Integrand size = 37, antiderivative size = 186 \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 c^3 d^3}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}-\frac {2 c^{7/2} d^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}} \]
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Time = 0.10 (sec) , antiderivative size = 186, 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, 53, 65, 214} \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=-\frac {2 c^{7/2} d^{7/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}+\frac {2 c^3 d^3}{\sqrt {d+e x} \left (c d^2-a e^2\right )^4}+\frac {2 c^2 d^2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}+\frac {2 c d}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}+\frac {2}{7 (d+e x)^{7/2} \left (c d^2-a e^2\right )} \]
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Rule 53
Rule 65
Rule 214
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a e+c d x) (d+e x)^{9/2}} \, dx \\ & = \frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {(c d) \int \frac {1}{(a e+c d x) (d+e x)^{7/2}} \, dx}{c d^2-a e^2} \\ & = \frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {\left (c^2 d^2\right ) \int \frac {1}{(a e+c d x) (d+e x)^{5/2}} \, dx}{\left (c d^2-a e^2\right )^2} \\ & = \frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {\left (c^3 d^3\right ) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{\left (c d^2-a e^2\right )^3} \\ & = \frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 c^3 d^3}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}+\frac {\left (c^4 d^4\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{\left (c d^2-a e^2\right )^4} \\ & = \frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 c^3 d^3}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}+\frac {\left (2 c^4 d^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 )}{e \left (c d^2-a e^2\right )^4} \\ & = \frac {2}{7 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}+\frac {2 c d}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}+\frac {2 c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}+\frac {2 c^3 d^3}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}-\frac {2 c^{7/2} d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {-30 a^3 e^6+6 a^2 c d e^4 (22 d+7 e x)-2 a c^2 d^2 e^2 \left (122 d^2+112 d e x+35 e^2 x^2\right )+2 c^3 d^3 \left (176 d^3+406 d^2 e x+350 d e^2 x^2+105 e^3 x^3\right )}{105 \left (c d^2-a e^2\right )^4 (d+e x)^{7/2}}+\frac {2 c^{7/2} d^{7/2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{\left (-c d^2+a e^2\right )^{9/2}} \]
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Time = 3.04 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(-\frac {2}{7 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 c^{2} d^{2}}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c^{3} d^{3}}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {e x +d}}+\frac {2 c d}{5 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 c^{4} d^{4} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(175\) |
default | \(-\frac {2}{7 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}-\frac {2 c^{2} d^{2}}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {2 c^{3} d^{3}}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {e x +d}}+\frac {2 c d}{5 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}+\frac {2 c^{4} d^{4} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(175\) |
pseudoelliptic | \(-\frac {2 \left (-7 c^{4} d^{4} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right ) \left (e x +d \right )^{\frac {7}{2}}+\left (\left (-7 x^{3} d^{3} e^{3}-\frac {70}{3} d^{4} e^{2} x^{2}-\frac {406}{15} d^{5} e x -\frac {176}{15} d^{6}\right ) c^{3}+\frac {122 e^{2} d^{2} a \left (\frac {35}{122} x^{2} e^{2}+\frac {56}{61} d e x +d^{2}\right ) c^{2}}{15}-\frac {22 e^{4} \left (\frac {7 e x}{22}+d \right ) d \,a^{2} c}{5}+e^{6} a^{3}\right ) \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\right )}{7 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\, \left (e^{2} a -c \,d^{2}\right )^{4}}\) | \(195\) |
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Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (158) = 316\).
