Optimal. Leaf size=175 \[ \frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a c f+5 b^2 f-8 b c e+16 c^2 d\right )}{64 c^3}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c (a f+2 b e)+5 b^2 f+16 c^2 d\right )}{128 c^{7/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (8 c e-5 b f)}{24 c^2}+\frac{f x \left (a+b x+c x^2\right )^{3/2}}{4 c} \]
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Rubi [A] time = 0.166976, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1661, 640, 612, 621, 206} \[ \frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a c f+5 b^2 f-8 b c e+16 c^2 d\right )}{64 c^3}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c (a f+2 b e)+5 b^2 f+16 c^2 d\right )}{128 c^{7/2}}+\frac{\left (a+b x+c x^2\right )^{3/2} (8 c e-5 b f)}{24 c^2}+\frac{f x \left (a+b x+c x^2\right )^{3/2}}{4 c} \]
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 (d+e x+f x^2\right ) \, dx &=\frac{f x \left (a+b x+c x^2\right )^{3/2}}{4 c}+\frac{\int \left (4 c d-a f+\frac{1}{2} (8 c e-5 b f) x\right ) \sqrt{a+b x+c x^2} \, dx}{4 c}\\ &=\frac{(8 c e-5 b f) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac{f x \left (a+b x+c x^2\right )^{3/2}}{4 c}+\frac{\left (16 c^2 d-8 b c e+5 b^2 f-4 a c f\right ) \int \sqrt{a+b x+c x^2} \, dx}{16 c^2}\\ &=\frac{\left (16 c^2 d-8 b c e+5 b^2 f-4 a c f\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3}+\frac{(8 c e-5 b f) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac{f x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac{\left (\left (b^2-4 a c\right ) \left (16 c^2 d+5 b^2 f-4 c (2 b e+a f)\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{128 c^3}\\ &=\frac{\left (16 c^2 d-8 b c e+5 b^2 f-4 a c f\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3}+\frac{(8 c e-5 b f) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac{f x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac{\left (\left (b^2-4 a c\right ) \left (16 c^2 d+5 b^2 f-4 c (2 b e+a f)\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 c^2 d-8 b c e+5 b^2 f-4 a c f\right ) (b+2 c x) \sqrt{a+b x+c x^2}}{64 c^3}+\frac{(8 c e-5 b f) \left (a+b x+c x^2\right )^{3/2}}{24 c^2}+\frac{f x \left (a+b x+c x^2\right )^{3/2}}{4 c}-\frac{\left (b^2-4 a c\right ) \left (16 c^2 d+5 b^2 f-4 c (2 b e+a f)\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 = 0.286353, size = 173, normalized size = 0.99 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 b c \left (2 c \left (6 d+2 e x+f x^2\right )-13 a f\right )+8 c^2 \left (a (8 e+3 f x)+2 c x \left (6 d+4 e x+3 f x^2\right )\right )-2 b^2 c (12 e+5 f x)+15 b^3 f\right )-3 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right ) \left (-4 c (a f+2 b e)+5 b^2 f+16 c^2 d\right )}{384 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 453, normalized size = 2.6 \begin{align*}{\frac{fx}{4\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,bf}{24\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}fx}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,f{b}^{3}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}fa}{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\,f{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{afx}{8\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{abf}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{{a}^{2}f}{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{e}{3\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{bxe}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{{b}^{2}e}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{aeb}{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{e{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}}}}+{\frac{dx}{2}\sqrt{c{x}^{2}+bx+a}}+{\frac{bd}{4\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{ad}{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{{b}^{2}d}{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 = 1.73372, size = 1061, normalized size = 6.06 \begin{align*} \left [\frac{3 \,{\left (16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d - 8 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} e +{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\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 (48 \, c^{4} f x^{3} + 48 \, b c^{3} d + 8 \,{\left (8 \, c^{4} e + b c^{3} f\right )} x^{2} - 8 \,{\left (3 \, b^{2} c^{2} - 8 \, a c^{3}\right )} e +{\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} f + 2 \,{\left (48 \, c^{4} d + 8 \, b c^{3} e -{\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} f\right )} x\right )} \sqrt{c x^{2} + b x + a}}{768 \, c^{4}}, \frac{3 \,{\left (16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d - 8 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} e +{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\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 (48 \, c^{4} f x^{3} + 48 \, b c^{3} d + 8 \,{\left (8 \, c^{4} e + b c^{3} f\right )} x^{2} - 8 \,{\left (3 \, b^{2} c^{2} - 8 \, a c^{3}\right )} e +{\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} f + 2 \,{\left (48 \, c^{4} d + 8 \, b c^{3} e -{\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} f\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 \sqrt{a + b x + c x^{2}} \left (d + e x + f x^{2}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24786, size = 286, normalized size = 1.63 \begin{align*} \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, f x + \frac{b c^{2} f + 8 \, c^{3} e}{c^{3}}\right )} x + \frac{48 \, c^{3} d - 5 \, b^{2} c f + 12 \, a c^{2} f + 8 \, b c^{2} e}{c^{3}}\right )} x + \frac{48 \, b c^{2} d + 15 \, b^{3} f - 52 \, a b c f - 24 \, b^{2} c e + 64 \, a c^{2} e}{c^{3}}\right )} + \frac{{\left (16 \, b^{2} c^{2} d - 64 \, a c^{3} d + 5 \, b^{4} f - 24 \, a b^{2} c f + 16 \, a^{2} c^{2} f - 8 \, b^{3} c e + 32 \, a b c^{2} e\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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