Optimal. Leaf size=261 \[ -\frac{\left (7 b \left (528 a^2 c^2-680 a b^2 c+165 b^4\right )-2 c x \left (400 a^2 c^2-1176 a b^2 c+385 b^4\right )\right ) \sqrt{a+b x+c x^2}}{2560 c^6}+\frac{\left (1680 a^2 b^2 c^2-320 a^3 c^3-1260 a b^4 c+231 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{13/2}}+\frac{x^3 \left (99 b^2-100 a c\right ) \sqrt{a+b x+c x^2}}{480 c^3}-\frac{b x^2 \left (77 b^2-156 a c\right ) \sqrt{a+b x+c x^2}}{320 c^4}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c} \]
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Rubi [A] time = 0.38183, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {742, 832, 779, 621, 206} \[ -\frac{\left (7 b \left (528 a^2 c^2-680 a b^2 c+165 b^4\right )-2 c x \left (400 a^2 c^2-1176 a b^2 c+385 b^4\right )\right ) \sqrt{a+b x+c x^2}}{2560 c^6}+\frac{\left (1680 a^2 b^2 c^2-320 a^3 c^3-1260 a b^4 c+231 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{13/2}}+\frac{x^3 \left (99 b^2-100 a c\right ) \sqrt{a+b x+c x^2}}{480 c^3}-\frac{b x^2 \left (77 b^2-156 a c\right ) \sqrt{a+b x+c x^2}}{320 c^4}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 832
Rule 779
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^6}{\sqrt{a+b x+c x^2}} \, dx &=\frac{x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\int \frac{x^4 \left (-5 a-\frac{11 b x}{2}\right )}{\sqrt{a+b x+c x^2}} \, dx}{6 c}\\ &=-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\int \frac{x^3 \left (22 a b+\frac{1}{4} \left (99 b^2-100 a c\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{30 c^2}\\ &=\frac{\left (99 b^2-100 a c\right ) x^3 \sqrt{a+b x+c x^2}}{480 c^3}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\int \frac{x^2 \left (-\frac{3}{4} a \left (99 b^2-100 a c\right )-\frac{9}{8} b \left (77 b^2-156 a c\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{120 c^3}\\ &=-\frac{b \left (77 b^2-156 a c\right ) x^2 \sqrt{a+b x+c x^2}}{320 c^4}+\frac{\left (99 b^2-100 a c\right ) x^3 \sqrt{a+b x+c x^2}}{480 c^3}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c}+\frac{\int \frac{x \left (\frac{9}{4} a b \left (77 b^2-156 a c\right )+\frac{9}{16} \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{360 c^4}\\ &=-\frac{b \left (77 b^2-156 a c\right ) x^2 \sqrt{a+b x+c x^2}}{320 c^4}+\frac{\left (99 b^2-100 a c\right ) x^3 \sqrt{a+b x+c x^2}}{480 c^3}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c}-\frac{\left (7 b \left (165 b^4-680 a b^2 c+528 a^2 c^2\right )-2 c \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right ) \sqrt{a+b x+c x^2}}{2560 c^6}+\frac{\left (231 b^6-1260 a b^4 c+1680 a^2 b^2 c^2-320 a^3 c^3\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{1024 c^6}\\ &=-\frac{b \left (77 b^2-156 a c\right ) x^2 \sqrt{a+b x+c x^2}}{320 c^4}+\frac{\left (99 b^2-100 a c\right ) x^3 \sqrt{a+b x+c x^2}}{480 c^3}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c}-\frac{\left (7 b \left (165 b^4-680 a b^2 c+528 a^2 c^2\right )-2 c \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right ) \sqrt{a+b x+c x^2}}{2560 c^6}+\frac{\left (231 b^6-1260 a b^4 c+1680 a^2 b^2 c^2-320 a^3 c^3\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 )}{512 c^6}\\ &=-\frac{b \left (77 b^2-156 a c\right ) x^2 \sqrt{a+b x+c x^2}}{320 c^4}+\frac{\left (99 b^2-100 a c\right ) x^3 \sqrt{a+b x+c x^2}}{480 c^3}-\frac{11 b x^4 \sqrt{a+b x+c x^2}}{60 c^2}+\frac{x^5 \sqrt{a+b x+c x^2}}{6 c}-\frac{\left (7 b \left (165 b^4-680 a b^2 c+528 a^2 c^2\right )-2 c \left (385 b^4-1176 a b^2 c+400 a^2 c^2\right ) x\right ) \sqrt{a+b x+c x^2}}{2560 c^6}+\frac{\left (231 b^6-1260 a b^4 c+1680 a^2 b^2 c^2-320 a^3 c^3\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.322678, size = 263, normalized size = 1.01 \[ \frac{8 a^2 c \left (-2268 b^2 c