Integrand size = 24, antiderivative size = 153 \[ \int \frac {x^5}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right ) x}+\frac {x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} \sqrt {a x^2+b x^3+c x^4}} \]
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Time = 0.11 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1937, 1963, 12, 1928, 635, 212} \[ \int \frac {x^5}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2} \sqrt {a x^2+b x^3+c x^4}}+\frac {2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c x \left (b^2-4 a c\right )} \]
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Rule 12
Rule 212
Rule 635
Rule 1928
Rule 1937
Rule 1963
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int \frac {x (2 a+b x)}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{b^2-4 a c} \\ & = \frac {2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right ) x}+\frac {2 \int \frac {\left (b^2-4 a c\right ) x}{2 \sqrt {a x^2+b x^3+c x^4}} \, dx}{c \left (b^2-4 a c\right )} \\ & = \frac {2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right ) x}+\frac {\int \frac {x}{\sqrt {a x^2+b x^3+c x^4}} \, dx}{c} \\ & = \frac {2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right ) x}+\frac {\left (x \sqrt {a+b x+c x^2}\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{c \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right ) x}+\frac {\left (2 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 )}{c \sqrt {a x^2+b x^3+c x^4}} \\ & = \frac {2 x^2 (2 a+b x)}{\left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 b \sqrt {a x^2+b x^3+c x^4}}{c \left (b^2-4 a c\right ) x}+\frac {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 )}{c^{3/2} \sqrt {a x^2+b x^3+c x^4}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.73 \[ \int \frac {x^5}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=-\frac {x \left (2 \sqrt {c} \left (-a b-b^2 x+2 a c x\right )+\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right )}{c^{3/2} \left (-b^2+4 a c\right ) \sqrt {x^2 (a+x (b+c x))}} \]
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Time = 0.15 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.55
method | result | size |
pseudoelliptic | \(-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}+\frac {b \left (b x +2 a \right )}{\sqrt {c \,x^{2}+b x +a}\, c \left (4 a c -b^{2}\right )}+\frac {\ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right )}{c^{\frac {3}{2}}}\) | \(84\) |
default | \(-\frac {x^{3} \left (c \,x^{2}+b x +a \right ) \left (4 c^{\frac {5}{2}} a x -2 c^{\frac {3}{2}} b^{2} x -2 c^{\frac {3}{2}} a b -4 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) \sqrt {c \,x^{2}+b x +a}\, a \,c^{2}+\ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) \sqrt {c \,x^{2}+b x +a}\, b^{2} c \right )}{c^{\frac {5}{2}} \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}\) | \(166\) |
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Time = 0.30 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.71 \[ \int \frac {x^5}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} + {\left (b^{3} - 4 \, a b c\right )} x^{2} + {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {c} \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (a b c + {\left (b^{2} c - 2 \, a c^{2}\right )} x\right )}}{2 \, {\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x\right )}}, -\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} + {\left (b^{3} - 4 \, a b c\right )} x^{2} + {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (a b c + {\left (b^{2} c - 2 \, a c^{2}\right )} x\right )}}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{3} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2} + {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x}\right ] \]
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\[ \int \frac {x^5}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {x^{5}}{\left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^5}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int { \frac {x^{5}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.35 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.07 \[ \int \frac {x^5}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {{\left (b^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 4 \, a c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 2 \, \sqrt {a} b \sqrt {c}\right )} \mathrm {sgn}\left (x\right )}{b^{2} c^{\frac {3}{2}} - 4 \, a c^{\frac {5}{2}}} - \frac {2 \, {\left (\frac {a b}{b^{2} c \mathrm {sgn}\left (x\right ) - 4 \, a c^{2} \mathrm {sgn}\left (x\right )} + \frac {{\left (b^{2} - 2 \, a c\right )} x}{b^{2} c \mathrm {sgn}\left (x\right ) - 4 \, a c^{2} \mathrm {sgn}\left (x\right )}\right )}}{\sqrt {c x^{2} + b x + a}} - \frac {\log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {x^5}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {x^5}{{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}} \,d x \]
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