Integrand size = 26, antiderivative size = 175 \[ \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx=4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}-2 \left (b^2-4 a c\right )^{9/4} d^{11/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \left (b^2-4 a c\right )^{9/4} d^{11/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]
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Time = 0.15 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {706, 708, 335, 218, 212, 209} \[ \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx=-2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+4 d^5 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}+\frac {4}{5} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 706
Rule 708
Rubi steps \begin{align*} \text {integral}& = \frac {4}{9} d (b d+2 c d x)^{9/2}+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx \\ & = \frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx \\ & = 4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\left (\left (b^2-4 a c\right )^3 d^6\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx \\ & = 4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\frac {\left (\left (b^2-4 a c\right )^3 d^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (a-\frac {b^2}{4 c}+\frac {x^2}{4 c d^2}\right )} \, dx,x,b d+2 c d x\right )}{2 c} \\ & = 4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}+\frac {\left (\left (b^2-4 a c\right )^3 d^5\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b^2}{4 c}+\frac {x^4}{4 c d^2}} \, dx,x,\sqrt {d (b+2 c x)}\right )}{c} \\ & = 4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}-\left (2 \left (b^2-4 a c\right )^{5/2} d^6\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 )^{5/2} d^6\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 ) \\ & = 4 \left (b^2-4 a c\right )^2 d^5 \sqrt {b d+2 c d x}+\frac {4}{5} \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{5/2}+\frac {4}{9} d (b d+2 c d x)^{9/2}-2 \left (b^2-4 a c\right )^{9/4} d^{11/2} \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 )^{9/4} d^{11/2} \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 = 0.39 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.45 \[ \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx=\frac {\left (\frac {1}{45}-\frac {i}{45}\right ) d (d (b+2 c x))^{9/2} \left ((2+2 i) \sqrt {b+2 c x} \left (45 b^4-360 a b^2 c+720 a^2 c^2+9 b^2 (b+2 c x)^2-36 a c (b+2 c x)^2+5 (b+2 c x)^4\right )+45 \left (b^2-4 a c\right )^{9/4} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-45 \left (b^2-4 a c\right )^{9/4} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-45 \left (b^2-4 a c\right )^{9/4} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )\right )}{(b+2 c x)^{9/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(145)=290\).
Time = 5.76 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.13
method | result | size |
pseudoelliptic | \(-\frac {4 d \left (\frac {d^{2} \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (4 a c -b^{2}\right ) \left (d \left (2 c x +b \right )\right )^{\frac {5}{2}}}{5}-\frac {\left (d \left (2 c x +b \right )\right )^{\frac {9}{2}} \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}}}{9}+\frac {d^{4} \left (4 a c -b^{2}\right )^{2} \left (-8 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \sqrt {d \left (2 c x +b \right )}+\sqrt {2}\, d^{2} \left (4 a c -b^{2}\right ) \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )\right )\right )}{8}\right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}}}\) | \(372\) |
derivativedivides | \(4 d \left (16 a^{2} c^{2} d^{4} \sqrt {2 c d x +b d}-8 a \,b^{2} c \,d^{4} \sqrt {2 c d x +b d}-\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+b^{4} d^{4} \sqrt {2 c d x +b d}+\frac {b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {\left (2 c d x +b d \right )^{\frac {9}{2}}}{9}-\frac {d^{6} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\) | \(377\) |
default | \(4 d \left (16 a^{2} c^{2} d^{4} \sqrt {2 c d x +b d}-8 a \,b^{2} c \,d^{4} \sqrt {2 c d x +b d}-\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+b^{4} d^{4} \sqrt {2 c d x +b d}+\frac {b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {\left (2 c d x +b d \right )^{\frac {9}{2}}}{9}-\frac {d^{6} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\) | \(377\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 1079, normalized size of antiderivative = 6.17 \[ \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (145) = 290\).
Time = 0.30 (sec) , antiderivative size = 733, normalized size of antiderivative = 4.19 \[ \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx=4 \, \sqrt {2 \, c d x + b d} b^{4} d^{5} - 32 \, \sqrt {2 \, c d x + b d} a b^{2} c d^{5} + 64 \, \sqrt {2 \, c d x + b d} a^{2} c^{2} d^{5} + \frac {4}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} d^{3} - \frac {16}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} a c d^{3} + \frac {4}{9} \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} d - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\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 {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\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 {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\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 {1}{4}} b^{4} d^{5} - 8 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a b^{2} c d^{5} + 16 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a^{2} c^{2} d^{5}\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 = 9.36 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.37 \[ \int \frac {(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx=\frac {4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{9/2}}{9}-\frac {4\,d^3\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (4\,a\,c-b^2\right )}{5}+4\,d^5\,\sqrt {b\,d+2\,c\,d\,x}\,{\left (4\,a\,c-b^2\right )}^2-2\,d^{11/2}\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}+16\,a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}-8\,a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}+d^{11/2}\,\mathrm {atan}\left (\frac {b^4\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}+a^2\,c^2\,\sqrt {b\,d+2\,c\,d\,x}\,16{}\mathrm {i}-a\,b^2\,c\,\sqrt {b\,d+2\,c\,d\,x}\,8{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{9/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{9/4}\,2{}\mathrm {i} \]
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