Integrand size = 37, antiderivative size = 244 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {63 e^2}{20 \left (c d^2-a e^2\right )^3 (d+e x)^{5/2}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{5/2}}+\frac {9 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{5/2}}+\frac {21 c d e^2}{4 \left (c d^2-a e^2\right )^4 (d+e x)^{3/2}}+\frac {63 c^2 d^2 e^2}{4 \left (c d^2-a e^2\right )^5 \sqrt {d+e x}}-\frac {63 c^{5/2} d^{5/2} e^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{11/2}} \]
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Time = 0.12 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {640, 44, 53, 65, 214} \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {63 c^{5/2} d^{5/2} e^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{11/2}}+\frac {63 c^2 d^2 e^2}{4 \sqrt {d+e x} \left (c d^2-a e^2\right )^5}+\frac {21 c d e^2}{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )^4}+\frac {9 e}{4 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac {1}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac {63 e^2}{20 (d+e x)^{5/2} \left (c d^2-a e^2\right )^3} \]
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Rule 44
Rule 53
Rule 65
Rule 214
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a e+c d x)^3 (d+e x)^{7/2}} \, dx \\ & = -\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{5/2}}-\frac {(9 e) \int \frac {1}{(a e+c d x)^2 (d+e x)^{7/2}} \, dx}{4 \left (c d^2-a e^2\right )} \\ & = -\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{5/2}}+\frac {9 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{5/2}}+\frac {\left (63 e^2\right ) \int \frac {1}{(a e+c d x) (d+e x)^{7/2}} \, dx}{8 \left (c d^2-a e^2\right )^2} \\ & = \frac {63 e^2}{20 \left (c d^2-a e^2\right )^3 (d+e x)^{5/2}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{5/2}}+\frac {9 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{5/2}}+\frac {\left (63 c d e^2\right ) \int \frac {1}{(a e+c d x) (d+e x)^{5/2}} \, dx}{8 \left (c d^2-a e^2\right )^3} \\ & = \frac {63 e^2}{20 \left (c d^2-a e^2\right )^3 (d+e x)^{5/2}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{5/2}}+\frac {9 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{5/2}}+\frac {21 c d e^2}{4 \left (c d^2-a e^2\right )^4 (d+e x)^{3/2}}+\frac {\left (63 c^2 d^2 e^2\right ) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{8 \left (c d^2-a e^2\right )^4} \\ & = \frac {63 e^2}{20 \left (c d^2-a e^2\right )^3 (d+e x)^{5/2}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{5/2}}+\frac {9 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{5/2}}+\frac {21 c d e^2}{4 \left (c d^2-a e^2\right )^4 (d+e x)^{3/2}}+\frac {63 c^2 d^2 e^2}{4 \left (c d^2-a e^2\right )^5 \sqrt {d+e x}}+\frac {\left (63 c^3 d^3 e^2\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{8 \left (c d^2-a e^2\right )^5} \\ & = \frac {63 e^2}{20 \left (c d^2-a e^2\right )^3 (d+e x)^{5/2}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{5/2}}+\frac {9 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{5/2}}+\frac {21 c d e^2}{4 \left (c d^2-a e^2\right )^4 (d+e x)^{3/2}}+\frac {63 c^2 d^2 e^2}{4 \left (c d^2-a e^2\right )^5 \sqrt {d+e x}}+\frac {\left (63 c^3 d^3 e\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 )}{4 \left (c d^2-a e^2\right )^5} \\ & = \frac {63 e^2}{20 \left (c d^2-a e^2\right )^3 (d+e x)^{5/2}}-\frac {1}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2 (d+e x)^{5/2}}+\frac {9 e}{4 \left (c d^2-a e^2\right )^2 (a e+c d x) (d+e x)^{5/2}}+\frac {21 c d e^2}{4 \left (c d^2-a e^2\right )^4 (d+e x)^{3/2}}+\frac {63 c^2 d^2 e^2}{4 \left (c d^2-a e^2\right )^5 \sqrt {d+e x}}-\frac {63 c^{5/2} d^{5/2} e^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{11/2}} \\ \end{align*}
Time = 0.86 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {8 a^4 e^8-8 a^3 c d e^6 (7 d+3 e x)+24 a^2 c^2 d^2 e^4 \left (12 d^2+17 d e x+7 e^2 x^2\right )+a c^3 d^3 e^2 \left (85 d^3+831 d^2 e x+1239 d e^2 x^2+525 e^3 x^3\right )+c^4 d^4 \left (-10 d^4+45 d^3 e x+483 d^2 e^2 x^2+735 d e^3 x^3+315 e^4 x^4\right )}{20 \left (c d^2-a e^2\right )^5 (a e+c d x)^2 (d+e x)^{5/2}}-\frac {63 c^{5/2} d^{5/2} e^2 \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{4 \left (-c d^2+a e^2\right )^{11/2}} \]
