Integrand size = 24, antiderivative size = 140 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {60 c^2}{\left (b^2-4 a c\right )^3 d^2 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^2}+\frac {5 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac {60 c^2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} d^2} \]
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Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {701, 707, 632, 212} \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=-\frac {60 c^2 \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{d^2 \left (b^2-4 a c\right )^{7/2}}+\frac {60 c^2}{d^2 \left (b^2-4 a c\right )^3 (b+2 c x)}+\frac {5 c}{d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac {1}{2 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^2} \]
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Rule 212
Rule 632
Rule 701
Rule 707
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^2}-\frac {(5 c) \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^2} \, dx}{b^2-4 a c} \\ & = -\frac {1}{2 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^2}+\frac {5 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \left (a+b x+c x^2\right )}+\frac {\left (30 c^2\right ) \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2} \\ & = \frac {60 c^2}{\left (b^2-4 a c\right )^3 d^2 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^2}+\frac {5 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \left (a+b x+c x^2\right )}+\frac {\left (30 c^2\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^3 d^2} \\ & = \frac {60 c^2}{\left (b^2-4 a c\right )^3 d^2 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^2}+\frac {5 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac {\left (60 c^2\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 d^2} \\ & = \frac {60 c^2}{\left (b^2-4 a c\right )^3 d^2 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^2}+\frac {5 c}{\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \left (a+b x+c x^2\right )}-\frac {60 c^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} d^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {64 c^2}{b+2 c x}-\frac {\left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac {14 c (b+2 c x)}{a+x (b+c x)}+\frac {120 c^2 \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{2 \left (b^2-4 a c\right )^3 d^2} \]
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Time = 2.37 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99
method | result | size |
default | \(\frac {-\frac {\frac {14 c^{3} x^{3}+21 b \,c^{2} x^{2}+6 c \left (3 a c +b^{2}\right ) x +\frac {b \left (18 a c -b^{2}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {60 c^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (4 a c -b^{2}\right )^{3}}-\frac {32 c^{2}}{\left (4 a c -b^{2}\right )^{3} \left (2 c x +b \right )}}{d^{2}}\) | \(139\) |
risch | \(\frac {-\frac {60 c^{4} x^{4}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {120 b \,c^{3} x^{3}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {5 c^{2} \left (20 a c +13 b^{2}\right ) x^{2}}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {5 c b \left (20 a c +b^{2}\right ) x}{64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}-\frac {64 a^{2} c^{2}+18 a \,b^{2} c -b^{4}}{2 \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}}{d^{2} \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{2}}+\frac {30 c^{2} \ln \left (-2 \left (-4 a c +b^{2}\right )^{\frac {7}{2}} c x -\left (-4 a c +b^{2}\right )^{\frac {7}{2}} b +256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}{d^{2} \left (-4 a c +b^{2}\right )^{\frac {7}{2}}}-\frac {30 c^{2} \ln \left (-2 \left (-4 a c +b^{2}\right )^{\frac {7}{2}} c x -\left (-4 a c +b^{2}\right )^{\frac {7}{2}} b -256 a^{4} c^{4}+256 a^{3} b^{2} c^{3}-96 a^{2} b^{4} c^{2}+16 a \,b^{6} c -b^{8}\right )}{d^{2} \left (-4 a c +b^{2}\right )^{\frac {7}{2}}}\) | \(444\) |
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Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (134) = 268\).
Time = 0.45 (sec) , antiderivative size = 1170, normalized size of antiderivative = 8.36 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 804 vs. \(2 (133) = 266\).
