Integrand size = 37, antiderivative size = 180 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^4 d^4}+\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3}+\frac {2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac {2 (d+e x)^{7/2}}{7 c d}-\frac {2 \left (c d^2-a e^2\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}} \]
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Time = 0.14 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {640, 52, 65, 214} \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {2 \left (c d^2-a e^2\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac {2 \sqrt {d+e x} \left (c d^2-a e^2\right )^3}{c^4 d^4}+\frac {2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 c^3 d^3}+\frac {2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^2 d^2}+\frac {2 (d+e x)^{7/2}}{7 c d} \]
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Rule 52
Rule 65
Rule 214
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^{7/2}}{a e+c d x} \, dx \\ & = \frac {2 (d+e x)^{7/2}}{7 c d}+\frac {\left (c d^2-a e^2\right ) \int \frac {(d+e x)^{5/2}}{a e+c d x} \, dx}{c d} \\ & = \frac {2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac {2 (d+e x)^{7/2}}{7 c d}+\frac {\left (c d^2-a e^2\right )^2 \int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx}{c^2 d^2} \\ & = \frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3}+\frac {2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac {2 (d+e x)^{7/2}}{7 c d}+\frac {\left (c d^2-a e^2\right )^3 \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{c^3 d^3} \\ & = \frac {2 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^4 d^4}+\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3}+\frac {2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac {2 (d+e x)^{7/2}}{7 c d}+\frac {\left (c d^2-a e^2\right )^4 \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{c^4 d^4} \\ & = \frac {2 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^4 d^4}+\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3}+\frac {2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac {2 (d+e x)^{7/2}}{7 c d}+\frac {\left (2 \left (c d^2-a e^2\right )^4\right ) \text {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{c^4 d^4 e} \\ & = \frac {2 \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^4 d^4}+\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 c^3 d^3}+\frac {2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^2 d^2}+\frac {2 (d+e x)^{7/2}}{7 c d}-\frac {2 \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \sqrt {d+e x} \left (-105 a^3 e^6+35 a^2 c d e^4 (10 d+e x)-7 a c^2 d^2 e^2 \left (58 d^2+16 d e x+3 e^2 x^2\right )+c^3 d^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )}{105 c^4 d^4}+\frac {2 \left (-c d^2+a e^2\right )^{7/2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{9/2} d^{9/2}} \]
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Time = 3.97 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.03
method | result | size |
pseudoelliptic | \(\frac {2 \left (e^{2} a -c \,d^{2}\right )^{4} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )-2 \left (-\frac {176 \left (\frac {15}{176} e^{3} x^{3}+\frac {3}{8} d \,e^{2} x^{2}+\frac {61}{88} d^{2} e x +d^{3}\right ) d^{3} c^{3}}{105}+\frac {58 \left (\frac {3}{58} x^{2} e^{2}+\frac {8}{29} d e x +d^{2}\right ) e^{2} d^{2} a \,c^{2}}{15}-\frac {10 \left (\frac {e x}{10}+d \right ) e^{4} d \,a^{2} c}{3}+e^{6} a^{3}\right ) \sqrt {e x +d}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}{c^{4} d^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(185\) |
derivativedivides | \(-\frac {2 \left (-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {\left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}} c d}{3}+\left (e^{2} a -c \,d^{2}\right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {e x +d}\right )}{c^{4} d^{4}}+\frac {2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{4} d^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(240\) |
default | \(-\frac {2 \left (-\frac {c^{3} d^{3} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {\left (e^{2} a -c \,d^{2}\right ) c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {\left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{\frac {3}{2}} c d}{3}+\left (e^{2} a -c \,d^{2}\right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {e x +d}\right )}{c^{4} d^{4}}+\frac {2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{4} d^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(240\) |
risch | \(-\frac {2 \left (-15 x^{3} c^{3} d^{3} e^{3}+21 x^{2} a \,c^{2} d^{2} e^{4}-66 x^{2} c^{3} d^{4} e^{2}-35 x \,a^{2} c d \,e^{5}+112 x a \,c^{2} d^{3} e^{3}-122 x \,c^{3} d^{5} e +105 e^{6} a^{3}-350 d^{2} e^{4} a^{2} c +406 d^{4} e^{2} c^{2} a -176 c^{3} d^{6}\right ) \sqrt {e x +d}}{105 c^{4} d^{4}}+\frac {2 \left (a^{4} e^{8}-4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}-4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{4} d^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(241\) |
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Time = 0.31 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.83 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\left [\frac {105 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, {\left (15 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 406 \, a c^{2} d^{4} e^{2} + 350 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \, {\left (22 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (122 \, c^{3} d^{5} e - 112 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{105 \, c^{4} d^{4}}, -\frac {2 \, {\left (105 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (15 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 406 \, a c^{2} d^{4} e^{2} + 350 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \, {\left (22 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + {\left (122 \, c^{3} d^{5} e - 112 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}\right )}}{105 \, c^{4} d^{4}}\right ] \]
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Timed out. \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.29 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.66 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{4} d^{4}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{6} d^{6} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{6} d^{7} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{6} d^{8} + 105 \, \sqrt {e x + d} c^{6} d^{9} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} a c^{5} d^{5} e^{2} - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{5} d^{6} e^{2} - 315 \, \sqrt {e x + d} a c^{5} d^{7} e^{2} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} c^{4} d^{4} e^{4} + 315 \, \sqrt {e x + d} a^{2} c^{4} d^{5} e^{4} - 105 \, \sqrt {e x + d} a^{3} c^{3} d^{3} e^{6}\right )}}{105 \, c^{7} d^{7}} \]
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Time = 0.07 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.15 \[ \int \frac {(d+e x)^{9/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2\,{\left (d+e\,x\right )}^{7/2}}{7\,c\,d}+\frac {2\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,c^3\,d^3}-\frac {2\,{\left (a\,e^2-c\,d^2\right )}^3\,\sqrt {d+e\,x}}{c^4\,d^4}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,{\left (a\,e^2-c\,d^2\right )}^{7/2}\,\sqrt {d+e\,x}}{a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8}\right )\,{\left (a\,e^2-c\,d^2\right )}^{7/2}}{c^{9/2}\,d^{9/2}}-\frac {2\,\left (a\,e^2-c\,d^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,c^2\,d^2} \]
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