Integrand size = 20, antiderivative size = 209 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}-\frac {3 \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{7/2}} \]
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Time = 0.19 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1921, 1965, 12, 1918, 212} \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=-\frac {3 \left (5 b^2-4 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{7/2}}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 x^2 \left (b^2-4 a c\right )}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 x^3 \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}} \]
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Rule 12
Rule 212
Rule 1918
Rule 1921
Rule 1965
Rubi steps \begin{align*} \text {integral}& = \frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int \frac {-2 \left (b^2-2 a c\right )+\frac {1}{2} \left (-b^2+4 a c\right )-2 b c x}{x^2 \sqrt {a x^2+b x^3+c x^4}} \, dx}{a \left (b^2-4 a c\right )} \\ & = \frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac {\int \frac {-\frac {1}{4} b \left (15 b^2-52 a c\right )-\frac {1}{2} c \left (5 b^2-12 a c\right ) x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx}{a^2 \left (b^2-4 a c\right )} \\ & = \frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}-\frac {\int -\frac {3 \left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )}{8 \sqrt {a x^2+b x^3+c x^4}} \, dx}{a^3 \left (b^2-4 a c\right )} \\ & = \frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}+\frac {\left (3 \left (5 b^2-4 a c\right )\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 a^3} \\ & = \frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}-\frac {\left (3 \left (5 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x (2 a+b x)}{\sqrt {a x^2+b x^3+c x^4}}\right )}{4 a^3} \\ & = \frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (5 b^2-12 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a^2 \left (b^2-4 a c\right ) x^3}+\frac {b \left (15 b^2-52 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{4 a^3 \left (b^2-4 a c\right ) x^2}-\frac {3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{8 a^{7/2}} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=-\frac {\sqrt {a} \left (-8 a^3 c-15 b^3 x^2 (b+c x)+2 a^2 \left (b^2+10 b c x-12 c^2 x^2\right )+a b x \left (-5 b^2+62 b c x+52 c^2 x^2\right )\right )-3 \left (5 b^4-24 a b^2 c+16 a^2 c^2\right ) x^2 \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{4 a^{7/2} \left (b^2-4 a c\right ) x \sqrt {x^2 (a+x (b+c x))}} \]
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Time = 0.22 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.83
method | result | size |
pseudoelliptic | \(\frac {-\frac {5 x b \left (-\frac {52}{5} c^{2} x^{2}-\frac {62}{5} b c x +b^{2}\right ) a^{\frac {3}{2}}}{4}+6 \left (-c^{2} x^{2}+\frac {5}{6} b c x +\frac {1}{12} b^{2}\right ) a^{\frac {5}{2}}-2 a^{\frac {7}{2}} c +6 \left (-\frac {5 b^{3} \left (c x +b \right ) \sqrt {a}}{8}+\sqrt {c \,x^{2}+b x +a}\, \left (-\ln \left (2\right )+\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )\right ) \left (a c -\frac {5 b^{2}}{4}\right ) \left (a c -\frac {b^{2}}{4}\right )\right ) x^{2}}{a^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, \left (4 a c -b^{2}\right ) x^{2}}\) | \(173\) |
default | \(-\frac {x \left (c \,x^{2}+b x +a \right ) \left (-104 a^{\frac {5}{2}} b \,c^{2} x^{3}+30 a^{\frac {3}{2}} b^{3} c \,x^{3}+48 a^{\frac {7}{2}} c^{2} x^{2}-124 a^{\frac {5}{2}} b^{2} c \,x^{2}+30 a^{\frac {3}{2}} b^{4} x^{2}-48 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{2} x^{2}+72 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c \,x^{2}-15 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} x^{2}-40 a^{\frac {7}{2}} b c x +10 a^{\frac {5}{2}} b^{3} x +16 a^{\frac {9}{2}} c -4 a^{\frac {7}{2}} b^{2}\right )}{8 \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {9}{2}} \left (4 a c -b^{2}\right )}\) | \(292\) |
risch | \(-\frac {\left (c \,x^{2}+b x +a \right ) \left (-7 b x +2 a \right )}{4 a^{3} x \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}+\frac {\left (-\frac {2 b^{3} c x}{a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {b^{4}}{a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {6 c^{2} b x}{a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {3 c \,b^{2}}{a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {c}{a^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {b^{2}}{a^{3} \sqrt {c \,x^{2}+b x +a}}+\frac {3 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) c}{2 a^{\frac {5}{2}}}-\frac {15 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b^{2}}{8 a^{\frac {7}{2}}}\right ) x \sqrt {c \,x^{2}+b x +a}}{\sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}\) | \(317\) |
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Time = 0.34 (sec) , antiderivative size = 630, normalized size of antiderivative = 3.01 \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{5} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{4} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{3}\right )} \sqrt {a} \log \left (-\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {a}}{x^{3}}\right ) + 4 \, {\left (2 \, a^{3} b^{2} - 8 \, a^{4} c - {\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{3} - {\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{2} - 5 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{16 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{5} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{4} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{3}\right )}}, \frac {3 \, {\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{5} + {\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{4} + {\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} x^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) - 2 \, {\left (2 \, a^{3} b^{2} - 8 \, a^{4} c - {\left (15 \, a b^{3} c - 52 \, a^{2} b c^{2}\right )} x^{3} - {\left (15 \, a b^{4} - 62 \, a^{2} b^{2} c + 24 \, a^{3} c^{2}\right )} x^{2} - 5 \, {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{8 \, {\left ({\left (a^{4} b^{2} c - 4 \, a^{5} c^{2}\right )} x^{5} + {\left (a^{4} b^{3} - 4 \, a^{5} b c\right )} x^{4} + {\left (a^{5} b^{2} - 4 \, a^{6} c\right )} x^{3}\right )}}\right ] \]
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\[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (a x^{2} + b x^{3} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}} \,d x \]
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