Integrand size = 26, antiderivative size = 147 \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}+2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]
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Time = 0.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {706, 708, 335, 304, 209, 212} \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+\frac {4}{3} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2} \]
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 706
Rule 708
Rubi steps \begin{align*} \text {integral}& = \frac {4}{7} d (b d+2 c d x)^{7/2}+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^{5/2}}{a+b x+c x^2} \, dx \\ & = \frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac {\sqrt {b d+2 c d x}}{a+b x+c x^2} \, dx \\ & = \frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}+\frac {\left (\left (b^2-4 a c\right )^2 d^3\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}} \, dx,x,b d+2 c d x\right )}{2 c} \\ & = \frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}+\frac {\left (\left (b^2-4 a c\right )^2 d^3\right ) \text {Subst}\left (\int \frac {x^2}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )}{c} \\ & = \frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}-\left (2 \left (b^2-4 a c\right )^2 d^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d-x^2} \, dx,x,\sqrt {d (b+2 c x)}\right )+\left (2 \left (b^2-4 a c\right )^2 d^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c} d+x^2} \, dx,x,\sqrt {d (b+2 c x)}\right ) \\ & = \frac {4}{3} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}+\frac {4}{7} d (b d+2 c d x)^{7/2}+2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \left (b^2-4 a c\right )^{7/4} d^{9/2} \tanh ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.48 \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\frac {\left (\frac {1}{21}+\frac {i}{21}\right ) (d (b+2 c x))^{9/2} \left ((2-2 i) (b+2 c x)^{3/2} \left (7 b^2-28 a c+3 (b+2 c x)^2\right )-21 \left (b^2-4 a c\right )^{7/4} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )+21 \left (b^2-4 a c\right )^{7/4} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-21 \left (b^2-4 a c\right )^{7/4} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )\right )}{(b+2 c x)^{9/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(302\) vs. \(2(119)=238\).
Time = 3.16 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.06
method | result | size |
derivativedivides | \(4 d \left (-\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+\frac {d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\) | \(303\) |
default | \(4 d \left (-\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+\frac {d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}\right )\) | \(303\) |
pseudoelliptic | \(\frac {4 \left (-\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} d^{2} \left (4 a c -b^{2}\right ) \left (d \left (2 c x +b \right )\right )^{\frac {3}{2}}}{3}+\frac {\left (d \left (2 c x +b \right )\right )^{\frac {7}{2}} \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{7}+\frac {\sqrt {2}\, d^{4} \left (4 a c -b^{2}\right )^{2} \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )\right )}{8}\right ) d}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\) | \(329\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1005, normalized size of antiderivative = 6.84 \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\frac {8}{21} \, {\left (12 \, c^{3} d^{4} x^{3} + 18 \, b c^{2} d^{4} x^{2} + 4 \, {\left (4 \, b^{2} c - 7 \, a c^{2}\right )} d^{4} x + {\left (5 \, b^{3} - 14 \, a b c\right )} d^{4}\right )} \sqrt {2 \, c d x + b d} + \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {1}{4}} \log \left (-{\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} \sqrt {2 \, c d x + b d} d^{13} + \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {3}{4}}\right ) - i \, \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {1}{4}} \log \left (-{\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} \sqrt {2 \, c d x + b d} d^{13} + i \, \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {3}{4}}\right ) + i \, \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {1}{4}} \log \left (-{\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} \sqrt {2 \, c d x + b d} d^{13} - i \, \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {3}{4}}\right ) - \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {1}{4}} \log \left (-{\left (b^{10} - 20 \, a b^{8} c + 160 \, a^{2} b^{6} c^{2} - 640 \, a^{3} b^{4} c^{3} + 1280 \, a^{4} b^{2} c^{4} - 1024 \, a^{5} c^{5}\right )} \sqrt {2 \, c d x + b d} d^{13} - \left ({\left (b^{14} - 28 \, a b^{12} c + 336 \, a^{2} b^{10} c^{2} - 2240 \, a^{3} b^{8} c^{3} + 8960 \, a^{4} b^{6} c^{4} - 21504 \, a^{5} b^{4} c^{5} + 28672 \, a^{6} b^{2} c^{6} - 16384 \, a^{7} c^{7}\right )} d^{18}\right )^{\frac {3}{4}}\right ) \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (119) = 238\).
Time = 0.29 (sec) , antiderivative size = 531, normalized size of antiderivative = 3.61 \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\frac {4}{3} \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{3} - \frac {16}{3} \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c d^{3} + \frac {4}{7} \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} d - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) + \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {3}{4}} a c d^{3}\right )} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) \]
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Time = 9.41 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.14 \[ \int \frac {(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx=\frac {4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}}{7}-\frac {4\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (4\,a\,c-b^2\right )}{3}+2\,d^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{7/4}+d^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}\,{\left (b^2-4\,a\,c\right )}^{7/4}\,1{}\mathrm {i}}{\sqrt {d}\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,{\left (b^2-4\,a\,c\right )}^{7/4}\,2{}\mathrm {i} \]
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