Optimal. Leaf size=144 \[ \frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-8 A b c+5 b^2 B\right )}{64 c^3}-\frac{\left (b^2-4 a c\right ) \left (-4 a B c-8 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2}}-\frac{\left (a+b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \]
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Rubi [A] time = 0.0643105, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {779, 612, 621, 206} \[ \frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-8 A b c+5 b^2 B\right )}{64 c^3}-\frac{\left (b^2-4 a c\right ) \left (-4 a B c-8 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2}}-\frac{\left (a+b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \]
Antiderivative was successfully verified.
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Rule 779
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int x (A+B x) \sqrt{a+b x+c x^2} \, dx &=-\frac{(5 b B-8 A c-6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac{\left (5 b^2 B-8 A b c-4 a B c\right ) \int \sqrt{a+b x+c x^2} \, dx}{16 c^2}\\ &=\frac{\left (5 b^2 B-8 A b c-4 a B c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3}-\frac{(5 b B-8 A c-6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}-\frac{\left (\left (b^2-4 a c\right ) \left (5 b^2 B-8 A b c-4 a B c\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{128 c^3}\\ &=\frac{\left (5 b^2 B-8 A b c-4 a B c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3}-\frac{(5 b B-8 A c-6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}-\frac{\left (\left (b^2-4 a c\right ) \left (5 b^2 B-8 A b c-4 a B 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 (5 b^2 B-8 A b c-4 a B c\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3}-\frac{(5 b B-8 A c-6 B c x) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}-\frac{\left (b^2-4 a c\right ) \left (5 b^2 B-8 A b c-4 a B 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 = 0.147003, size = 127, normalized size = 0.88 \[ \frac{(a+x (b+c x))^{3/2} (8 A c-5 b B+6 B c x)-\frac{3 \left (-4 a B c-8 A b c+5 b^2 B\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{16 c^{3/2}}}{24 c^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 352, normalized size = 2.4 \begin{align*}{\frac{Bx}{4\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,bB}{24\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}Bx}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{b}^{3}B}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,Ba{b}^{2}}{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\,{b}^{4}B}{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{aBx}{8\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{abB}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{B{a}^{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}}}}+{\frac{A}{3\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{Abx}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{A{b}^{2}}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{Aab}{4}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{A{b}^{3}}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{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 = 1.79027, size = 915, normalized size = 6.35 \begin{align*} \left [\frac{3 \,{\left (5 \, B b^{4} + 16 \,{\left (B a^{2} + 2 \, A a b\right )} c^{2} - 8 \,{\left (3 \, B a b^{2} + A b^{3}\right )} c\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 \, B c^{4} x^{3} + 15 \, B b^{3} c + 64 \, A a c^{3} - 4 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} c^{2} + 8 \,{\left (B b c^{3} + 8 \, A c^{4}\right )} x^{2} - 2 \,{\left (5 \, B b^{2} c^{2} - 4 \,{\left (3 \, B a + 2 \, A b\right )} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{768 \, c^{4}}, \frac{3 \,{\left (5 \, B b^{4} + 16 \,{\left (B a^{2} + 2 \, A a b\right )} c^{2} - 8 \,{\left (3 \, B a b^{2} + A b^{3}\right )} c\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 \, B c^{4} x^{3} + 15 \, B b^{3} c + 64 \, A a c^{3} - 4 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} c^{2} + 8 \,{\left (B b c^{3} + 8 \, A c^{4}\right )} x^{2} - 2 \,{\left (5 \, B b^{2} c^{2} - 4 \,{\left (3 \, B a + 2 \, A b\right )} c^{3}\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 x \left (A + B 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 [A] time = 1.22934, size = 240, normalized size = 1.67 \begin{align*} \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, B x + \frac{B b c^{2} + 8 \, A c^{3}}{c^{3}}\right )} x - \frac{5 \, B b^{2} c - 12 \, B a c^{2} - 8 \, A b c^{2}}{c^{3}}\right )} x + \frac{15 \, B b^{3} - 52 \, B a b c - 24 \, A b^{2} c + 64 \, A a c^{2}}{c^{3}}\right )} + \frac{{\left (5 \, B b^{4} - 24 \, B a b^{2} c - 8 \, A b^{3} c + 16 \, B a^{2} c^{2} + 32 \, A a b c^{2}\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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