Integrand size = 18, antiderivative size = 115 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^4} \, dx=\frac {x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 a b+\left (b^2+2 a c\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {2 \left (b^2+2 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1368, 752, 652, 632, 212} \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^4} \, dx=-\frac {2 \left (2 a c+b^2\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {x \left (2 a c+b^2\right )+3 a b}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]
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Rule 212
Rule 632
Rule 652
Rule 752
Rule 1368
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2}{\left (a+b x+c x^2\right )^3} \, dx \\ & = \frac {x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int \frac {2 a-2 b x}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )} \\ & = \frac {x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 a b+\left (b^2+2 a c\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {\left (b^2+2 a c\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2} \\ & = \frac {x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 a b+\left (b^2+2 a c\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (2 \left (b^2+2 a c\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2} \\ & = \frac {x (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 a b+\left (b^2+2 a c\right ) x}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {2 \left (b^2+2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^4} \, dx=\frac {1}{2} \left (\frac {b^2 x+a (b-2 c x)}{c \left (-b^2+4 a c\right ) (a+x (b+c x))^2}+\frac {\left (b^2+2 a c\right ) (b+2 c x)}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {4 \left (b^2+2 a c\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.83
method | result | size |
default | \(\frac {\frac {c \left (2 a c +b^{2}\right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {3 b \left (2 a c +b^{2}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (2 a c -5 b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {3 a^{2} b}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {2 \left (2 a c +b^{2}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}\) | \(210\) |
risch | \(\frac {\frac {c \left (2 a c +b^{2}\right ) x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {3 b \left (2 a c +b^{2}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a \left (2 a c -5 b^{2}\right ) x}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}+\frac {3 a^{2} b}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {2 \ln \left (\left (32 a^{2} c^{3}-16 b^{2} a \,c^{2}+2 b^{4} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) a c}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left (\left (32 a^{2} c^{3}-16 b^{2} a \,c^{2}+2 b^{4} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right ) b^{2}}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {2 \ln \left (\left (-32 a^{2} c^{3}+16 b^{2} a \,c^{2}-2 b^{4} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) a c}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left (\left (-32 a^{2} c^{3}+16 b^{2} a \,c^{2}-2 b^{4} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right ) b^{2}}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\) | \(436\) |
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Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (109) = 218\).
Time = 0.28 (sec) , antiderivative size = 887, normalized size of antiderivative = 7.71 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^4} \, dx=\left [\frac {6 \, a^{2} b^{3} - 24 \, a^{3} b c + 2 \, {\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} x^{3} + 3 \, {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} x^{2} + 2 \, {\left ({\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{4} + a^{2} b^{2} + 2 \, a^{3} c + 2 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x^{3} + {\left (b^{4} + 4 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (a b^{3} + 2 \, a^{2} b c\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \, {\left (5 \, a b^{4} - 22 \, a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} x}{2 \, {\left (a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{4} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{3} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{2} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} x\right )}}, \frac {6 \, a^{2} b^{3} - 24 \, a^{3} b c + 2 \, {\left (b^{4} c - 2 \, a b^{2} c^{2} - 8 \, a^{2} c^{3}\right )} x^{3} + 3 \, {\left (b^{5} - 2 \, a b^{3} c - 8 \, a^{2} b c^{2}\right )} x^{2} - 4 \, {\left ({\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{4} + a^{2} b^{2} + 2 \, a^{3} c + 2 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x^{3} + {\left (b^{4} + 4 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (a b^{3} + 2 \, a^{2} b c\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (5 \, a b^{4} - 22 \, a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} x}{2 \, {\left (a^{2} b^{6} - 12 \, a^{3} b^{4} c + 48 \, a^{4} b^{2} c^{2} - 64 \, a^{5} c^{3} + {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} x^{4} + 2 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} x^{3} + {\left (b^{8} - 10 \, a b^{6} c + 24 \, a^{2} b^{4} c^{2} + 32 \, a^{3} b^{2} c^{3} - 128 \, a^{4} c^{4}\right )} x^{2} + 2 \, {\left (a b^{7} - 12 \, a^{2} b^{5} c + 48 \, a^{3} b^{3} c^{2} - 64 \, a^{4} b c^{3}\right )} x\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 570 vs. \(2 (107) = 214\).
Time = 0.75 (sec) , antiderivative size = 570, normalized size of antiderivative = 4.96 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^4} \, dx=- \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) \log {\left (x + \frac {- 64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) - 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + 2 a b c + b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + b^{3}}{4 a c^{2} + 2 b^{2} c} \right )} + \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) \log {\left (x + \frac {64 a^{3} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) - 48 a^{2} b^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + 12 a b^{4} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + 2 a b c - b^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \cdot \left (2 a c + b^{2}\right ) + b^{3}}{4 a c^{2} + 2 b^{2} c} \right )} + \frac {6 a^{2} b + x^{3} \cdot \left (4 a c^{2} + 2 b^{2} c\right ) + x^{2} \cdot \left (6 a b c + 3 b^{3}\right ) + x \left (- 4 a^{2} c + 10 a b^{2}\right )}{32 a^{4} c^{2} - 16 a^{3} b^{2} c + 2 a^{2} b^{4} + x^{4} \cdot \left (32 a^{2} c^{4} - 16 a b^{2} c^{3} + 2 b^{4} c^{2}\right ) + x^{3} \cdot \left (64 a^{2} b c^{3} - 32 a b^{3} c^{2} + 4 b^{5} c\right ) + x^{2} \cdot \left (64 a^{3} c^{3} - 12 a b^{4} c + 2 b^{6}\right ) + x \left (64 a^{3} b c^{2} - 32 a^{2} b^{3} c + 4 a b^{5}\right )} \]
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Exception generated. \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^4} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.34 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^4} \, dx=\frac {2 \, {\left (b^{2} + 2 \, a c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {2 \, b^{2} c x^{3} + 4 \, a c^{2} x^{3} + 3 \, b^{3} x^{2} + 6 \, a b c x^{2} + 10 \, a b^{2} x - 4 \, a^{2} c x + 6 \, a^{2} b}{2 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (c x^{2} + b x + a\right )}^{2}} \]
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Time = 8.41 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.72 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^4} \, dx=\frac {\frac {3\,a^2\,b}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}-\frac {a\,x\,\left (2\,a\,c-5\,b^2\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {3\,b\,x^2\,\left (b^2+2\,a\,c\right )}{2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {c\,x^3\,\left (b^2+2\,a\,c\right )}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}+\frac {2\,\mathrm {atan}\left (\frac {\left (\frac {\left (b^2+2\,a\,c\right )\,\left (16\,a^2\,b\,c^2-8\,a\,b^3\,c+b^5\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {2\,c\,x\,\left (b^2+2\,a\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{b^2+2\,a\,c}\right )\,\left (b^2+2\,a\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}} \]
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