c} \]
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Rubi [A] time = 0.123895, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1661, 640, 612, 621, 206} \[ \frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a c C+16 A c^2+5 b^2 C\right )}{64 c^3}-\frac{\left (b^2-4 a c\right ) \left (-4 a c C+16 A c^2+5 b^2 C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2}}-\frac{5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac{C x \left (a+b x+c x^2\right )^{3/2}}{4 c} \]
Antiderivative was successfully verified.
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Rule 1661
Rule 640
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \sqrt{a+b x+c x^2} \left (A+C x^2\right ) \, dx &=\frac{C x \left (a+b x+c x^2\right )^{3/2}}{4 c}+\frac{\int \left (4 A c-a C-\frac{5 b C x}{2}\right ) \sqrt{a+b x+c x^2} \, dx}{4 c}\\ &=-\frac{5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac{C x \left (a+b x+c x^2\right )^{3/2}}{4 c}+\frac{\left (\frac{5 b^2 C}{2}+2 c (4 A c-a C)\right ) \int \sqrt{a+b x+c x^2} \, dx}{8 c^2}\\ &=\frac{\left (16 A c^2+5 b^2 C-4 a c C\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3}-\frac{5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac{C x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac{\left (\left (b^2-4 a c\right ) \left (16 A c^2+5 b^2 C-4 a c C\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{128 c^3}\\ &=\frac{\left (16 A c^2+5 b^2 C-4 a c C\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3}-\frac{5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac{C x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac{\left (\left (b^2-4 a c\right ) \left (16 A c^2+5 b^2 C-4 a c C\right )\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 )}{64 c^3}\\ &=\frac{\left (16 A c^2+5 b^2 C-4 a c C\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3}-\frac{5 b C \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac{C x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac{\left (b^2-4 a c\right ) \left (16 A c^2+5 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 )}{128 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.20037, size = 144, normalized size = 0.92 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (C \left (b \left (8 c^2 x^2-52 a c\right )+24 c^2 x \left (a+2 c x^2\right )-10 b^2 c x+15 b^3\right )+48 A c^2 (b+2 c x)\right )-3 \left (b^2-4 a c\right ) \left (-4 a c C+16 A c^2+5 b^2 C\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{384 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 327, normalized size = 2.1 \begin{align*}{\frac{Cx}{4\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,bC}{24\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,C{b}^{2}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,C{b}^{3}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,C{b}^{2}a}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{5\,C{b}^{4}}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}-{\frac{Cax}{8\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{aCb}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{{a}^{2}C}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{Ax}{2}\sqrt{c{x}^{2}+bx+a}}+{\frac{Ab}{4\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{Aa}{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{A{b}^{2}}{8}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{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 [A] time = 2.09802, size = 846, normalized size = 5.39 \begin{align*} \left [-\frac{3 \,{\left (5 \, C b^{4} - 24 \, C a b^{2} c - 64 \, A a c^{3} + 16 \,{\left (C a^{2} + A b^{2}\right )} 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 (48 \, C c^{4} x^{3} + 8 \, C b c^{3} x^{2} + 15 \, C b^{3} c - 52 \, C a b c^{2} + 48 \, A b c^{3} - 2 \,{\left (5 \, C b^{2} c^{2} - 12 \, C a c^{3} - 48 \, A c^{4}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{768 \, c^{4}}, \frac{3 \,{\left (5 \, C b^{4} - 24 \, C a b^{2} c - 64 \, A a c^{3} + 16 \,{\left (C a^{2} + A b^{2}\right )} 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 (48 \, C c^{4} x^{3} + 8 \, C b c^{3} x^{2} + 15 \, C b^{3} c - 52 \, C a b c^{2} + 48 \, A b c^{3} - 2 \,{\left (5 \, C b^{2} c^{2} - 12 \, C a c^{3} - 48 \, A c^{4}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{384 \, c^{4}}\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 \left (A + C x^{2}\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 [A] time = 1.25234, size = 216, normalized size = 1.38 \begin{align*} \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, C x + \frac{C b}{c}\right )} x - \frac{5 \, C b^{2} c - 12 \, C a c^{2} - 48 \, A c^{3}}{c^{3}}\right )} x + \frac{15 \, C b^{3} - 52 \, C a b c + 48 \, A b c^{2}}{c^{3}}\right )} + \frac{{\left (5 \, C b^{4} - 24 \, C a b^{2} c + 16 \, C a^{2} c^{2} + 16 \, A b^{2} c^{2} - 64 \, A a c^{3}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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