Time = 2.13 (sec) , antiderivative size = 804, normalized size of antiderivative = 5.74 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {30 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {- 7680 a^{4} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 7680 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 2880 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 480 a b^{6} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 30 b^{8} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b c^{2}}{60 c^{3}} \right )}}{d^{2}} - \frac {30 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} \log {\left (x + \frac {7680 a^{4} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 7680 a^{3} b^{2} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 2880 a^{2} b^{4} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} - 480 a b^{6} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b^{8} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{7}}} + 30 b c^{2}}{60 c^{3}} \right )}}{d^{2}} + \frac {- 64 a^{2} c^{2} - 18 a b^{2} c + b^{4} - 240 b c^{3} x^{3} - 120 c^{4} x^{4} + x^{2} \left (- 200 a c^{3} - 130 b^{2} c^{2}\right ) + x \left (- 200 a b c^{2} - 10 b^{3} c\right )}{128 a^{5} b c^{3} d^{2} - 96 a^{4} b^{3} c^{2} d^{2} + 24 a^{3} b^{5} c d^{2} - 2 a^{2} b^{7} d^{2} + x^{5} \cdot \left (256 a^{3} c^{6} d^{2} - 192 a^{2} b^{2} c^{5} d^{2} + 48 a b^{4} c^{4} d^{2} - 4 b^{6} c^{3} d^{2}\right ) + x^{4} \cdot \left (640 a^{3} b c^{5} d^{2} - 480 a^{2} b^{3} c^{4} d^{2} + 120 a b^{5} c^{3} d^{2} - 10 b^{7} c^{2} d^{2}\right ) + x^{3} \cdot \left (512 a^{4} c^{5} d^{2} + 128 a^{3} b^{2} c^{4} d^{2} - 288 a^{2} b^{4} c^{3} d^{2} + 88 a b^{6} c^{2} d^{2} - 8 b^{8} c d^{2}\right ) + x^{2} \cdot \left (768 a^{4} b c^{4} d^{2} - 448 a^{3} b^{3} c^{3} d^{2} + 48 a^{2} b^{5} c^{2} d^{2} + 12 a b^{7} c d^{2} - 2 b^{9} d^{2}\right ) + x \left (256 a^{5} c^{4} d^{2} + 64 a^{4} b^{2} c^{3} d^{2} - 144 a^{3} b^{4} c^{2} d^{2} + 44 a^{2} b^{6} c d^{2} - 4 a b^{8} d^{2}\right )} \]
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Exception generated. \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (134) = 268\).
Time = 0.27 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.16 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {32 \, c^{8} d^{11}}{{\left (b^{6} c^{6} d^{12} - 12 \, a b^{4} c^{7} d^{12} + 48 \, a^{2} b^{2} c^{8} d^{12} - 64 \, a^{3} c^{9} d^{12}\right )} {\left (2 \, c d x + b d\right )}} - \frac {60 \, c^{2} \arctan \left (-\frac {\frac {b^{2} d}{2 \, c d x + b d} - \frac {4 \, a c d}{2 \, c d x + b d}}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c} d^{2}} - \frac {4 \, {\left (\frac {9 \, b^{2} c^{2} d}{{\left (2 \, c d x + b d\right )}^{3}} - \frac {36 \, a c^{3} d}{{\left (2 \, c d x + b d\right )}^{3}} - \frac {7 \, c^{2}}{{\left (2 \, c d x + b d\right )} d}\right )}}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (\frac {b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {4 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )}^{2}} \]
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Time = 9.80 (sec) , antiderivative size = 494, normalized size of antiderivative = 3.53 \[ \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^3} \, dx=\frac {60\,c^2\,\mathrm {atan}\left (\frac {\frac {30\,c^2\,\left (-64\,a^3\,b\,c^3\,d^2+48\,a^2\,b^3\,c^2\,d^2-12\,a\,b^5\,c\,d^2+b^7\,d^2\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{7/2}}+\frac {60\,c^3\,x\,\left (-64\,a^3\,c^3\,d^2+48\,a^2\,b^2\,c^2\,d^2-12\,a\,b^4\,c\,d^2+b^6\,d^2\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{7/2}}}{30\,c^2}\right )}{d^2\,{\left (4\,a\,c-b^2\right )}^{7/2}}-\frac {\frac {64\,a^2\,c^2+18\,a\,b^2\,c-b^4}{2\,\left (4\,a\,c-b^2\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {60\,c^4\,x^4}{\left (4\,a\,c-b^2\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {5\,c\,x\,\left (b^3+20\,a\,c\,b\right )}{\left (4\,a\,c-b^2\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {5\,c\,x^2\,\left (13\,b^2\,c+20\,a\,c^2\right )}{\left (4\,a\,c-b^2\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {120\,b\,c^3\,x^3}{\left (4\,a\,c-b^2\right )\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^3\,\left (4\,b^2\,c\,d^2+4\,a\,c^2\,d^2\right )+x\,\left (2\,c\,a^2\,d^2+2\,a\,b^2\,d^2\right )+x^2\,\left (b^3\,d^2+6\,a\,c\,b\,d^2\right )+a^2\,b\,d^2+2\,c^3\,d^2\,x^5+5\,b\,c^2\,d^2\,x^4} \]
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