Integrand size = 16, antiderivative size = 146 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=\frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {10 a^2 x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {40 a^3 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \]
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Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {736, 632, 212} \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=\frac {40 a^3 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}+\frac {10 a^2 x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \]
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Rule 212
Rule 632
Rule 736
Rubi steps \begin{align*} \text {integral}& = \frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {(10 a) \int \frac {x^4}{\left (a+b x+c x^2\right )^3} \, dx}{3 \left (b^2-4 a c\right )} \\ & = \frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {\left (10 a^2\right ) \int \frac {x^2}{\left (a+b x+c x^2\right )^2} \, dx}{\left (b^2-4 a c\right )^2} \\ & = \frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {10 a^2 x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {\left (20 a^3\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3} \\ & = \frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {10 a^2 x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {\left (40 a^3\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 )^3} \\ & = \frac {x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac {10 a^2 x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac {40 a^3 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(314\) vs. \(2(146)=292\).
Time = 0.12 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.15 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=\frac {b^7-12 a b^5 c+48 a^2 b^3 c^2-59 a^3 b c^3-4 b^6 c x+33 a b^4 c^2 x-72 a^2 b^2 c^3 x+26 a^3 c^4 x}{3 c^5 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac {-b^7+12 a b^5 c-48 a^2 b^3 c^2+74 a^3 b c^3+b^6 c x-12 a b^4 c^2 x+48 a^2 b^2 c^3 x-44 a^3 c^4 x}{c^4 \left (-b^2+4 a c\right )^3 (a+x (b+c x))}+\frac {b^6 x+a b^4 (b-6 c x)+a^3 c^2 (5 b-2 c x)+a^2 b^2 c (-5 b+9 c x)}{3 c^5 \left (-b^2+4 a c\right ) (a+x (b+c x))^3}+\frac {40 a^3 \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(530\) vs. \(2(138)=276\).
Time = 13.19 (sec) , antiderivative size = 531, normalized size of antiderivative = 3.64
method | result | size |
default | \(\frac {-\frac {\left (44 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{5}}{c \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {b \left (14 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{4}}{\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{2}}-\frac {\left (160 a^{4} c^{4}-286 a^{3} b^{2} c^{3}+12 a^{2} b^{4} c^{2}+7 a \,b^{6} c -b^{8}\right ) x^{3}}{3 c^{3} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b a \left (16 c^{3} a^{3}+53 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right ) x^{2}}{c^{3} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {a^{2} \left (20 c^{3} a^{3}-66 a^{2} b^{2} c^{2}+13 a \,b^{4} c -b^{6}\right ) x}{\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{3}}+\frac {\left (66 a^{2} c^{2}-13 a \,b^{2} c +b^{4}\right ) a^{3} b}{3 c^{3} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {40 a^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {4 a c -b^{2}}}\) | \(531\) |
risch | \(\frac {-\frac {\left (44 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{5}}{c \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {b \left (14 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) x^{4}}{\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{2}}-\frac {\left (160 a^{4} c^{4}-286 a^{3} b^{2} c^{3}+12 a^{2} b^{4} c^{2}+7 a \,b^{6} c -b^{8}\right ) x^{3}}{3 c^{3} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}+\frac {b a \left (16 c^{3} a^{3}+53 a^{2} b^{2} c^{2}-12 a \,b^{4} c +b^{6}\right ) x^{2}}{c^{3} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}-\frac {a^{2} \left (20 c^{3} a^{3}-66 a^{2} b^{2} c^{2}+13 a \,b^{4} c -b^{6}\right ) x}{\left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) c^{3}}+\frac {\left (66 a^{2} c^{2}-13 a \,b^{2} c +b^{4}\right ) a^{3} b}{3 c^{3} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {20 a^{3} \ln \left (\left (-128 a^{3} c^{4}+96 a^{2} b^{2} c^{3}-24 c^{2} a \,b^{4}+2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}-64 a^{3} c^{3} b +48 a^{2} c^{2} b^{3}-12 a \,b^{5} c +b^{7}\right )}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {20 a^{3} \ln \left (\left (128 a^{3} c^{4}-96 a^{2} b^{2} c^{3}+24 c^{2} a \,b^{4}-2 b^{6} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}}+64 a^{3} c^{3} b -48 a^{2} c^{2} b^{3}+12 a \,b^{5} c -b^{7}\right )}{\left (-4 a c +b^{2}\right )^{\frac {7}{2}}}\) | \(650\) |
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Leaf count of result is larger than twice the leaf count of optimal. 827 vs. \(2 (138) = 276\).
Time = 0.30 (sec) , antiderivative size = 1675, normalized size of antiderivative = 11.47 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 938 vs. \(2 (139) = 278\).