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Time = 2.96 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(2 e^{2} \left (-\frac {1}{5 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}-\frac {6 c^{2} d^{2}}{\left (e^{2} a -c \,d^{2}\right )^{5} \sqrt {e x +d}}+\frac {c d}{\left (e^{2} a -c \,d^{2}\right )^{4} \left (e x +d \right )^{\frac {3}{2}}}-\frac {c^{3} d^{3} \left (\frac {\frac {15 c d \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {17 e^{2} a}{8}-\frac {17 c \,d^{2}}{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {63 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{5}}\right )\) | \(209\) |
default | \(2 e^{2} \left (-\frac {1}{5 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {5}{2}}}-\frac {6 c^{2} d^{2}}{\left (e^{2} a -c \,d^{2}\right )^{5} \sqrt {e x +d}}+\frac {c d}{\left (e^{2} a -c \,d^{2}\right )^{4} \left (e x +d \right )^{\frac {3}{2}}}-\frac {c^{3} d^{3} \left (\frac {\frac {15 c d \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {17 e^{2} a}{8}-\frac {17 c \,d^{2}}{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {63 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{5}}\right )\) | \(209\) |
pseudoelliptic | \(-\frac {2 \left (\frac {315 c^{3} d^{3} e^{2} \left (e x +d \right )^{\frac {5}{2}} \left (c d x +a e \right )^{2} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8}+\left (-\frac {5 \left (-\frac {63}{2} e^{4} x^{4}-\frac {147}{2} d \,e^{3} x^{3}-\frac {483}{10} d^{2} e^{2} x^{2}-\frac {9}{2} d^{3} e x +d^{4}\right ) d^{4} c^{4}}{4}+\frac {85 \left (\frac {105}{17} e^{3} x^{3}+\frac {1239}{85} d \,e^{2} x^{2}+\frac {831}{85} d^{2} e x +d^{3}\right ) e^{2} d^{3} a \,c^{3}}{8}+36 \left (\frac {7}{12} x^{2} e^{2}+\frac {17}{12} d e x +d^{2}\right ) e^{4} d^{2} a^{2} c^{2}-7 e^{6} d \,a^{3} \left (\frac {3 e x}{7}+d \right ) c +a^{4} e^{8}\right ) \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\right )}{5 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\, \left (e^{2} a -c \,d^{2}\right )^{5} \left (c d x +a e \right )^{2}}\) | \(268\) |
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Leaf count of result is larger than twice the leaf count of optimal. 996 vs. \(2 (208) = 416\).
Time = 1.03 (sec) , antiderivative size = 2007, normalized size of antiderivative = 8.23 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (208) = 416\).
Time = 0.29 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.77 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {63 \, c^{3} d^{3} e^{2} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{4 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} + \frac {15 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{4} d^{4} e^{2} - 17 \, \sqrt {e x + d} c^{4} d^{5} e^{2} + 17 \, \sqrt {e x + d} a c^{3} d^{3} e^{4}}{4 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} {\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}^{2}} + \frac {2 \, {\left (30 \, {\left (e x + d\right )}^{2} c^{2} d^{2} e^{2} + 5 \, {\left (e x + d\right )} c^{2} d^{3} e^{2} + c^{2} d^{4} e^{2} - 5 \, {\left (e x + d\right )} a c d e^{4} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )}}{5 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} {\left (e x + d\right )}^{\frac {5}{2}}} \]
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Time = 10.31 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {\frac {2\,e^2}{5\,\left (a\,e^2-c\,d^2\right )}-\frac {6\,c\,d\,e^2\,\left (d+e\,x\right )}{5\,{\left (a\,e^2-c\,d^2\right )}^2}+\frac {42\,c^2\,d^2\,e^2\,{\left (d+e\,x\right )}^2}{5\,{\left (a\,e^2-c\,d^2\right )}^3}+\frac {105\,c^3\,d^3\,e^2\,{\left (d+e\,x\right )}^3}{4\,{\left (a\,e^2-c\,d^2\right )}^4}+\frac {63\,c^4\,d^4\,e^2\,{\left (d+e\,x\right )}^4}{4\,{\left (a\,e^2-c\,d^2\right )}^5}}{{\left (d+e\,x\right )}^{5/2}\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )-\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{7/2}+c^2\,d^2\,{\left (d+e\,x\right )}^{9/2}}-\frac {63\,c^{5/2}\,d^{5/2}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\left (a^5\,e^{10}-5\,a^4\,c\,d^2\,e^8+10\,a^3\,c^2\,d^4\,e^6-10\,a^2\,c^3\,d^6\,e^4+5\,a\,c^4\,d^8\,e^2-c^5\,d^{10}\right )}{{\left (a\,e^2-c\,d^2\right )}^{11/2}}\right )}{4\,{\left (a\,e^2-c\,d^2\right )}^{11/2}} \]
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