Integrand size = 24, antiderivative size = 122 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=2 \left (b^2-4 a c\right )^3 d^8 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^3+\frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7-2 \left (b^2-4 a c\right )^{7/2} d^8 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {706, 632, 212} \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=-2 d^8 \left (b^2-4 a c\right )^{7/2} \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+\frac {2}{5} d^8 \left (b^2-4 a c\right ) (b+2 c x)^5+\frac {2}{3} d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^3+2 d^8 \left (b^2-4 a c\right )^3 (b+2 c x)+\frac {2}{7} d^8 (b+2 c x)^7 \]
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Rule 212
Rule 632
Rule 706
Rubi steps \begin{align*} \text {integral}& = \frac {2}{7} d^8 (b+2 c x)^7+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^6}{a+b x+c x^2} \, dx \\ & = \frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac {(b d+2 c d x)^4}{a+b x+c x^2} \, dx \\ & = \frac {2}{3} \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^3+\frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7+\left (\left (b^2-4 a c\right )^3 d^6\right ) \int \frac {(b d+2 c d x)^2}{a+b x+c x^2} \, dx \\ & = 2 \left (b^2-4 a c\right )^3 d^8 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^3+\frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7+\left (\left (b^2-4 a c\right )^4 d^8\right ) \int \frac {1}{a+b x+c x^2} \, dx \\ & = 2 \left (b^2-4 a c\right )^3 d^8 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^3+\frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7-\left (2 \left (b^2-4 a c\right )^4 d^8\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right ) \\ & = 2 \left (b^2-4 a c\right )^3 d^8 (b+2 c x)+\frac {2}{3} \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^3+\frac {2}{5} \left (b^2-4 a c\right ) d^8 (b+2 c x)^5+\frac {2}{7} d^8 (b+2 c x)^7-2 \left (b^2-4 a c\right )^{7/2} d^8 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.54 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=d^8 \left (\frac {16}{105} c x \left (105 b^6+315 b^5 c x+420 b^3 c^2 x \left (-2 a+3 c x^2\right )+70 b^4 c \left (-9 a+11 c x^2\right )+840 b c^3 x \left (a^2-a c x^2+c^2 x^4\right )+112 b^2 c^2 \left (15 a^2-10 a c x^2+12 c^2 x^4\right )+16 c^3 \left (-105 a^3+35 a^2 c x^2-21 a c^2 x^4+15 c^3 x^6\right )\right )+2 \left (-b^2+4 a c\right )^{7/2} \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(245\) vs. \(2(112)=224\).
Time = 2.49 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.02
method | result | size |
default | \(d^{8} \left (\frac {256 c^{7} x^{7}}{7}+128 b \,c^{6} x^{6}-\frac {256 a \,c^{6} x^{5}}{5}+\frac {1024 b^{2} c^{5} x^{5}}{5}-128 a b \,c^{5} x^{4}+192 b^{3} c^{4} x^{4}+\frac {256 a^{2} c^{5} x^{3}}{3}-\frac {512 a \,b^{2} c^{4} x^{3}}{3}+\frac {352 b^{4} c^{3} x^{3}}{3}+128 a^{2} b \,c^{4} x^{2}-128 b^{3} c^{3} a \,x^{2}+48 b^{5} c^{2} x^{2}-256 a^{3} c^{4} x +256 a^{2} b^{2} c^{3} x -96 c^{2} a \,b^{4} x +16 b^{6} c x +\frac {2 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}\right )\) | \(246\) |
risch | \(\frac {256 c^{7} d^{8} x^{7}}{7}+128 c^{6} d^{8} b \,x^{6}-\frac {256 c^{6} d^{8} x^{5} a}{5}+\frac {1024 c^{5} d^{8} x^{5} b^{2}}{5}-128 c^{5} d^{8} a b \,x^{4}+192 c^{4} d^{8} b^{3} x^{4}-\frac {512 c^{4} d^{8} a \,b^{2} x^{3}}{3}+\frac {352 c^{3} d^{8} b^{4} x^{3}}{3}+\frac {256 c^{5} d^{8} x^{3} a^{2}}{3}+128 c^{4} d^{8} a^{2} b \,x^{2}-128 c^{3} d^{8} a \,b^{3} x^{2}+48 c^{2} d^{8} b^{5} x^{2}-256 c^{4} d^{8} a^{3} x +256 c^{3} d^{8} a^{2} b^{2} x -96 c^{2} d^{8} a \,b^{4} x +16 c \,d^{8} b^{6} x -d^{8} \left (-4 a c +b^{2}\right )^{\frac {7}{2}} \ln \left (2 \left (-4 a c +b^{2}\right )^{\frac {7}{2}} c x +\left (-4 a c +b^{2}\right )^{\frac {7}{2}} b +256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )+d^{8} \left (-4 a c +b^{2}\right )^{\frac {7}{2}} \ln \left (-2 \left (-4 a c +b^{2}\right )^{\frac {7}{2}} c x -\left (-4 a c +b^{2}\right )^{\frac {7}{2}} b +256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )\) | \(381\) |
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Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (112) = 224\).