Time = 1.83 (sec) , antiderivative size = 938, normalized size of antiderivative = 6.42 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=- 20 a^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {- 5120 a^{7} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 5120 a^{6} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 1920 a^{5} b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 320 a^{4} b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 20 a^{3} b^{8} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 20 a^{3} b}{40 a^{3} c} \right )} + 20 a^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {5120 a^{7} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 5120 a^{6} b^{2} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 1920 a^{5} b^{4} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 320 a^{4} b^{6} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 20 a^{3} b^{8} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 20 a^{3} b}{40 a^{3} c} \right )} + \frac {66 a^{5} b c^{2} - 13 a^{4} b^{3} c + a^{3} b^{5} + x^{5} \left (- 132 a^{3} c^{5} + 144 a^{2} b^{2} c^{4} - 36 a b^{4} c^{3} + 3 b^{6} c^{2}\right ) + x^{4} \left (- 42 a^{3} b c^{4} + 144 a^{2} b^{3} c^{3} - 36 a b^{5} c^{2} + 3 b^{7} c\right ) + x^{3} \left (- 160 a^{4} c^{4} + 286 a^{3} b^{2} c^{3} - 12 a^{2} b^{4} c^{2} - 7 a b^{6} c + b^{8}\right ) + x^{2} \cdot \left (48 a^{4} b c^{3} + 159 a^{3} b^{3} c^{2} - 36 a^{2} b^{5} c + 3 a b^{7}\right ) + x \left (- 60 a^{5} c^{3} + 198 a^{4} b^{2} c^{2} - 39 a^{3} b^{4} c + 3 a^{2} b^{6}\right )}{192 a^{6} c^{6} - 144 a^{5} b^{2} c^{5} + 36 a^{4} b^{4} c^{4} - 3 a^{3} b^{6} c^{3} + x^{6} \cdot \left (192 a^{3} c^{9} - 144 a^{2} b^{2} c^{8} + 36 a b^{4} c^{7} - 3 b^{6} c^{6}\right ) + x^{5} \cdot \left (576 a^{3} b c^{8} - 432 a^{2} b^{3} c^{7} + 108 a b^{5} c^{6} - 9 b^{7} c^{5}\right ) + x^{4} \cdot \left (576 a^{4} c^{8} + 144 a^{3} b^{2} c^{7} - 324 a^{2} b^{4} c^{6} + 99 a b^{6} c^{5} - 9 b^{8} c^{4}\right ) + x^{3} \cdot \left (1152 a^{4} b c^{7} - 672 a^{3} b^{3} c^{6} + 72 a^{2} b^{5} c^{5} + 18 a b^{7} c^{4} - 3 b^{9} c^{3}\right ) + x^{2} \cdot \left (576 a^{5} c^{7} + 144 a^{4} b^{2} c^{6} - 324 a^{3} b^{4} c^{5} + 99 a^{2} b^{6} c^{4} - 9 a b^{8} c^{3}\right ) + x \left (576 a^{5} b c^{6} - 432 a^{4} b^{3} c^{5} + 108 a^{3} b^{5} c^{4} - 9 a^{2} b^{7} c^{3}\right )} \]
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Exception generated. \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (138) = 276\).
Time = 0.27 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.64 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {40 \, a^{3} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {3 \, b^{6} c^{2} x^{5} - 36 \, a b^{4} c^{3} x^{5} + 144 \, a^{2} b^{2} c^{4} x^{5} - 132 \, a^{3} c^{5} x^{5} + 3 \, b^{7} c x^{4} - 36 \, a b^{5} c^{2} x^{4} + 144 \, a^{2} b^{3} c^{3} x^{4} - 42 \, a^{3} b c^{4} x^{4} + b^{8} x^{3} - 7 \, a b^{6} c x^{3} - 12 \, a^{2} b^{4} c^{2} x^{3} + 286 \, a^{3} b^{2} c^{3} x^{3} - 160 \, a^{4} c^{4} x^{3} + 3 \, a b^{7} x^{2} - 36 \, a^{2} b^{5} c x^{2} + 159 \, a^{3} b^{3} c^{2} x^{2} + 48 \, a^{4} b c^{3} x^{2} + 3 \, a^{2} b^{6} x - 39 \, a^{3} b^{4} c x + 198 \, a^{4} b^{2} c^{2} x - 60 \, a^{5} c^{3} x + a^{3} b^{5} - 13 \, a^{4} b^{3} c + 66 \, a^{5} b c^{2}}{3 \, {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} {\left (c x^{2} + b x + a\right )}^{3}} \]
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Time = 9.99 (sec) , antiderivative size = 656, normalized size of antiderivative = 4.49 \[ \int \frac {x^6}{\left (a+b x+c x^2\right )^4} \, dx=-\frac {\frac {a^3\,\left (66\,a^2\,b\,c^2-13\,a\,b^3\,c+b^5\right )}{3\,c^3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^5\,\left (-44\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{c\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}-\frac {x^3\,\left (160\,a^4\,c^4-286\,a^3\,b^2\,c^3+12\,a^2\,b^4\,c^2+7\,a\,b^6\,c-b^8\right )}{3\,c^3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {x^4\,\left (-14\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{c^2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {a\,x^2\,\left (16\,a^3\,b\,c^3+53\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{c^3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {a^2\,x\,\left (-20\,a^3\,c^3+66\,a^2\,b^2\,c^2-13\,a\,b^4\,c+b^6\right )}{c^3\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}}{x^2\,\left (3\,c\,a^2+3\,a\,b^2\right )+x^4\,\left (3\,b^2\,c+3\,a\,c^2\right )+a^3+x^3\,\left (b^3+6\,a\,c\,b\right )+c^3\,x^6+3\,b\,c^2\,x^5+3\,a^2\,b\,x}-\frac {40\,a^3\,\mathrm {atan}\left (\frac {\left (\frac {20\,a^3\,\left (-64\,a^3\,b\,c^3+48\,a^2\,b^3\,c^2-12\,a\,b^5\,c+b^7\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {40\,a^3\,c\,x}{{\left (4\,a\,c-b^2\right )}^{7/2}}\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}{20\,a^3}\right )}{{\left (4\,a\,c-b^2\right )}^{7/2}} \]
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