Integrand size = 37, antiderivative size = 178 \[ \int \frac {(d+e x)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {7 e \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{c^4 d^4}+\frac {7 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^3 d^3}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}-\frac {7 e \left (c d^2-a e^2\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}} \]
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Time = 0.07 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {7 e \left (c d^2-a e^2\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{9/2} d^{9/2}}+\frac {7 e \sqrt {d+e x} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac {7 e (d+e x)^{3/2} \left (c d^2-a e^2\right )}{3 c^3 d^3}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2} \]
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Rule 43
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)^2} \, dx \\ & = -\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {(7 e) \int \frac {(d+e x)^{5/2}}{a e+c d x} \, dx}{2 c d} \\ & = \frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {\left (7 e \left (c d^2-a e^2\right )\right ) \int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx}{2 c^2 d^2} \\ & = \frac {7 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^3 d^3}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {\left (7 e \left (c d^2-a e^2\right )^2\right ) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{2 c^3 d^3} \\ & = \frac {7 e \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{c^4 d^4}+\frac {7 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^3 d^3}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {\left (7 e \left (c d^2-a e^2\right )^3\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 c^4 d^4} \\ & = \frac {7 e \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{c^4 d^4}+\frac {7 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^3 d^3}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}+\frac {\left (7 \left (c d^2-a e^2\right )^3\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} \\ & = \frac {7 e \left (c d^2-a e^2\right )^2 \sqrt {d+e x}}{c^4 d^4}+\frac {7 e \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{3 c^3 d^3}+\frac {7 e (d+e x)^{5/2}}{5 c^2 d^2}-\frac {(d+e x)^{7/2}}{c d (a e+c d x)}-\frac {7 e \left (c d^2-a e^2\right )^{5/2} \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 = 0.33 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\sqrt {d+e x} \left (105 a^3 e^6-35 a^2 c d e^4 (7 d-2 e x)+7 a c^2 d^2 e^2 \left (23 d^2-24 d e x-2 e^2 x^2\right )+c^3 d^3 \left (-15 d^3+116 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )}{15 c^4 d^4 (a e+c d x)}-\frac {7 e \left (-c d^2+a e^2\right )^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{c^{9/2} d^{9/2}} \]
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Time = 3.53 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.15
method | result | size |
pseudoelliptic | \(-\frac {7 \left (e \left (e^{2} a -c \,d^{2}\right )^{3} \left (c d x +a e \right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )-\sqrt {e x +d}\, \left (-\frac {\left (-\frac {2}{5} e^{3} x^{3}-\frac {32}{15} d \,e^{2} x^{2}-\frac {116}{15} d^{2} e x +d^{3}\right ) d^{3} c^{3}}{7}+\frac {23 \left (-\frac {2}{23} x^{2} e^{2}-\frac {24}{23} d e x +d^{2}\right ) e^{2} d^{2} a \,c^{2}}{15}-\frac {7 \left (-\frac {2 e x}{7}+d \right ) e^{4} d \,a^{2} c}{3}+e^{6} a^{3}\right ) \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\right )}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\, c^{4} d^{4} \left (c d x +a e \right )}\) | \(204\) |
risch | \(\frac {2 e \left (3 x^{2} c^{2} d^{2} e^{2}-10 x a c d \,e^{3}+16 x \,c^{2} d^{3} e +45 a^{2} e^{4}-100 a c \,d^{2} e^{2}+58 c^{2} d^{4}\right ) \sqrt {e x +d}}{15 d^{4} c^{4}}-\frac {\left (2 e^{6} a^{3}-6 d^{2} e^{4} a^{2} c +6 d^{4} e^{2} c^{2} a -2 c^{3} d^{6}\right ) e \left (-\frac {\sqrt {e x +d}}{2 \left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )}+\frac {7 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{4} d^{4}}\) | \(207\) |
