Integrand size = 24, antiderivative size = 271 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^4 \left (b^2-4 a c\right ) x^2}+\frac {5 b \left (7 b^2-12 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{16 a^{9/2}} \]
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Time = 0.29 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1938, 1965, 12, 1918, 212} \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {5 b \left (7 b^2-12 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{16 a^{9/2}}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{12 a^3 x^3 \left (b^2-4 a c\right )}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a^2 x^4 \left (b^2-4 a c\right )}-\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^4 x^2 \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}} \]
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Rule 12
Rule 212
Rule 1918
Rule 1938
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^2 \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int \frac {-\frac {7 b^2}{2}+8 a c-3 b c x}{x^3 \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^2 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac {2 \int \frac {-\frac {1}{4} b \left (35 b^2-116 a c\right )-c \left (7 b^2-16 a c\right ) x}{x^2 \sqrt {a x^2+b x^3+c x^4}} \, dx}{3 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^2 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac {\int \frac {\frac {1}{8} \left (-105 b^4+460 a b^2 c-256 a^2 c^2\right )-\frac {1}{4} b c \left (35 b^2-116 a c\right ) x}{x \sqrt {a x^2+b x^3+c x^4}} \, dx}{3 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^2 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^4 \left (b^2-4 a c\right ) x^2}+\frac {\int -\frac {15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )}{16 \sqrt {a x^2+b x^3+c x^4}} \, dx}{3 a^4 \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^2 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^4 \left (b^2-4 a c\right ) x^2}-\frac {\left (5 b \left (7 b^2-12 a c\right )\right ) \int \frac {1}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{16 a^4} \\ & = \frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^4 \left (b^2-4 a c\right ) x^2}+\frac {\left (5 b \left (7 b^2-12 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 )}{8 a^4} \\ & = \frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^4 \left (b^2-4 a c\right ) x^2}+\frac {5 b \left (7 b^2-12 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 )}{16 a^{9/2}} \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {a} \left (-32 a^4 c+105 b^4 x^3 (b+c x)+5 a b^2 x^2 \left (7 b^2-106 b c x-92 c^2 x^2\right )+8 a^3 \left (b^2+7 b c x+16 c^2 x^2\right )+2 a^2 x \left (-7 b^3-86 b^2 c x+244 b c^2 x^2+128 c^3 x^3\right )\right )+15 b \left (7 b^4-40 a b^2 c+48 a^2 c^2\right ) x^3 \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{24 a^{9/2} \left (-b^2+4 a c\right ) x^2 \sqrt {x^2 (a+x (b+c x))}} \]
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Time = 0.27 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(-\frac {15 \left (-\frac {7 x^{2} b^{2} \left (-\frac {92}{7} c^{2} x^{2}-\frac {106}{7} b c x +b^{2}\right ) a^{\frac {3}{2}}}{72}+\frac {7 x \left (-\frac {128}{7} c^{3} x^{3}-\frac {244}{7} b \,c^{2} x^{2}+\frac {86}{7} b^{2} c x +b^{3}\right ) a^{\frac {5}{2}}}{180}+\frac {\left (-16 c^{2} x^{2}-7 b c x -b^{2}\right ) a^{\frac {7}{2}}}{45}+\frac {4 a^{\frac {9}{2}} c}{45}+\left (-\frac {7 b^{3} \left (c x +b \right ) \sqrt {a}}{24}+\sqrt {c \,x^{2}+b x +a}\, \left (-\ln \left (2\right )+\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )\right ) \left (a c -\frac {7 b^{2}}{12}\right ) \left (a c -\frac {b^{2}}{4}\right )\right ) x^{3} b \right )}{\sqrt {c \,x^{2}+b x +a}\, a^{\frac {9}{2}} \left (4 a c -b^{2}\right ) x^{3}}\) | \(213\) |
