Integrand size = 18, antiderivative size = 107 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^3} \, dx=-\frac {x^3 (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 b x (2 a+b x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {6 a b \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 107, 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, 742, 736, 632, 212} \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^3} \, dx=\frac {6 a b \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {3 b x (2 a+b x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {x^3 (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]
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Rule 212
Rule 632
Rule 736
Rule 742
Rule 1368
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{\left (a+b x+c x^2\right )^3} \, dx \\ & = -\frac {x^3 (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {(3 b) \int \frac {x^2}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )} \\ & = -\frac {x^3 (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 b x (2 a+b x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {(3 a b) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2} \\ & = -\frac {x^3 (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 b x (2 a+b x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {(6 a b) \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^3 (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {3 b x (2 a+b x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac {6 a b \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 = 0.14 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^3} \, dx=-\frac {8 a^3 c+b^4 x^2+a b x \left (2 b^2+b c x+6 c^2 x^2\right )+a^2 \left (b^2+10 b c x+16 c^2 x^2\right )}{2 c \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}-\frac {6 a b \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(222\) vs. \(2(99)=198\).
Time = 0.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.08
method | result | size |
default | \(\frac {-\frac {3 a b c \,x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {\left (16 a^{2} c^{2}+a \,b^{2} c +b^{4}\right ) x^{2}}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (5 a c +b^{2}\right ) a b x}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a^{2} \left (8 a c +b^{2}\right )}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {6 a b \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {4 a c -b^{2}}}\) | \(223\) |
risch | \(\frac {-\frac {3 a b c \,x^{3}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {\left (16 a^{2} c^{2}+a \,b^{2} c +b^{4}\right ) x^{2}}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {\left (5 a c +b^{2}\right ) a b x}{c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}-\frac {a^{2} \left (8 a c +b^{2}\right )}{2 c \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {3 a b \ln \left (\left (32 a^{2} c^{3}-16 b^{2} a \,c^{2}+2 b^{4} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}+16 a^{2} b \,c^{2}-8 a \,b^{3} c +b^{5}\right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}+\frac {3 a b \ln \left (\left (-32 a^{2} c^{3}+16 b^{2} a \,c^{2}-2 b^{4} c \right ) x -\left (-4 a c +b^{2}\right )^{\frac {5}{2}}-16 a^{2} b \,c^{2}+8 a \,b^{3} c -b^{5}\right )}{\left (-4 a c +b^{2}\right )^{\frac {5}{2}}}\) | \(314\) |
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Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (99) = 198\).
Time = 0.29 (sec) , antiderivative size = 872, normalized size of antiderivative = 8.15 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^3} \, dx=\left [-\frac {a^{2} b^{4} + 4 \, a^{3} b^{2} c - 32 \, a^{4} c^{2} + 6 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 3 \, a b^{4} c + 12 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} x^{2} - 6 \, {\left (a b c^{3} x^{4} + 2 \, a b^{2} c^{2} x^{3} + 2 \, a^{2} b^{2} c x + a^{3} b c + {\left (a b^{3} c + 2 \, a^{2} b c^{2}\right )} x^{2}\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 (a b^{5} + a^{2} b^{3} c - 20 \, a^{3} b c^{2}\right )} x}{2 \, {\left (a^{2} b^{6} c - 12 \, a^{3} b^{4} c^{2} + 48 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4} + {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} x^{4} + 2 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} x^{3} + {\left (b^{8} c - 10 \, a b^{6} c^{2} + 24 \, a^{2} b^{4} c^{3} + 32 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5}\right )} x^{2} + 2 \, {\left (a b^{7} c - 12 \, a^{2} b^{5} c^{2} + 48 \, a^{3} b^{3} c^{3} - 64 \, a^{4} b c^{4}\right )} x\right )}}, -\frac {a^{2} b^{4} + 4 \, a^{3} b^{2} c - 32 \, a^{4} c^{2} + 6 \, {\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 3 \, a b^{4} c + 12 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} x^{2} - 12 \, {\left (a b c^{3} x^{4} + 2 \, a b^{2} c^{2} x^{3} + 2 \, a^{2} b^{2} c x + a^{3} b c + {\left (a b^{3} c + 2 \, a^{2} b c^{2}\right )} x^{2}\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 (a b^{5} + a^{2} b^{3} c - 20 \, a^{3} b c^{2}\right )} x}{2 \, {\left (a^{2} b^{6} c - 12 \, a^{3} b^{4} c^{2} + 48 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4} + {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} x^{4} + 2 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} x^{3} + {\left (b^{8} c - 10 \, a b^{6} c^{2} + 24 \, a^{2} b^{4} c^{3} + 32 \, a^{3} b^{2} c^{4} - 128 \, a^{4} c^{5}\right )} x^{2} + 2 \, {\left (a b^{7} c - 12 \, a^{2} b^{5} c^{2} + 48 \, a^{3} b^{3} c^{3} - 64 \, a^{4} b c^{4}\right )} x\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (102) = 204\).