Time = 0.42 (sec) , antiderivative size = 536, normalized size of antiderivative = 4.39 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=\left [\frac {256}{7} \, c^{7} d^{8} x^{7} + 128 \, b c^{6} d^{8} x^{6} + \frac {256}{5} \, {\left (4 \, b^{2} c^{5} - a c^{6}\right )} d^{8} x^{5} + 64 \, {\left (3 \, b^{3} c^{4} - 2 \, a b c^{5}\right )} d^{8} x^{4} + \frac {32}{3} \, {\left (11 \, b^{4} c^{3} - 16 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} d^{8} x^{3} + 16 \, {\left (3 \, b^{5} c^{2} - 8 \, a b^{3} c^{3} + 8 \, a^{2} b c^{4}\right )} d^{8} x^{2} - {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {b^{2} - 4 \, a c} d^{8} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 16 \, {\left (b^{6} c - 6 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} d^{8} x, \frac {256}{7} \, c^{7} d^{8} x^{7} + 128 \, b c^{6} d^{8} x^{6} + \frac {256}{5} \, {\left (4 \, b^{2} c^{5} - a c^{6}\right )} d^{8} x^{5} + 64 \, {\left (3 \, b^{3} c^{4} - 2 \, a b c^{5}\right )} d^{8} x^{4} + \frac {32}{3} \, {\left (11 \, b^{4} c^{3} - 16 \, a b^{2} c^{4} + 8 \, a^{2} c^{5}\right )} d^{8} x^{3} + 16 \, {\left (3 \, b^{5} c^{2} - 8 \, a b^{3} c^{3} + 8 \, a^{2} b c^{4}\right )} d^{8} x^{2} - 2 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c} d^{8} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 16 \, {\left (b^{6} c - 6 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3} - 16 \, a^{3} c^{4}\right )} d^{8} x\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (124) = 248\).
Time = 0.67 (sec) , antiderivative size = 502, normalized size of antiderivative = 4.11 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=128 b c^{6} d^{8} x^{6} + \frac {256 c^{7} d^{8} x^{7}}{7} - d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}} \log {\left (x + \frac {64 a^{3} b c^{3} d^{8} - 48 a^{2} b^{3} c^{2} d^{8} + 12 a b^{5} c d^{8} - b^{7} d^{8} - d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}}}{128 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 24 a b^{4} c^{2} d^{8} - 2 b^{6} c d^{8}} \right )} + d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}} \log {\left (x + \frac {64 a^{3} b c^{3} d^{8} - 48 a^{2} b^{3} c^{2} d^{8} + 12 a b^{5} c d^{8} - b^{7} d^{8} + d^{8} \sqrt {- \left (4 a c - b^{2}\right )^{7}}}{128 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 24 a b^{4} c^{2} d^{8} - 2 b^{6} c d^{8}} \right )} + x^{5} \left (- \frac {256 a c^{6} d^{8}}{5} + \frac {1024 b^{2} c^{5} d^{8}}{5}\right ) + x^{4} \left (- 128 a b c^{5} d^{8} + 192 b^{3} c^{4} d^{8}\right ) + x^{3} \cdot \left (\frac {256 a^{2} c^{5} d^{8}}{3} - \frac {512 a b^{2} c^{4} d^{8}}{3} + \frac {352 b^{4} c^{3} d^{8}}{3}\right ) + x^{2} \cdot \left (128 a^{2} b c^{4} d^{8} - 128 a b^{3} c^{3} d^{8} + 48 b^{5} c^{2} d^{8}\right ) + x \left (- 256 a^{3} c^{4} d^{8} + 256 a^{2} b^{2} c^{3} d^{8} - 96 a b^{4} c^{2} d^{8} + 16 b^{6} c d^{8}\right ) \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 313 vs. \(2 (112) = 224\).