default | \(-\frac {\left (c \,x^{2}+b x +a \right ) \left (-512 a^{\frac {7}{2}} c^{3} x^{4}+920 a^{\frac {5}{2}} b^{2} c^{2} x^{4}-210 a^{\frac {3}{2}} b^{4} c \,x^{4}-976 a^{\frac {7}{2}} b \,c^{2} x^{3}+1060 a^{\frac {5}{2}} b^{3} c \,x^{3}-210 a^{\frac {3}{2}} b^{5} x^{3}+720 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3} b \,c^{2} x^{3}-600 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} c \,x^{3}+105 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a \,b^{5} x^{3}-256 a^{\frac {9}{2}} c^{2} x^{2}+344 a^{\frac {7}{2}} b^{2} c \,x^{2}-70 a^{\frac {5}{2}} b^{4} x^{2}-112 a^{\frac {9}{2}} b c x +28 a^{\frac {7}{2}} b^{3} x +64 a^{\frac {11}{2}} c -16 a^{\frac {9}{2}} b^{2}\right )}{48 \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {11}{2}} \left (4 a c -b^{2}\right )}\) | \(340\) |
risch | \(-\frac {\left (c \,x^{2}+b x +a \right ) \left (-40 a c \,x^{2}+57 b^{2} x^{2}-22 a b x +8 a^{2}\right )}{24 a^{4} x^{2} \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}+\frac {\left (\frac {2 b^{4} c x}{a^{4} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {b^{5}}{a^{4} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {4 c^{3} x}{a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {2 c^{2} b}{a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {8 b^{2} c^{2} x}{a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {4 b^{3} c}{a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {b^{3}}{a^{4} \sqrt {c \,x^{2}+b x +a}}+\frac {2 b c}{a^{3} \sqrt {c \,x^{2}+b x +a}}-\frac {15 b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) c}{4 a^{\frac {7}{2}}}+\frac {35 b^{3} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{16 a^{\frac {9}{2}}}\right ) x \sqrt {c \,x^{2}+b x +a}}{\sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}\) | \(405\) |
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Time = 0.41 (sec) , antiderivative size = 716, normalized size of antiderivative = 2.64 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\left [-\frac {15 \, {\left ({\left (7 \, b^{5} c - 40 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x^{6} + {\left (7 \, b^{6} - 40 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2}\right )} x^{5} + {\left (7 \, a b^{5} - 40 \, a^{2} b^{3} c + 48 \, a^{3} b c^{2}\right )} x^{4}\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 (8 \, a^{4} b^{2} - 32 \, a^{5} c + {\left (105 \, a b^{4} c - 460 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3}\right )} x^{4} + {\left (105 \, a b^{5} - 530 \, a^{2} b^{3} c + 488 \, a^{3} b c^{2}\right )} x^{3} + {\left (35 \, a^{2} b^{4} - 172 \, a^{3} b^{2} c + 128 \, a^{4} c^{2}\right )} x^{2} - 14 \, {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{96 \, {\left ({\left (a^{5} b^{2} c - 4 \, a^{6} c^{2}\right )} x^{6} + {\left (a^{5} b^{3} - 4 \, a^{6} b c\right )} x^{5} + {\left (a^{6} b^{2} - 4 \, a^{7} c\right )} x^{4}\right )}}, -\frac {15 \, {\left ({\left (7 \, b^{5} c - 40 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x^{6} + {\left (7 \, b^{6} - 40 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2}\right )} x^{5} + {\left (7 \, a b^{5} - 40 \, a^{2} b^{3} c + 48 \, a^{3} b c^{2}\right )} x^{4}\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 (8 \, a^{4} b^{2} - 32 \, a^{5} c + {\left (105 \, a b^{4} c - 460 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3}\right )} x^{4} + {\left (105 \, a b^{5} - 530 \, a^{2} b^{3} c + 488 \, a^{3} b c^{2}\right )} x^{3} + {\left (35 \, a^{2} b^{4} - 172 \, a^{3} b^{2} c + 128 \, a^{4} c^{2}\right )} x^{2} - 14 \, {\left (a^{3} b^{3} - 4 \, a^{4} b c\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{48 \, {\left ({\left (a^{5} b^{2} c - 4 \, a^{6} c^{2}\right )} x^{6} + {\left (a^{5} b^{3} - 4 \, a^{6} b c\right )} x^{5} + {\left (a^{6} b^{2} - 4 \, a^{7} c\right )} x^{4}\right )}}\right ] \]
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\[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x \left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}} x} \,d x } \]
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Timed out. \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}} \,d x \]
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