Time = 0.39 (sec) , antiderivative size = 1157, normalized size of antiderivative = 6.22 \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\left [\frac {105 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {\frac {c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) + 2 \, {\left (105 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 122 \, a c^{2} d^{4} e^{2} + 66 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6} + 35 \, {\left (10 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \, {\left (58 \, c^{3} d^{5} e - 16 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (c^{4} d^{12} - 4 \, a c^{3} d^{10} e^{2} + 6 \, a^{2} c^{2} d^{8} e^{4} - 4 \, a^{3} c d^{6} e^{6} + a^{4} d^{4} e^{8} + {\left (c^{4} d^{8} e^{4} - 4 \, a c^{3} d^{6} e^{6} + 6 \, a^{2} c^{2} d^{4} e^{8} - 4 \, a^{3} c d^{2} e^{10} + a^{4} e^{12}\right )} x^{4} + 4 \, {\left (c^{4} d^{9} e^{3} - 4 \, a c^{3} d^{7} e^{5} + 6 \, a^{2} c^{2} d^{5} e^{7} - 4 \, a^{3} c d^{3} e^{9} + a^{4} d e^{11}\right )} x^{3} + 6 \, {\left (c^{4} d^{10} e^{2} - 4 \, a c^{3} d^{8} e^{4} + 6 \, a^{2} c^{2} d^{6} e^{6} - 4 \, a^{3} c d^{4} e^{8} + a^{4} d^{2} e^{10}\right )} x^{2} + 4 \, {\left (c^{4} d^{11} e - 4 \, a c^{3} d^{9} e^{3} + 6 \, a^{2} c^{2} d^{7} e^{5} - 4 \, a^{3} c d^{5} e^{7} + a^{4} d^{3} e^{9}\right )} x\right )}}, -\frac {2 \, {\left (105 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac {{\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}}}{c d e x + c d^{2}}\right ) - {\left (105 \, c^{3} d^{3} e^{3} x^{3} + 176 \, c^{3} d^{6} - 122 \, a c^{2} d^{4} e^{2} + 66 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6} + 35 \, {\left (10 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 7 \, {\left (58 \, c^{3} d^{5} e - 16 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}\right )}}{105 \, {\left (c^{4} d^{12} - 4 \, a c^{3} d^{10} e^{2} + 6 \, a^{2} c^{2} d^{8} e^{4} - 4 \, a^{3} c d^{6} e^{6} + a^{4} d^{4} e^{8} + {\left (c^{4} d^{8} e^{4} - 4 \, a c^{3} d^{6} e^{6} + 6 \, a^{2} c^{2} d^{4} e^{8} - 4 \, a^{3} c d^{2} e^{10} + a^{4} e^{12}\right )} x^{4} + 4 \, {\left (c^{4} d^{9} e^{3} - 4 \, a c^{3} d^{7} e^{5} + 6 \, a^{2} c^{2} d^{5} e^{7} - 4 \, a^{3} c d^{3} e^{9} + a^{4} d e^{11}\right )} x^{3} + 6 \, {\left (c^{4} d^{10} e^{2} - 4 \, a c^{3} d^{8} e^{4} + 6 \, a^{2} c^{2} d^{6} e^{6} - 4 \, a^{3} c d^{4} e^{8} + a^{4} d^{2} e^{10}\right )} x^{2} + 4 \, {\left (c^{4} d^{11} e - 4 \, a c^{3} d^{9} e^{3} + 6 \, a^{2} c^{2} d^{7} e^{5} - 4 \, a^{3} c d^{5} e^{7} + a^{4} d^{3} e^{9}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.67 \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {2 \, c^{4} d^{4} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} + \frac {2 \, {\left (105 \, {\left (e x + d\right )}^{3} c^{3} d^{3} + 35 \, {\left (e x + d\right )}^{2} c^{3} d^{4} + 21 \, {\left (e x + d\right )} c^{3} d^{5} + 15 \, c^{3} d^{6} - 35 \, {\left (e x + d\right )}^{2} a c^{2} d^{2} e^{2} - 42 \, {\left (e x + d\right )} a c^{2} d^{3} e^{2} - 45 \, a c^{2} d^{4} e^{2} + 21 \, {\left (e x + d\right )} a^{2} c d e^{4} + 45 \, a^{2} c d^{2} e^{4} - 15 \, a^{3} e^{6}\right )}}{105 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left (e x + d\right )}^{\frac {7}{2}}} \]
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Time = 10.13 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx=\frac {2\,c^{7/2}\,d^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{{\left (a\,e^2-c\,d^2\right )}^{9/2}}\right )}{{\left (a\,e^2-c\,d^2\right )}^{9/2}}-\frac {\frac {2}{7\,\left (a\,e^2-c\,d^2\right )}+\frac {2\,c^2\,d^2\,{\left (d+e\,x\right )}^2}{3\,{\left (a\,e^2-c\,d^2\right )}^3}-\frac {2\,c^3\,d^3\,{\left (d+e\,x\right )}^3}{{\left (a\,e^2-c\,d^2\right )}^4}-\frac {2\,c\,d\,\left (d+e\,x\right )}{5\,{\left (a\,e^2-c\,d^2\right )}^2}}{{\left (d+e\,x\right )}^{7/2}} \]
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