Optimal. Leaf size=195 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (8 a e+5 b d)+5 b^2 e^2+6 c e x (2 c d-b e)+16 c^2 d^2\right )}{60 c^2}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{32 c^3}-\frac{e \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{7/2}}+\frac{2}{5} (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \]
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Rubi [A] time = 0.328284, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {832, 779, 612, 621, 206} \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (8 a e+5 b d)+5 b^2 e^2+6 c e x (2 c d-b e)+16 c^2 d^2\right )}{60 c^2}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{32 c^3}-\frac{e \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{7/2}}+\frac{2}{5} (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 832
Rule 779
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (b+2 c x) (d+e x)^2 \sqrt{a+b x+c x^2} \, dx &=\frac{2}{5} (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}+\frac{\int (d+e x) (2 c (b d-2 a e)+2 c (2 c d-b e) x) \sqrt{a+b x+c x^2} \, dx}{5 c}\\ &=\frac{2}{5} (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}+\frac{\left (16 c^2 d^2+5 b^2 e^2-2 c e (5 b d+8 a e)+6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{60 c^2}+\frac{\left (\left (b^2-4 a c\right ) e (2 c d-b e)\right ) \int \sqrt{a+b x+c x^2} \, dx}{8 c^2}\\ &=\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{32 c^3}+\frac{2}{5} (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}+\frac{\left (16 c^2 d^2+5 b^2 e^2-2 c e (5 b d+8 a e)+6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{60 c^2}-\frac{\left (\left (b^2-4 a c\right )^2 e (2 c d-b e)\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{64 c^3}\\ &=\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{32 c^3}+\frac{2}{5} (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}+\frac{\left (16 c^2 d^2+5 b^2 e^2-2 c e (5 b d+8 a e)+6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{60 c^2}-\frac{\left (\left (b^2-4 a c\right )^2 e (2 c d-b e)\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 )}{32 c^3}\\ &=\frac{\left (b^2-4 a c\right ) e (2 c d-b e) (b+2 c x) \sqrt{a+b x+c x^2}}{32 c^3}+\frac{2}{5} (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}+\frac{\left (16 c^2 d^2+5 b^2 e^2-2 c e (5 b d+8 a e)+6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{60 c^2}-\frac{\left (b^2-4 a c\right )^2 e (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.262841, size = 177, normalized size = 0.91 \[ \frac{(a+x (b+c x))^{3/2} \left (-2 c e (8 a e+5 b d+3 b e x)+5 b^2 e^2+4 c^2 d (4 d+3 e x)\right )}{60 c^2}+\frac{e \left (b^2-4 a c\right ) (b e-2 c d) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)}\right )}{64 c^{7/2}}+\frac{2}{5} (d+e x)^2 (a+x (b+c x))^{3/2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 535, normalized size = 2.7 \begin{align*}{\frac{2\,{d}^{2}}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{bde}{6\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{3}de}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{axde}{2}\sqrt{c{x}^{2}+bx+a}}-{\frac{{b}^{3}{e}^{2}x}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{{e}^{2}{b}^{5}}{64}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{{b}^{2}{e}^{2}a}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{b{e}^{2}x}{10\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}dae}{4}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+x \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}de+{\frac{{b}^{2}{e}^{2}}{12\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{2}{b}^{4}}{32\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{4\,a{e}^{2}}{15\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}xde}{8\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{{b}^{4}de}{32}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{abde}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{{a}^{2}de}{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{ab{e}^{2}x}{4\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{b{e}^{2}{a}^{2}}{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{{b}^{3}{e}^{2}a}{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{2\,{e}^{2}{x}^{2}}{5} \left ( c{x}^{2}+bx+a \right ) ^{{\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 = 1.55265, size = 1388, normalized size = 7.12 \begin{align*} \left [-\frac{15 \,{\left (2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d e -{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} e^{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 (192 \, c^{5} e^{2} x^{4} + 320 \, a c^{4} d^{2} + 48 \,{\left (10 \, c^{5} d e + 3 \, b c^{4} e^{2}\right )} x^{3} + 10 \,{\left (3 \, b^{3} c^{2} - 20 \, a b c^{3}\right )} d e -{\left (15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3}\right )} e^{2} + 8 \,{\left (40 \, c^{5} d^{2} + 50 \, b c^{4} d e -{\left (b^{2} c^{3} - 8 \, a c^{4}\right )} e^{2}\right )} x^{2} + 2 \,{\left (160 \, b c^{4} d^{2} - 10 \,{\left (b^{2} c^{3} - 12 \, a c^{4}\right )} d e +{\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{1920 \, c^{4}}, \frac{15 \,{\left (2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d e -{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} e^{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 (192 \, c^{5} e^{2} x^{4} + 320 \, a c^{4} d^{2} + 48 \,{\left (10 \, c^{5} d e + 3 \, b c^{4} e^{2}\right )} x^{3} + 10 \,{\left (3 \, b^{3} c^{2} - 20 \, a b c^{3}\right )} d e -{\left (15 \, b^{4} c - 100 \, a b^{2} c^{2} + 128 \, a^{2} c^{3}\right )} e^{2} + 8 \,{\left (40 \, c^{5} d^{2} + 50 \, b c^{4} d e -{\left (b^{2} c^{3} - 8 \, a c^{4}\right )} e^{2}\right )} x^{2} + 2 \,{\left (160 \, b c^{4} d^{2} - 10 \,{\left (b^{2} c^{3} - 12 \, a c^{4}\right )} d e +{\left (5 \, b^{3} c^{2} - 28 \, a b c^{3}\right )} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{960 \, 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 (b + 2 c x\right ) \left (d + e x\right )^{2} \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 = 1.20749, size = 416, normalized size = 2.13 \begin{align*} \frac{1}{480} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (4 \, c x e^{2} + \frac{10 \, c^{5} d e + 3 \, b c^{4} e^{2}}{c^{4}}\right )} x + \frac{40 \, c^{5} d^{2} + 50 \, b c^{4} d e - b^{2} c^{3} e^{2} + 8 \, a c^{4} e^{2}}{c^{4}}\right )} x + \frac{160 \, b c^{4} d^{2} - 10 \, b^{2} c^{3} d e + 120 \, a c^{4} d e + 5 \, b^{3} c^{2} e^{2} - 28 \, a b c^{3} e^{2}}{c^{4}}\right )} x + \frac{320 \, a c^{4} d^{2} + 30 \, b^{3} c^{2} d e - 200 \, a b c^{3} d e - 15 \, b^{4} c e^{2} + 100 \, a b^{2} c^{2} e^{2} - 128 \, a^{2} c^{3} e^{2}}{c^{4}}\right )} + \frac{{\left (2 \, b^{4} c d e - 16 \, a b^{2} c^{2} d e + 32 \, a^{2} c^{3} d e - b^{5} e^{2} + 8 \, a b^{3} c e^{2} - 16 \, a^{2} b c^{2} e^{2}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{64 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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