Optimal. Leaf size=288 \[ -\frac {b \left (1296 a^2 c^2-760 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{7680 c^4 x}+\frac {\left (240 a^2 c^2-216 a b^2 c+35 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{3840 c^3}+\frac {x \left (7 b^2-4 a c\right ) \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2} \sqrt {a x^2+b x^3+c x^4}}-\frac {x \left (6 c x \left (7 b^2-20 a c\right )+b \left (12 a c+7 b^2\right )\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^2}+\frac {(3 b+10 c x) \left (a x^2+b x^3+c x^4\right )^{3/2}}{60 c x} \]
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Rubi [A] time = 0.52, antiderivative size = 288, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 621
Rule 1914
Rule 1919
Rule 1934
Rule 1949
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*}
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Mathematica [A] time = 0.22, size = 180, normalized size = 0.62 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.42, size = 220, normalized size = 0.76 \begin {gather*} \frac {\sqrt {a x^2+b x^3+c x^4} \left (-1296 a^2 b c^2+480 a^2 c^3 x+760 a b^3 c-432 a b^2 c^2 x+288 a b c^3 x^2+2240 a c^4 x^3-105 b^5+70 b^4 c x-56 b^3 c^2 x^2+48 b^2 c^3 x^3+1664 b c^4 x^4+1280 c^5 x^5\right )}{7680 c^4 x}+\frac {\left (64 a^3 c^3-144 a^2 b^2 c^2+60 a b^4 c-7 b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a} x-\sqrt {a x^2+b x^3+c x^4}}\right )}{512 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 474, normalized size = 1.65 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 365, normalized size = 1.27 \begin {gather*} \frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c x \mathrm {sgn}\relax (x) + 13 \, b \mathrm {sgn}\relax (x)\right )} x + \frac {3 \, b^{2} c^{4} \mathrm {sgn}\relax (x) + 140 \, a c^{5} \mathrm {sgn}\relax (x)}{c^{5}}\right )} x - \frac {7 \, b^{3} c^{3} \mathrm {sgn}\relax (x) - 36 \, a b c^{4} \mathrm {sgn}\relax (x)}{c^{5}}\right )} x + \frac {35 \, b^{4} c^{2} \mathrm {sgn}\relax (x) - 216 \, a b^{2} c^{3} \mathrm {sgn}\relax (x) + 240 \, a^{2} c^{4} \mathrm {sgn}\relax (x)}{c^{5}}\right )} x - \frac {105 \, b^{5} c \mathrm {sgn}\relax (x) - 760 \, a b^{3} c^{2} \mathrm {sgn}\relax (x) + 1296 \, a^{2} b c^{3} \mathrm {sgn}\relax (x)}{c^{5}}\right )} - \frac {{\left (7 \, b^{6} \mathrm {sgn}\relax (x) - 60 \, a b^{4} c \mathrm {sgn}\relax (x) + 144 \, a^{2} b^{2} c^{2} \mathrm {sgn}\relax (x) - 64 \, a^{3} c^{3} \mathrm {sgn}\relax (x)\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}\relax (x)}{15360 \, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 431, normalized size = 1.50 \begin {gather*} \frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (-960 a^{3} c^{4} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+2160 a^{2} b^{2} c^{3} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-900 a \,b^{4} c^{2} \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )+105 b^{6} c \ln \left (\frac {2 c x +b +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{2 \sqrt {c}}\right )-960 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{\frac {9}{2}} x +1920 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{\frac {7}{2}} x -420 \sqrt {c \,x^{2}+b x +a}\, b^{4} c^{\frac {5}{2}} x -480 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{\frac {7}{2}}+960 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} c^{\frac {5}{2}}-640 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,c^{\frac {9}{2}} x -210 \sqrt {c \,x^{2}+b x +a}\, b^{5} c^{\frac {3}{2}}+1120 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} c^{\frac {7}{2}} x -320 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b \,c^{\frac {7}{2}}+560 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} c^{\frac {5}{2}}+2560 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c^{\frac {9}{2}} x -1792 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b \,c^{\frac {7}{2}}\right )}{15360 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {11}{2}} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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