Optimal. Leaf size=202 \[ \frac{\left (1024 a^2 c^2-14 b c x \left (45 b^2-92 a c\right )-2940 a b^2 c+945 b^4\right ) \sqrt{a+b x+c x^2}}{1920 c^5}-\frac{b \left (240 a^2 c^2-280 a b^2 c+63 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac{x^2 \left (63 b^2-64 a c\right ) \sqrt{a+b x+c x^2}}{240 c^3}-\frac{9 b x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{x^4 \sqrt{a+b x+c x^2}}{5 c} \]
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Rubi [A] time = 0.2165, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{\left (1024 a^2 c^2-14 b c x \left (45 b^2-92 a c\right )-2940 a b^2 c+945 b^4\right ) \sqrt{a+b x+c x^2}}{1920 c^5}-\frac{b \left (240 a^2 c^2-280 a b^2 c+63 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac{x^2 \left (63 b^2-64 a c\right ) \sqrt{a+b x+c x^2}}{240 c^3}-\frac{9 b x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{x^4 \sqrt{a+b x+c x^2}}{5 c} \]
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^5}{\sqrt{a+b x+c x^2}} \, dx &=\frac{x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x^3 \left (-4 a-\frac{9 b x}{2}\right )}{\sqrt{a+b x+c x^2}} \, dx}{5 c}\\ &=-\frac{9 b x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x^2 \left (\frac{27 a b}{2}+\frac{1}{4} \left (63 b^2-64 a c\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{20 c^2}\\ &=\frac{\left (63 b^2-64 a c\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}-\frac{9 b x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x \left (-\frac{1}{2} a \left (63 b^2-64 a c\right )-\frac{7}{8} b \left (45 b^2-92 a c\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{60 c^3}\\ &=\frac{\left (63 b^2-64 a c\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}-\frac{9 b x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}-\frac{\left (b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right )\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{256 c^5}\\ &=\frac{\left (63 b^2-64 a c\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}-\frac{9 b x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}-\frac{\left (b \left (63 b^4-280 a b^2 c+240 a^2 c^2\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 )}{128 c^5}\\ &=\frac{\left (63 b^2-64 a c\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}-\frac{9 b x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\left (945 b^4-2940 a b^2 c+1024 a^2 c^2-14 b c \left (45 b^2-92 a c\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}-\frac{b \left (63 b^4-280 a b^2 c+240 a^2 c^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.207688, size = 213, normalized size = 1.05 \[ \frac{4 a^2 c \left (-735 b^2+578 b c x+128 c^2 x^2\right )+1024 a^3 c^2+a \left (-1148 b^2 c^2 x^2-3570 b^3 c x+945 b^4+344 b c^3 x^3-128 c^4 x^4\right )+3 x \left (-42 b^3 c^2 x^2+24 b^2 c^3 x^3+105 b^4 c x+315 b^5-16 b c^4 x^4+128 c^5 x^5\right )}{1920 c^5 \sqrt{a+x (b+c x)}}-\frac{b \left (240 a^2 c^2-280 a b^2 c+63 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{256 c^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 290, normalized size = 1.4 \begin{align*}{\frac{{x}^{4}}{5\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{9\,b{x}^{3}}{40\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{21\,{b}^{2}{x}^{2}}{80\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{21\,{b}^{3}x}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{63\,{b}^{4}}{128\,{c}^{5}}\sqrt{c{x}^{2}+bx+a}}-{\frac{63\,{b}^{5}}{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{35\,a{b}^{3}}{32}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}-{\frac{49\,{b}^{2}a}{32\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{161\,abx}{240\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{15\,b{a}^{2}}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}-{\frac{4\,a{x}^{2}}{15\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{8\,{a}^{2}}{15\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}} \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 = 2.74862, size = 848, normalized size = 4.2 \begin{align*} \left [\frac{15 \,{\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b 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 (384 \, c^{5} x^{4} - 432 \, b c^{4} x^{3} + 945 \, b^{4} c - 2940 \, a b^{2} c^{2} + 1024 \, a^{2} c^{3} + 8 \,{\left (63 \, b^{2} c^{3} - 64 \, a c^{4}\right )} x^{2} - 14 \,{\left (45 \, b^{3} c^{2} - 92 \, a b c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{7680 \, c^{6}}, \frac{15 \,{\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} b 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 (384 \, c^{5} x^{4} - 432 \, b c^{4} x^{3} + 945 \, b^{4} c - 2940 \, a b^{2} c^{2} + 1024 \, a^{2} c^{3} + 8 \,{\left (63 \, b^{2} c^{3} - 64 \, a c^{4}\right )} x^{2} - 14 \,{\left (45 \, b^{3} c^{2} - 92 \, a b c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3840 \, c^{6}}\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^{5}}{\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 = 1.10125, size = 217, normalized size = 1.07 \begin{align*} \frac{1}{1920} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, x{\left (\frac{8 \, x}{c} - \frac{9 \, b}{c^{2}}\right )} + \frac{63 \, b^{2} c^{2} - 64 \, a c^{3}}{c^{5}}\right )} x - \frac{7 \,{\left (45 \, b^{3} c - 92 \, a b c^{2}\right )}}{c^{5}}\right )} x + \frac{945 \, b^{4} - 2940 \, a b^{2} c + 1024 \, a^{2} c^{2}}{c^{5}}\right )} + \frac{{\left (63 \, b^{5} - 280 \, a b^{3} c + 240 \, a^{2} 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 )}{256 \, c^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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