x+1785 b^3-618 b c^2 x^2+100 c^3 x^3\right )+48 a^3 c^2 (50 c x-231 b)+a \left (5376 b^3 c^2 x^2-1728 b^2 c^3 x^3+16590 b^4 c x-3465 b^5+736 b c^4 x^4-320 c^5 x^5\right )+x \left (462 b^4 c^2 x^2-264 b^3 c^3 x^3+176 b^2 c^4 x^4-1155 b^5 c x-3465 b^6-128 b c^5 x^5+1280 c^6 x^6\right )}{7680 c^6 \sqrt{a+x (b+c x)}}+\frac{\left (1680 a^2 b^2 c^2-320 a^3 c^3-1260 a b^4 c+231 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{1024 c^{13/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 394, normalized size = 1.5 \begin{align*}{\frac{{x}^{5}}{6\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{11\,b{x}^{4}}{60\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{33\,{b}^{2}{x}^{3}}{160\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{77\,{b}^{3}{x}^{2}}{320\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{77\,{b}^{4}x}{256\,{c}^{5}}\sqrt{c{x}^{2}+bx+a}}-{\frac{231\,{b}^{5}}{512\,{c}^{6}}\sqrt{c{x}^{2}+bx+a}}+{\frac{231\,{b}^{6}}{1024}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{13}{2}}}}-{\frac{315\,{b}^{4}a}{256}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{11}{2}}}}+{\frac{119\,a{b}^{3}}{64\,{c}^{5}}\sqrt{c{x}^{2}+bx+a}}-{\frac{147\,{b}^{2}ax}{160\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{105\,{b}^{2}{a}^{2}}{64}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}+{\frac{39\,ab{x}^{2}}{80\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{231\,b{a}^{2}}{160\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,a{x}^{3}}{24\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{a}^{2}x}{16\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,{a}^{3}}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{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 = 2.9075, size = 1075, normalized size = 4.12 \begin{align*} \left [-\frac{15 \,{\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\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 (1280 \, c^{6} x^{5} - 1408 \, b c^{5} x^{4} - 3465 \, b^{5} c + 14280 \, a b^{3} c^{2} - 11088 \, a^{2} b c^{3} + 16 \,{\left (99 \, b^{2} c^{4} - 100 \, a c^{5}\right )} x^{3} - 24 \,{\left (77 \, b^{3} c^{3} - 156 \, a b c^{4}\right )} x^{2} + 6 \,{\left (385 \, b^{4} c^{2} - 1176 \, a b^{2} c^{3} + 400 \, a^{2} c^{4}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{30720 \, c^{7}}, -\frac{15 \,{\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\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 (1280 \, c^{6} x^{5} - 1408 \, b c^{5} x^{4} - 3465 \, b^{5} c + 14280 \, a b^{3} c^{2} - 11088 \, a^{2} b c^{3} + 16 \,{\left (99 \, b^{2} c^{4} - 100 \, a c^{5}\right )} x^{3} - 24 \,{\left (77 \, b^{3} c^{3} - 156 \, a b c^{4}\right )} x^{2} + 6 \,{\left (385 \, b^{4} c^{2} - 1176 \, a b^{2} c^{3} + 400 \, a^{2} c^{4}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{15360 \, c^{7}}\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 \frac{x^{6}}{\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 = 1.16743, size = 281, normalized size = 1.08 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \, x{\left (\frac{10 \, x}{c} - \frac{11 \, b}{c^{2}}\right )} + \frac{99 \, b^{2} c^{3} - 100 \, a c^{4}}{c^{6}}\right )} x - \frac{3 \,{\left (77 \, b^{3} c^{2} - 156 \, a b c^{3}\right )}}{c^{6}}\right )} x + \frac{3 \,{\left (385 \, b^{4} c - 1176 \, a b^{2} c^{2} + 400 \, a^{2} c^{3}\right )}}{c^{6}}\right )} x - \frac{21 \,{\left (165 \, b^{5} - 680 \, a b^{3} c + 528 \, a^{2} b c^{2}\right )}}{c^{6}}\right )} - \frac{{\left (231 \, b^{6} - 1260 \, a b^{4} c + 1680 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3}\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{13}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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