Integrand size = 26, antiderivative size = 145 \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=4 \left (b^2-4 a c\right ) d^3 \sqrt {b d+2 c d x}+\frac {4}{5} d (b d+2 c d x)^{5/2}-2 \left (b^2-4 a c\right )^{5/4} d^{7/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-2 \left (b^2-4 a c\right )^{5/4} d^{7/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 145, 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, 218, 212, 209} \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=-2 d^{7/2} \left (b^2-4 a c\right )^{5/4} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{7/2} \left (b^2-4 a c\right )^{5/4} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )+4 d^3 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}+\frac {4}{5} d (b d+2 c d x)^{5/2} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 706
Rule 708
Rubi steps \begin{align*} \text {integral}& = \frac {4}{5} d (b d+2 c d x)^{5/2}+\left (\left (b^2-4 a c\right ) d^2\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 ) d^3 \sqrt {b d+2 c d x}+\frac {4}{5} d (b d+2 c d x)^{5/2}+\left (\left (b^2-4 a c\right )^2 d^4\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 ) d^3 \sqrt {b d+2 c d x}+\frac {4}{5} d (b d+2 c d x)^{5/2}+\frac {\left (\left (b^2-4 a c\right )^2 d^3\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 ) d^3 \sqrt {b d+2 c d x}+\frac {4}{5} d (b d+2 c d x)^{5/2}+\frac {\left (\left (b^2-4 a c\right )^2 d^3\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 ) d^3 \sqrt {b d+2 c d x}+\frac {4}{5} d (b d+2 c d x)^{5/2}-\left (2 \left (b^2-4 a c\right )^{3/2} d^4\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 )^{3/2} d^4\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 ) d^3 \sqrt {b d+2 c d x}+\frac {4}{5} d (b d+2 c d x)^{5/2}-2 \left (b^2-4 a c\right )^{5/4} d^{7/2} \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 )^{5/4} d^{7/2} \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 = 0.32 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.48 \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=\frac {\left (\frac {1}{5}-\frac {i}{5}\right ) (d (b+2 c x))^{7/2} \left ((2+2 i) \sqrt {b+2 c x} \left (5 b^2-20 a c+(b+2 c x)^2\right )+5 \left (b^2-4 a c\right )^{5/4} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-5 \left (b^2-4 a c\right )^{5/4} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-5 \left (b^2-4 a c\right )^{5/4} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )\right )}{(b+2 c x)^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(301\) vs. \(2(119)=238\).
Time = 3.93 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.08
method | result | size |
derivativedivides | \(4 d \left (-4 a c \,d^{2} \sqrt {2 c d x +b d}+b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\) | \(302\) |
default | \(4 d \left (-4 a c \,d^{2} \sqrt {2 c d x +b d}+b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}+\frac {d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\) | \(302\) |
pseudoelliptic | \(-\frac {4 d \left (-\frac {\left (d \left (2 c x +b \right )\right )^{\frac {5}{2}} \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}}}{5}+\frac {d^{2} \left (4 a c -b^{2}\right ) \left (8 \left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \sqrt {d \left (2 c x +b \right )}+\sqrt {2}\, d^{2} \left (4 a c -b^{2}\right ) \left (2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )\right )\right )}{8}\right )}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}}}\) | \(331\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 629, normalized size of antiderivative = 4.34 \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=\frac {8}{5} \, {\left (2 \, c^{2} d^{3} x^{2} + 2 \, b c d^{3} x + {\left (3 \, b^{2} - 10 \, a c\right )} d^{3}\right )} \sqrt {2 \, c d x + b d} + \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 )} d^{14}\right )^{\frac {1}{4}} \log \left (-\sqrt {2 \, c d x + b d} {\left (b^{2} - 4 \, a c\right )} d^{3} + \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 )} d^{14}\right )^{\frac {1}{4}}\right ) + i \, \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 )} d^{14}\right )^{\frac {1}{4}} \log \left (-\sqrt {2 \, c d x + b d} {\left (b^{2} - 4 \, a c\right )} d^{3} + i \, \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 )} d^{14}\right )^{\frac {1}{4}}\right ) - i \, \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 )} d^{14}\right )^{\frac {1}{4}} \log \left (-\sqrt {2 \, c d x + b d} {\left (b^{2} - 4 \, a c\right )} d^{3} - i \, \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 )} d^{14}\right )^{\frac {1}{4}}\right ) - \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 )} d^{14}\right )^{\frac {1}{4}} \log \left (-\sqrt {2 \, c d x + b d} {\left (b^{2} - 4 \, a c\right )} d^{3} - \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 )} d^{14}\right )^{\frac {1}{4}}\right ) \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (119) = 238\).
Time = 0.30 (sec) , antiderivative size = 531, normalized size of antiderivative = 3.66 \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=4 \, \sqrt {2 \, c d x + b d} b^{2} d^{3} - 16 \, \sqrt {2 \, c d x + b d} a c d^{3} + \frac {4}{5} \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} d - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c d^{3}\right )} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {1}{2} \, {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} b^{2} d^{3} - 4 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} a c d^{3}\right )} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) \]
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Time = 9.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.15 \[ \int \frac {(b d+2 c d x)^{7/2}}{a+b x+c x^2} \, dx=\frac {4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}}{5}-4\,d^3\,\sqrt {b\,d+2\,c\,d\,x}\,\left (4\,a\,c-b^2\right )-2\,d^{7/2}\,\mathrm {atan}\left (\frac {b^2\,\sqrt {b\,d+2\,c\,d\,x}-4\,a\,c\,\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{5/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}+d^{7/2}\,\mathrm {atan}\left (\frac {b^2\,\sqrt {b\,d+2\,c\,d\,x}\,1{}\mathrm {i}-a\,c\,\sqrt {b\,d+2\,c\,d\,x}\,4{}\mathrm {i}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{5/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{5/4}\,2{}\mathrm {i} \]
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