Time = 0.66 (sec) , antiderivative size = 513, normalized size of antiderivative = 4.79 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^3} \, dx=3 a b \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {- 192 a^{4} b c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{3} b^{3} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a^{2} b^{5} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a b^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a b^{2}}{6 a b c} \right )} - 3 a b \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} \log {\left (x + \frac {192 a^{4} b c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{3} b^{3} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a^{2} b^{5} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} - 3 a b^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a b^{2}}{6 a b c} \right )} + \frac {- 8 a^{3} c - a^{2} b^{2} - 6 a b c^{2} x^{3} + x^{2} \left (- 16 a^{2} c^{2} - a b^{2} c - b^{4}\right ) + x \left (- 10 a^{2} b c - 2 a b^{3}\right )}{32 a^{4} c^{3} - 16 a^{3} b^{2} c^{2} + 2 a^{2} b^{4} c + x^{4} \cdot \left (32 a^{2} c^{5} - 16 a b^{2} c^{4} + 2 b^{4} c^{3}\right ) + x^{3} \cdot \left (64 a^{2} b c^{4} - 32 a b^{3} c^{3} + 4 b^{5} c^{2}\right ) + x^{2} \cdot \left (64 a^{3} c^{4} - 12 a b^{4} c^{2} + 2 b^{6} c\right ) + x \left (64 a^{3} b c^{3} - 32 a^{2} b^{3} c^{2} + 4 a b^{5} c\right )} \]
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Exception generated. \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^3} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.52 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^3} \, dx=-\frac {6 \, a b \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {6 \, a b c^{2} x^{3} + b^{4} x^{2} + a b^{2} c x^{2} + 16 \, a^{2} c^{2} x^{2} + 2 \, a b^{3} x + 10 \, a^{2} b c x + a^{2} b^{2} + 8 \, a^{3} c}{2 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} {\left (c x^{2} + b x + a\right )}^{2}} \]
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Time = 8.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.53 \[ \int \frac {1}{\left (c+\frac {a}{x^2}+\frac {b}{x}\right )^3 x^3} \, dx=-\frac {\frac {x^2\,\left (16\,a^2\,c^2+a\,b^2\,c+b^4\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {a^2\,\left (b^2+8\,a\,c\right )}{2\,c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {3\,a\,b\,c\,x^3}{16\,a^2\,c^2-8\,a\,b^2\,c+b^4}+\frac {a\,b\,x\,\left (b^2+5\,a\,c\right )}{c\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}-\frac {6\,a\,b\,\mathrm {atan}\left (\frac {\left (\frac {3\,a\,b^2}{{\left (4\,a\,c-b^2\right )}^{5/2}}+\frac {6\,a\,b\,c\,x}{{\left (4\,a\,c-b^2\right )}^{5/2}}\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}{3\,a\,b}\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}} \]
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