Optimal. Leaf size=281 \[ \frac{\sqrt{a+b x+c x^2} \left (1024 a^2 B c^2-2 c x \left (360 a A c^2-644 a b B c-350 A b^2 c+315 b^3 B\right )+2200 a A b c^2-2940 a b^2 B c-1050 A b^3 c+945 b^4 B\right )}{1920 c^5}-\frac{\left (-96 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-280 a b^3 B c-70 A b^4 c+63 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac{x^2 \sqrt{a+b x+c x^2} \left (-64 a B c-70 A b c+63 b^2 B\right )}{240 c^3}-\frac{x^3 \sqrt{a+b x+c x^2} (9 b B-10 A c)}{40 c^2}+\frac{B x^4 \sqrt{a+b x+c x^2}}{5 c} \]
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Rubi [A] time = 0.421644, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {832, 779, 621, 206} \[ \frac{\sqrt{a+b x+c x^2} \left (1024 a^2 B c^2-2 c x \left (360 a A c^2-644 a b B c-350 A b^2 c+315 b^3 B\right )+2200 a A b c^2-2940 a b^2 B c-1050 A b^3 c+945 b^4 B\right )}{1920 c^5}-\frac{\left (-96 a^2 A c^3+240 a^2 b B c^2+240 a A b^2 c^2-280 a b^3 B c-70 A b^4 c+63 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{11/2}}+\frac{x^2 \sqrt{a+b x+c x^2} \left (-64 a B c-70 A b c+63 b^2 B\right )}{240 c^3}-\frac{x^3 \sqrt{a+b x+c x^2} (9 b B-10 A c)}{40 c^2}+\frac{B x^4 \sqrt{a+b x+c x^2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 832
Rule 779
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{x^4 (A+B x)}{\sqrt{a+b x+c x^2}} \, dx &=\frac{B x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x^3 \left (-4 a B-\frac{1}{2} (9 b B-10 A c) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{5 c}\\ &=-\frac{(9 b B-10 A c) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{B x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x^2 \left (\frac{3}{2} a (9 b B-10 A c)+\frac{1}{4} \left (63 b^2 B-70 A b c-64 a B c\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{20 c^2}\\ &=\frac{\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}-\frac{(9 b B-10 A c) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{B x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\int \frac{x \left (-\frac{1}{2} a \left (63 b^2 B-70 A b c-64 a B c\right )-\frac{1}{8} \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right )}{\sqrt{a+b x+c x^2}} \, dx}{60 c^3}\\ &=\frac{\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}-\frac{(9 b B-10 A c) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{B x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\left (945 b^4 B-1050 A b^3 c-2940 a b^2 B c+2200 a A b c^2+1024 a^2 B c^2-2 c \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}-\frac{\left (63 b^5 B-70 A b^4 c-280 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{256 c^5}\\ &=\frac{\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}-\frac{(9 b B-10 A c) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{B x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\left (945 b^4 B-1050 A b^3 c-2940 a b^2 B c+2200 a A b c^2+1024 a^2 B c^2-2 c \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}-\frac{\left (63 b^5 B-70 A b^4 c-280 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A 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 )}{128 c^5}\\ &=\frac{\left (63 b^2 B-70 A b c-64 a B c\right ) x^2 \sqrt{a+b x+c x^2}}{240 c^3}-\frac{(9 b B-10 A c) x^3 \sqrt{a+b x+c x^2}}{40 c^2}+\frac{B x^4 \sqrt{a+b x+c x^2}}{5 c}+\frac{\left (945 b^4 B-1050 A b^3 c-2940 a b^2 B c+2200 a A b c^2+1024 a^2 B c^2-2 c \left (315 b^3 B-350 A b^2 c-644 a b B c+360 a A c^2\right ) x\right ) \sqrt{a+b x+c x^2}}{1920 c^5}-\frac{\left (63 b^5 B-70 A b^4 c-280 a b^3 B c+240 a A b^2 c^2+240 a^2 b B c^2-96 a^2 A c^3\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 = 0.407577, size = 225, normalized size = 0.8 \[ \frac{\sqrt{a+x (b+c x)} \left (16 c^2 \left (64 a^2 B-a c x (45 A+32 B x)+6 c^2 x^3 (5 A+4 B x)\right )+28 b^2 c (c x (25 A+18 B x)-105 a B)+8 b c^2 \left (a (275 A+161 B x)-2 c x^2 (35 A+27 B x)\right )-210 b^3 c (5 A+3 B x)+945 b^4 B\right )}{1920 c^5}+\frac{\left (96 a^2 