Time = 0.30 (sec) , antiderivative size = 313, normalized size of antiderivative = 2.57 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=\frac {2 \, {\left (b^{8} d^{8} - 16 \, a b^{6} c d^{8} + 96 \, a^{2} b^{4} c^{2} d^{8} - 256 \, a^{3} b^{2} c^{3} d^{8} + 256 \, a^{4} c^{4} d^{8}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c}} + \frac {16 \, {\left (240 \, c^{14} d^{8} x^{7} + 840 \, b c^{13} d^{8} x^{6} + 1344 \, b^{2} c^{12} d^{8} x^{5} - 336 \, a c^{13} d^{8} x^{5} + 1260 \, b^{3} c^{11} d^{8} x^{4} - 840 \, a b c^{12} d^{8} x^{4} + 770 \, b^{4} c^{10} d^{8} x^{3} - 1120 \, a b^{2} c^{11} d^{8} x^{3} + 560 \, a^{2} c^{12} d^{8} x^{3} + 315 \, b^{5} c^{9} d^{8} x^{2} - 840 \, a b^{3} c^{10} d^{8} x^{2} + 840 \, a^{2} b c^{11} d^{8} x^{2} + 105 \, b^{6} c^{8} d^{8} x - 630 \, a b^{4} c^{9} d^{8} x + 1680 \, a^{2} b^{2} c^{10} d^{8} x - 1680 \, a^{3} c^{11} d^{8} x\right )}}{105 \, c^{7}} \]
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Time = 9.45 (sec) , antiderivative size = 761, normalized size of antiderivative = 6.24 \[ \int \frac {(b d+2 c d x)^8}{a+b x+c x^2} \, dx=x^3\,\left (\frac {1120\,b^4\,c^3\,d^8}{3}-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{3\,c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{3\,c}\right )-x^5\,\left (\frac {256\,a\,c^6\,d^8}{5}-\frac {1024\,b^2\,c^5\,d^8}{5}\right )-x^2\,\left (\frac {a\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{2\,c}+\frac {b\,\left (1120\,b^4\,c^3\,d^8-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}\right )}{2\,c}-224\,b^5\,c^2\,d^8\right )+x^4\,\left (448\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{4\,c}-192\,a\,b\,c^5\,d^8\right )+x\,\left (112\,b^6\,c\,d^8-\frac {a\,\left (1120\,b^4\,c^3\,d^8-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}\right )}{c}+\frac {b\,\left (\frac {a\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {b\,\left (1120\,b^4\,c^3\,d^8-\frac {b\,\left (1792\,b^3\,c^4\,d^8+\frac {b\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}-768\,a\,b\,c^5\,d^8\right )}{c}+\frac {a\,\left (256\,a\,c^6\,d^8-1024\,b^2\,c^5\,d^8\right )}{c}\right )}{c}-448\,b^5\,c^2\,d^8\right )}{c}\right )+2\,d^8\,\mathrm {atan}\left (\frac {b\,d^8\,{\left (4\,a\,c-b^2\right )}^{7/2}+2\,c\,d^8\,x\,{\left (4\,a\,c-b^2\right )}^{7/2}}{256\,a^4\,c^4\,d^8-256\,a^3\,b^2\,c^3\,d^8+96\,a^2\,b^4\,c^2\,d^8-16\,a\,b^6\,c\,d^8+b^8\,d^8}\right )\,{\left (4\,a\,c-b^2\right )}^{7/2}+\frac {256\,c^7\,d^8\,x^7}{7}+128\,b\,c^6\,d^8\,x^6 \]
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