Integrand size = 24, antiderivative size = 288 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\frac {\left (35 b^4-216 a b^2 c+240 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{3840 c^3}-\frac {b \left (105 b^4-760 a b^2 c+1296 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{7680 c^4 x}-\frac {x \left (b \left (7 b^2+12 a c\right )+6 c \left (7 b^2-20 a c\right ) x\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^2}+\frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x}+\frac {\left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2} \sqrt {a x^2+b x^3+c x^4}} \]
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Time = 0.34 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1933, 1948, 1963, 12, 1928, 635, 212} \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=-\frac {b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{7680 c^4 x}+\frac {\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{3840 c^3}+\frac {x \left (7 b^2-4 a c\right ) \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2} \sqrt {a x^2+b x^3+c x^4}}-\frac {x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^2}+\frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x} \]
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Rule 12
Rule 212
Rule 635
Rule 1928
Rule 1933
Rule 1948
Rule 1963
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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*}
Time = 0.54 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\frac {x \sqrt {a+x (b+c x)} \left (2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^5+70 b^4 c x+8 b^3 c \left (95 a-7 c x^2\right )+48 b^2 c^2 x \left (-9 a+c x^2\right )+160 c^3 x \left (3 a^2+14 a c x^2+8 c^2 x^4\right )+16 b c^2 \left (-81 a^2+18 a c x^2+104 c^2 x^4\right )\right )-15 \left (b^2-4 a c\right )^2 \left (7 b^2-4 a c\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{15360 c^{9/2} \sqrt {x^2 (a+x (b+c x))}} \]
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Time = 0.75 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {\left (-1280 c^{5} x^{5}-1664 b \,c^{4} x^{4}-2240 a \,c^{4} x^{3}-48 b^{2} c^{3} x^{3}-288 a b \,c^{3} x^{2}+56 c^{2} x^{2} b^{3}-480 a^{2} c^{3} x +432 a \,b^{2} c^{2} x -70 x c \,b^{4}+1296 a^{2} b \,c^{2}-760 a \,b^{3} c +105 b^{5}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{7680 c^{4} x}-\frac {\left (64 c^{3} a^{3}-144 a^{2} b^{2} c^{2}+60 a \,b^{4} c -7 b^{6}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{1024 c^{\frac {9}{2}} x \sqrt {c \,x^{2}+b x +a}}\) | \(225\) |
default | \(\frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (2560 x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c^{\frac {9}{2}}-640 c^{\frac {9}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a x -960 c^{\frac {9}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{2} x -1792 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b +1120 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} x -320 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b +1920 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} x -480 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a^{2} b +560 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3}-420 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{4} x +960 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{3}-210 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{5}-960 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{3} c^{4}+2160 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{2} b^{2} c^{3}-900 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,b^{4} c^{2}+105 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) b^{6} c \right )}{15360 x^{3} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {11}{2}}}\) | \(431\) |
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Time = 0.31 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.65 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\left [-\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, {\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{30720 \, c^{5} x}, -\frac {15 \, {\left (7 \, b^{6} - 60 \, a b^{4} c + 144 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, {\left (1280 \, c^{6} x^{5} + 1664 \, b c^{5} x^{4} - 105 \, b^{5} c + 760 \, a b^{3} c^{2} - 1296 \, a^{2} b c^{3} + 16 \, {\left (3 \, b^{2} c^{4} + 140 \, a c^{5}\right )} x^{3} - 8 \, {\left (7 \, b^{3} c^{3} - 36 \, a b c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{2} - 216 \, a b^{2} c^{3} + 240 \, a^{2} c^{4}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{15360 \, c^{5} x}\right ] \]
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\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\int \frac {\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}{x}\, dx \]
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\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\int { \frac {{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c x \mathrm {sgn}\left (x\right ) + 13 \, b \mathrm {sgn}\left (x\right )\right )} x + \frac {3 \, b^{2} c^{4} \mathrm {sgn}\left (x\right ) + 140 \, a c^{5} \mathrm {sgn}\left (x\right )}{c^{5}}\right )} x - \frac {7 \, b^{3} c^{3} \mathrm {sgn}\left (x\right ) - 36 \, a b c^{4} \mathrm {sgn}\left (x\right )}{c^{5}}\right )} x + \frac {35 \, b^{4} c^{2} \mathrm {sgn}\left (x\right ) - 216 \, a b^{2} c^{3} \mathrm {sgn}\left (x\right ) + 240 \, a^{2} c^{4} \mathrm {sgn}\left (x\right )}{c^{5}}\right )} x - \frac {105 \, b^{5} c \mathrm {sgn}\left (x\right ) - 760 \, a b^{3} c^{2} \mathrm {sgn}\left (x\right ) + 1296 \, a^{2} b c^{3} \mathrm {sgn}\left (x\right )}{c^{5}}\right )} - \frac {{\left (7 \, b^{6} \mathrm {sgn}\left (x\right ) - 60 \, a b^{4} c \mathrm {sgn}\left (x\right ) + 144 \, a^{2} b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 64 \, a^{3} c^{3} \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{1024 \, c^{\frac {9}{2}}} + \frac {{\left (105 \, b^{6} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 900 \, a b^{4} c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 2160 \, a^{2} b^{2} c^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 960 \, a^{3} c^{3} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 210 \, \sqrt {a} b^{5} \sqrt {c} - 1520 \, a^{\frac {3}{2}} b^{3} c^{\frac {3}{2}} + 2592 \, a^{\frac {5}{2}} b c^{\frac {5}{2}}\right )} \mathrm {sgn}\left (x\right )}{15360 \, c^{\frac {9}{2}}} \]
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Timed out. \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x} \, dx=\int \frac {{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}}{x} \,d x \]
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