A c^3-240 a^2 b B c^2-240 a A b^2 c^2+280 a b^3 B c+70 A b^4 c-63 b^5 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )}{256 c^{11/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 531, normalized size = 1.9 \begin{align*}{\frac{{x}^{4}B}{5\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{9\,bB{x}^{3}}{40\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{21\,{b}^{2}B{x}^{2}}{80\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{21\,{b}^{3}Bx}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{63\,{b}^{4}B}{128\,{c}^{5}}\sqrt{c{x}^{2}+bx+a}}-{\frac{63\,B{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\,Ba{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\,Ba{b}^{2}}{32\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{161\,abBx}{240\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{15\,B{a}^{2}b}{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\,aB{x}^{2}}{15\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{8\,B{a}^{2}}{15\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{A{x}^{3}}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{7\,Ab{x}^{2}}{24\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,A{b}^{2}x}{96\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{35\,A{b}^{3}}{64\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,A{b}^{4}}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}-{\frac{15\,A{b}^{2}a}{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{55\,Aba}{48\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,aAx}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,A{a}^{2}}{8}\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 = 2.37912, size = 1249, normalized size = 4.44 \begin{align*} \left [-\frac{15 \,{\left (63 \, B b^{5} - 96 \, A a^{2} c^{3} + 240 \,{\left (B a^{2} b + A a b^{2}\right )} c^{2} - 70 \,{\left (4 \, B a b^{3} + A b^{4}\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 (384 \, B c^{5} x^{4} + 945 \, B b^{4} c + 8 \,{\left (128 \, B a^{2} + 275 \, A a b\right )} c^{3} - 48 \,{\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} - 210 \,{\left (14 \, B a b^{2} + 5 \, A b^{3}\right )} c^{2} + 8 \,{\left (63 \, B b^{2} c^{3} - 2 \,{\left (32 \, B a + 35 \, A b\right )} c^{4}\right )} x^{2} - 2 \,{\left (315 \, B b^{3} c^{2} + 360 \, A a c^{4} - 14 \,{\left (46 \, B a b + 25 \, A b^{2}\right )} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{7680 \, c^{6}}, \frac{15 \,{\left (63 \, B b^{5} - 96 \, A a^{2} c^{3} + 240 \,{\left (B a^{2} b + A a b^{2}\right )} c^{2} - 70 \,{\left (4 \, B a b^{3} + A b^{4}\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 (384 \, B c^{5} x^{4} + 945 \, B b^{4} c + 8 \,{\left (128 \, B a^{2} + 275 \, A a b\right )} c^{3} - 48 \,{\left (9 \, B b c^{4} - 10 \, A c^{5}\right )} x^{3} - 210 \,{\left (14 \, B a b^{2} + 5 \, A b^{3}\right )} c^{2} + 8 \,{\left (63 \, B b^{2} c^{3} - 2 \,{\left (32 \, B a + 35 \, A b\right )} c^{4}\right )} x^{2} - 2 \,{\left (315 \, B b^{3} c^{2} + 360 \, A a c^{4} - 14 \,{\left (46 \, B a b + 25 \, A b^{2}\right )} 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^{4} \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 = 1.36876, size = 336, normalized size = 1.2 \begin{align*} \frac{1}{1920} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (\frac{8 \, B x}{c} - \frac{9 \, B b c^{3} - 10 \, A c^{4}}{c^{5}}\right )} x + \frac{63 \, B b^{2} c^{2} - 64 \, B a c^{3} - 70 \, A b c^{3}}{c^{5}}\right )} x - \frac{315 \, B b^{3} c - 644 \, B a b c^{2} - 350 \, A b^{2} c^{2} + 360 \, A a c^{3}}{c^{5}}\right )} x + \frac{945 \, B b^{4} - 2940 \, B a b^{2} c - 1050 \, A b^{3} c + 1024 \, B a^{2} c^{2} + 2200 \, A a b c^{2}}{c^{5}}\right )} + \frac{{\left (63 \, B b^{5} - 280 \, B a b^{3} c - 70 \, A b^{4} c + 240 \, B a^{2} b c^{2} + 240 \, A a b^{2} c^{2} - 96 \, A a^{2} c^{3}\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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