Integrand size = 26, antiderivative size = 150 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=20 c d^3 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}-10 c \sqrt [4]{b^2-4 a c} d^{7/2} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-10 c \sqrt [4]{b^2-4 a c} d^{7/2} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {700, 706, 708, 335, 218, 212, 209} \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=-10 c d^{7/2} \sqrt [4]{b^2-4 a c} \arctan \left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-10 c d^{7/2} \sqrt [4]{b^2-4 a c} \text {arctanh}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt {d} \sqrt [4]{b^2-4 a c}}\right )-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}+20 c d^3 \sqrt {b d+2 c d x} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 700
Rule 706
Rule 708
Rubi steps \begin{align*} \text {integral}& = -\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}+\left (5 c d^2\right ) \int \frac {(b d+2 c d x)^{3/2}}{a+b x+c x^2} \, dx \\ & = 20 c d^3 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}+\left (5 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {1}{\sqrt {b d+2 c d x} \left (a+b x+c x^2\right )} \, dx \\ & = 20 c d^3 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}+\frac {1}{2} \left (5 \left (b^2-4 a c\right ) 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 ) \\ & = 20 c d^3 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}+\left (5 \left (b^2-4 a c\right ) 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 ) \\ & = 20 c d^3 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}-\left (10 c \sqrt {b^2-4 a c} 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 (10 c \sqrt {b^2-4 a c} 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 ) \\ & = 20 c d^3 \sqrt {b d+2 c d x}-\frac {d (b d+2 c d x)^{5/2}}{a+b x+c x^2}-10 c \sqrt [4]{b^2-4 a c} d^{7/2} \tan ^{-1}\left (\frac {\sqrt {d (b+2 c x)}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right )-10 c \sqrt [4]{b^2-4 a c} 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 = 0.74 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.71 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=(1+i) c (d (b+2 c x))^{7/2} \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (-5 b^2+20 a c+4 (b+2 c x)^2\right )}{c (b+2 c x)^3 (a+x (b+c x))}-\frac {5 i \sqrt [4]{b^2-4 a c} \arctan \left (1-\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{7/2}}+\frac {5 i \sqrt [4]{b^2-4 a c} \arctan \left (1+\frac {(1+i) \sqrt {b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{7/2}}+\frac {5 i \sqrt [4]{b^2-4 a c} \text {arctanh}\left (\frac {(1+i) \sqrt [4]{b^2-4 a c} \sqrt {b+2 c x}}{\sqrt {b^2-4 a c}+i (b+2 c x)}\right )}{(b+2 c x)^{7/2}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(311\) vs. \(2(126)=252\).
Time = 2.60 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.08
method | result | size |
derivativedivides | \(16 c \,d^{3} \left (\sqrt {2 c d x +b d}-d^{2} \left (\frac {\left (-\frac {a c}{4}+\frac {b^{2}}{16}\right ) \sqrt {2 c d x +b d}}{a c \,d^{2}-\frac {b^{2} d^{2}}{4}+\frac {\left (2 c d x +b d \right )^{2}}{4}}+\frac {5 \left (-\frac {b^{2}}{4}+a c \right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\) | \(312\) |
default | \(16 c \,d^{3} \left (\sqrt {2 c d x +b d}-d^{2} \left (\frac {\left (-\frac {a c}{4}+\frac {b^{2}}{16}\right ) \sqrt {2 c d x +b d}}{a c \,d^{2}-\frac {b^{2} d^{2}}{4}+\frac {\left (2 c d x +b d \right )^{2}}{4}}+\frac {5 \left (-\frac {b^{2}}{4}+a c \right ) \sqrt {2}\, \left (\ln \left (\frac {2 c d x +b d +\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}{2 c d x +b d -\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}} \sqrt {2 c d x +b d}\, \sqrt {2}+\sqrt {4 a c \,d^{2}-b^{2} d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {2 c d x +b d}}{\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (4 a c \,d^{2}-b^{2} d^{2}\right )^{\frac {3}{4}}}\right )\right )\) | \(312\) |
pseudoelliptic | \(\frac {40 \left (\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (\frac {4 c^{2} x^{2}}{5}+\left (\frac {4 b x}{5}+a \right ) c -\frac {b^{2}}{20}\right ) \sqrt {d \left (2 c x +b \right )}-\frac {c \,d^{2} \sqrt {2}\, \left (c \,x^{2}+b x +a \right ) \left (-\frac {b^{2}}{4}+a c \right ) \left (2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )+\ln \left (\frac {\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+\sqrt {d^{2} \left (4 a c -b^{2}\right )}+d \left (2 c x +b \right )}{\sqrt {d^{2} \left (4 a c -b^{2}\right )}-\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}} \sqrt {d \left (2 c x +b \right )}\, \sqrt {2}+d \left (2 c x +b \right )}\right )-2 \arctan \left (\frac {-\sqrt {2}\, \sqrt {d \left (2 c x +b \right )}+\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {1}{4}}}\right )\right )}{2}\right ) d^{3}}{\left (d^{2} \left (4 a c -b^{2}\right )\right )^{\frac {3}{4}} \left (2 c \,x^{2}+2 b x +2 a \right )}\) | \(335\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.41 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {5 \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac {1}{4}} {\left (c x^{2} + b x + a\right )} \log \left (5 \, \sqrt {2 \, c d x + b d} c d^{3} + 5 \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac {1}{4}}\right ) - 5 \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac {1}{4}} {\left (-i \, c x^{2} - i \, b x - i \, a\right )} \log \left (5 \, \sqrt {2 \, c d x + b d} c d^{3} + 5 i \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac {1}{4}}\right ) - 5 \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac {1}{4}} {\left (i \, c x^{2} + i \, b x + i \, a\right )} \log \left (5 \, \sqrt {2 \, c d x + b d} c d^{3} - 5 i \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac {1}{4}}\right ) - 5 \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac {1}{4}} {\left (c x^{2} + b x + a\right )} \log \left (5 \, \sqrt {2 \, c d x + b d} c d^{3} - 5 \, \left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{14}\right )^{\frac {1}{4}}\right ) - {\left (16 \, c^{2} d^{3} x^{2} + 16 \, b c d^{3} x - {\left (b^{2} - 20 \, a c\right )} d^{3}\right )} \sqrt {2 \, c d x + b d}}{c x^{2} + b x + a} \]
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Timed out. \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (126) = 252\).
Time = 0.31 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.94 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=-5 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c d^{3} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} + 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - 5 \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c d^{3} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} - 2 \, \sqrt {2 \, c d x + b d}\right )}}{2 \, {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}}}\right ) - \frac {5}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c d^{3} \log \left (2 \, c d x + b d + \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac {5}{2} \, \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} c d^{3} \log \left (2 \, c d x + b d - \sqrt {2} {\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac {1}{4}} \sqrt {2 \, c d x + b d} + \sqrt {-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + 16 \, \sqrt {2 \, c d x + b d} c d^{3} + \frac {4 \, {\left (\sqrt {2 \, c d x + b d} b^{2} c d^{5} - 4 \, \sqrt {2 \, c d x + b d} a c^{2} d^{5}\right )}}{b^{2} d^{2} - 4 \, a c d^{2} - {\left (2 \, c d x + b d\right )}^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 597, normalized size of antiderivative = 3.98 \[ \int \frac {(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^2} \, dx=16\,c\,d^3\,\sqrt {b\,d+2\,c\,d\,x}+\frac {\sqrt {b\,d+2\,c\,d\,x}\,\left (16\,a\,c^2\,d^5-4\,b^2\,c\,d^5\right )}{{\left (b\,d+2\,c\,d\,x\right )}^2-b^2\,d^2+4\,a\,c\,d^2}-10\,c\,d^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,d+2\,c\,d\,x}}{\sqrt {d}\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}-c\,d^{7/2}\,\mathrm {atan}\left (\frac {c\,d^{7/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (25600\,a^2\,c^4\,d^{10}-12800\,a\,b^2\,c^3\,d^{10}+1600\,b^4\,c^2\,d^{10}\right )-5\,c\,d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (5120\,a^2\,c^3\,d^7-2560\,a\,b^2\,c^2\,d^7+320\,b^4\,c\,d^7\right )\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}\,5{}\mathrm {i}+c\,d^{7/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (25600\,a^2\,c^4\,d^{10}-12800\,a\,b^2\,c^3\,d^{10}+1600\,b^4\,c^2\,d^{10}\right )+5\,c\,d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (5120\,a^2\,c^3\,d^7-2560\,a\,b^2\,c^2\,d^7+320\,b^4\,c\,d^7\right )\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}\,5{}\mathrm {i}}{5\,c\,d^{7/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (25600\,a^2\,c^4\,d^{10}-12800\,a\,b^2\,c^3\,d^{10}+1600\,b^4\,c^2\,d^{10}\right )-5\,c\,d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (5120\,a^2\,c^3\,d^7-2560\,a\,b^2\,c^2\,d^7+320\,b^4\,c\,d^7\right )\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}-5\,c\,d^{7/2}\,\left (\sqrt {b\,d+2\,c\,d\,x}\,\left (25600\,a^2\,c^4\,d^{10}-12800\,a\,b^2\,c^3\,d^{10}+1600\,b^4\,c^2\,d^{10}\right )+5\,c\,d^{7/2}\,{\left (b^2-4\,a\,c\right )}^{1/4}\,\left (5120\,a^2\,c^3\,d^7-2560\,a\,b^2\,c^2\,d^7+320\,b^4\,c\,d^7\right )\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}}\right )\,{\left (b^2-4\,a\,c\right )}^{1/4}\,10{}\mathrm {i} \]
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