Integrand size = 24, antiderivative size = 219 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\frac {3 (3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}-\frac {3 \sqrt {a} b x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a x^2+b x^3+c x^4}}+\frac {3 \left (b^2+4 a c\right ) x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} \sqrt {a x^2+b x^3+c x^4}} \]
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Time = 0.16 (sec) , antiderivative size = 219, 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 = {1934, 1959, 1947, 857, 635, 212, 738} \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\frac {3 x \left (4 a c+b^2\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} \sqrt {a x^2+b x^3+c x^4}}-\frac {3 \sqrt {a} b x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}+\frac {3 (3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x} \]
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Rule 212
Rule 635
Rule 738
Rule 857
Rule 1934
Rule 1947
Rule 1959
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}+\frac {3}{2} \int \frac {(b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{x^2} \, dx \\ & = \frac {3 (3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}+\frac {3 \int \frac {4 a b c+c \left (b^2+4 a c\right ) x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{8 c} \\ & = \frac {3 (3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}+\frac {\left (3 x \sqrt {a+b x+c x^2}\right ) \int \frac {4 a b c+c \left (b^2+4 a c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx}{8 c \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {3 (3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}+\frac {\left (3 a b x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{2 \sqrt {a x^2+b x^3+c x^4}}+\frac {\left (3 \left (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}{8 \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {3 (3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}-\frac {\left (3 a b x \sqrt {a+b x+c x^2}\right ) \text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{\sqrt {a x^2+b x^3+c x^4}}+\frac {\left (3 \left (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 )}{4 \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {3 (3 b+2 c x) \sqrt {a x^2+b x^3+c x^4}}{4 x}-\frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^4}-\frac {3 \sqrt {a} b x \sqrt {a+b x+c x^2} \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {a x^2+b x^3+c x^4}}+\frac {3 \left (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 )}{8 \sqrt {c} \sqrt {a x^2+b x^3+c x^4}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.71 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\frac {\sqrt {a+x (b+c x)} \left (2 \sqrt {c} \sqrt {a+x (b+c x)} (-4 a+x (5 b+2 c x))+24 \sqrt {a} b \sqrt {c} x \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-3 \left (b^2+4 a c\right ) x \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{8 \sqrt {c} \sqrt {x^2 (a+x (b+c x))}} \]
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Time = 0.17 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.82
method | result | size |
pseudoelliptic | \(\frac {4 c^{\frac {3}{2}} x^{2} \sqrt {c \,x^{2}+b x +a}-12 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right ) b x \sqrt {a}\, \sqrt {c}+12 \ln \left (2\right ) b x \sqrt {a}\, \sqrt {c}+10 b \sqrt {c \,x^{2}+b x +a}\, x \sqrt {c}+12 \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) a c x +3 \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right ) b^{2} x -8 a \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}}{8 x \sqrt {c}}\) | \(180\) |
risch | \(-\frac {a \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{x^{2}}+\frac {\left (\frac {3 b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 \sqrt {c}}+\frac {3 a \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}+\frac {c \sqrt {c \,x^{2}+b x +a}\, x}{2}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, b}{4}-\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{x \sqrt {c \,x^{2}+b x +a}}\) | \(187\) |
default | \(\frac {\left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (8 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} x^{2}+12 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,x^{2}-12 c^{\frac {3}{2}} a^{\frac {3}{2}} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) b x -8 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c^{\frac {3}{2}}+8 c^{\frac {3}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b x +18 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, a b x +12 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{2} c^{2} x +3 c \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,b^{2} x \right )}{8 x^{4} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a \,c^{\frac {3}{2}}}\) | \(254\) |
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Time = 0.34 (sec) , antiderivative size = 757, normalized size of antiderivative = 3.46 \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\left [\frac {12 \, \sqrt {a} b c x^{2} \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 ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x^{2} \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 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{16 \, c x^{2}}, \frac {6 \, \sqrt {a} b c x^{2} \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 ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x^{2} \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 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{8 \, c x^{2}}, \frac {24 \, \sqrt {-a} b c x^{2} \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 ) + 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {c} x^{2} \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 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{16 \, c x^{2}}, \frac {12 \, \sqrt {-a} b c x^{2} \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 ) - 3 \, {\left (b^{2} + 4 \, a c\right )} \sqrt {-c} x^{2} \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 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c^{2} x^{2} + 5 \, b c x - 4 \, a c\right )}}{8 \, c x^{2}}\right ] \]
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\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\int \frac {\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}{x^{5}}\, dx \]
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\[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\int { \frac {{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{5}} \,d x } \]
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Exception generated. \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (a x^2+b x^3+c x^4\right )^{3/2}}{x^5} \, dx=\int \frac {{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}}{x^5} \,d x \]
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