derivativedivides | \(2 e \left (\frac {\frac {c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 a c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a^{2} e^{4} \sqrt {e x +d}-6 a c \,d^{2} e^{2} \sqrt {e x +d}+3 c^{2} d^{4} \sqrt {e x +d}}{c^{4} d^{4}}-\frac {\frac {\left (-\frac {1}{2} e^{6} a^{3}+\frac {3}{2} d^{2} e^{4} a^{2} c -\frac {3}{2} d^{4} e^{2} c^{2} a +\frac {1}{2} c^{3} d^{6}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {7 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{4} d^{4}}\right )\) | \(272\) |
default | \(2 e \left (\frac {\frac {c^{2} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 a c d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 c^{2} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a^{2} e^{4} \sqrt {e x +d}-6 a c \,d^{2} e^{2} \sqrt {e x +d}+3 c^{2} d^{4} \sqrt {e x +d}}{c^{4} d^{4}}-\frac {\frac {\left (-\frac {1}{2} e^{6} a^{3}+\frac {3}{2} d^{2} e^{4} a^{2} c -\frac {3}{2} d^{4} e^{2} c^{2} a +\frac {1}{2} c^{3} d^{6}\right ) \sqrt {e x +d}}{c d \left (e x +d \right )+e^{2} a -c \,d^{2}}+\frac {7 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{4} d^{4}}\right )\) | \(272\) |
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Time = 0.33 (sec) , antiderivative size = 586, normalized size of antiderivative = 3.29 \[ \int \frac {(d+e x)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\left [\frac {105 \, {\left (a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, {\left (6 \, c^{3} d^{3} e^{3} x^{3} - 15 \, c^{3} d^{6} + 161 \, a c^{2} d^{4} e^{2} - 245 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \, {\left (16 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (58 \, c^{3} d^{5} e - 84 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{30 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}, -\frac {105 \, {\left (a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} + {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (6 \, c^{3} d^{3} e^{3} x^{3} - 15 \, c^{3} d^{6} + 161 \, a c^{2} d^{4} e^{2} - 245 \, a^{2} c d^{2} e^{4} + 105 \, a^{3} e^{6} + 2 \, {\left (16 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (58 \, c^{3} d^{5} e - 84 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (c^{5} d^{5} x + a c^{4} d^{4} e\right )}}\right ] \]
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Timed out. \[ \int \frac {(d+e x)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+e x)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (152) = 304\).
Time = 0.28 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.75 \[ \int \frac {(d+e x)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {7 \, {\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{\sqrt {-c^{2} d^{3} + a c d e^{2}} c^{4} d^{4}} - \frac {\sqrt {e x + d} c^{3} d^{6} e - 3 \, \sqrt {e x + d} a c^{2} d^{4} e^{3} + 3 \, \sqrt {e x + d} a^{2} c d^{2} e^{5} - \sqrt {e x + d} a^{3} e^{7}}{{\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )} c^{4} d^{4}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{8} d^{8} e + 10 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{8} d^{9} e + 45 \, \sqrt {e x + d} c^{8} d^{10} e - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{7} d^{7} e^{3} - 90 \, \sqrt {e x + d} a c^{7} d^{8} e^{3} + 45 \, \sqrt {e x + d} a^{2} c^{6} d^{6} e^{5}\right )}}{15 \, c^{10} d^{10}} \]
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Time = 9.71 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.63 \[ \int \frac {(d+e x)^{11/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {\sqrt {d+e\,x}\,\left (a^3\,e^7-3\,a^2\,c\,d^2\,e^5+3\,a\,c^2\,d^4\,e^3-c^3\,d^6\,e\right )}{c^5\,d^5\,\left (d+e\,x\right )-c^5\,d^6+a\,c^4\,d^4\,e^2}-\left (\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2}{c^4\,d^4}-\frac {2\,e\,{\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}^2}{c^6\,d^6}\right )\,\sqrt {d+e\,x}+\frac {2\,e\,{\left (d+e\,x\right )}^{5/2}}{5\,c^2\,d^2}+\frac {2\,e\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,c^4\,d^4}-\frac {7\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e\,{\left (a\,e^2-c\,d^2\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^7-3\,a^2\,c\,d^2\,e^5+3\,a\,c^2\,d^4\,e^3-c^3\,d^6\,e}\right )\,{\left (a\,e^2-c\,d^2\right )}^{5/2}}{c^{9/2}\,d^{9/2}} \]
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