Optimal. Leaf size=288 \[ \frac{\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{x \left (7 b^2-4 a c\right ) \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x} \]
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Rubi [A] time = 0.519235, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1919, 1934, 1949, 12, 1914, 621, 206} \[ \frac{\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{x \left (7 b^2-4 a c\right ) \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x} \]
Antiderivative was successfully verified.
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Rule 1919
Rule 1934
Rule 1949
Rule 12
Rule 1914
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx &=\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\int \left (-2 a b+\frac{1}{2} \left (-7 b^2+20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4} \, dx}{20 c}\\ &=-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\int \frac{x^2 \left (-16 a^2 b c-a b \left (-7 b^2+20 a c\right )+\left (-8 a b^2 c-\frac{5}{4} b^2 \left (-7 b^2+20 a c\right )+3 a c \left (-7 b^2+20 a c\right )\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{480 c^2}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}-\frac{\int \frac{x \left (\frac{1}{4} a \left (35 b^4-216 a b^2 c+240 a^2 c^2\right )+\frac{1}{8} b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) x\right )}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{960 c^3}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\int \frac{15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) x}{16 \sqrt{a x^2+b x^3+c x^4}} \, dx}{960 c^4}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right )\right ) \int \frac{x}{\sqrt{a x^2+b x^3+c x^4}} \, dx}{1024 c^4}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) x \sqrt{a+b x+c x^2}\right ) \int \frac{1}{\sqrt{a+b x+c x^2}} \, dx}{1024 c^4 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\left (\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) x \sqrt{a+b x+c x^2}\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^4 \sqrt{a x^2+b x^3+c x^4}}\\ &=\frac{\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{3840 c^3}-\frac{b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt{a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac{x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt{a x^2+b x^3+c x^4}}{960 c^2}+\frac{(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac{\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) x \sqrt{a+b x+c x^2} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{1024 c^{9/2} \sqrt{a x^2+b x^3+c x^4}}\\ \end{align*}
Mathematica [A] time = 0.242008, size = 180, normalized size = 0.62 \[ \frac{\left (x^2 (a+x (b+c x))\right )^{3/2} \left (\frac{\left (7 b^2-4 a c\right ) \left (2 \sqrt{c} (b+2 c x) \sqrt{a+x (b+c x)} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+x (b+c x)}}\right )\right )}{512 c^{7/2} (a+x (b+c x))^{3/2}}+x (a+x (b+c x))-\frac{7 b (a+x (b+c x))}{10 c}\right )}{6 c x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 431, normalized size = 1.5 \begin{align*}{\frac{1}{15360\,{x}^{3}} \left ( c{x}^{4}+b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 2560\,x \left ( c{x}^{2}+bx+a \right ) ^{5/2}{c}^{9/2}-1792\,{c}^{7/2} \left ( c{x}^{2}+bx+a \right ) ^{5/2}b-640\,{c}^{9/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}xa+1120\,{c}^{7/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}x{b}^{2}-320\,{c}^{7/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}ab+560\,{c}^{5/2} \left ( c{x}^{2}+bx+a \right ) ^{3/2}{b}^{3}-960\,{c}^{9/2}\sqrt{c{x}^{2}+bx+a}x{a}^{2}+1920\,{c}^{7/2}\sqrt{c{x}^{2}+bx+a}xa{b}^{2}-420\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}x{b}^{4}-480\,{c}^{7/2}\sqrt{c{x}^{2}+bx+a}{a}^{2}b+960\,{c}^{5/2}\sqrt{c{x}^{2}+bx+a}a{b}^{3}-210\,{c}^{3/2}\sqrt{c{x}^{2}+bx+a}{b}^{5}-960\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{3}{c}^{4}+2160\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){a}^{2}{b}^{2}{c}^{3}-900\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ) a{b}^{4}{c}^{2}+105\,\ln \left ( 1/2\,{\frac{2\,\sqrt{c{x}^{2}+bx+a}\sqrt{c}+2\,cx+b}{\sqrt{c}}} \right ){b}^{6}c \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac{3}{2}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.83513, size = 1099, normalized size = 3.82 \begin{align*} \left [-\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{c} x \log \left (-\frac{8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{c} +{\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \,{\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \,{\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \,{\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{30720 \, c^{5} x}, -\frac{15 \,{\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \,{\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \,{\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \,{\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \,{\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \,{\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} \sqrt{c x^{4} + b x^{3} + a x^{2}}}{15360 \, c^{5} x}\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{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac{3}{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32493, size = 493, normalized size = 1.71 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, c x \mathrm{sgn}\left (x\right ) + 13 \, b \mathrm{sgn}\left (x\right )\right )} x + \frac{3 \, b^{2} c^{4} \mathrm{sgn}\left (x\right ) + 140 \, a c^{5} \mathrm{sgn}\left (x\right )}{c^{5}}\right )} x - \frac{7 \, b^{3} c^{3} \mathrm{sgn}\left (x\right ) - 36 \, a b c^{4} \mathrm{sgn}\left (x\right )}{c^{5}}\right )} x + \frac{35 \, b^{4} c^{2} \mathrm{sgn}\left (x\right ) - 216 \, a b^{2} c^{3} \mathrm{sgn}\left (x\right ) + 240 \, a^{2} c^{4} \mathrm{sgn}\left (x\right )}{c^{5}}\right )} x - \frac{105 \, b^{5} c \mathrm{sgn}\left (x\right ) - 760 \, a b^{3} c^{2} \mathrm{sgn}\left (x\right ) + 1296 \, a^{2} b c^{3} \mathrm{sgn}\left (x\right )}{c^{5}}\right )} - \frac{{\left (7 \, b^{6} \mathrm{sgn}\left (x\right ) - 60 \, a b^{4} c \mathrm{sgn}\left (x\right ) + 144 \, a^{2} b^{2} c^{2} \mathrm{sgn}\left (x\right ) - 64 \, a^{3} c^{3} \mathrm{sgn}\left (x\right )\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{9}{2}}} + \frac{{\left (105 \, b^{6} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 900 \, a b^{4} c \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 2160 \, a^{2} b^{2} c^{2} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) - 960 \, a^{3} c^{3} \log \left ({\left | -b + 2 \, \sqrt{a} \sqrt{c} \right |}\right ) + 210 \, \sqrt{a} b^{5} \sqrt{c} - 1520 \, a^{\frac{3}{2}} b^{3} c^{\frac{3}{2}} + 2592 \, a^{\frac{5}{2}} b c^{\frac{5}{2}}\right )} \mathrm{sgn}\left (x\right )}{15360 \, c^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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