Integrand size = 37, antiderivative size = 192 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {7 c d e}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {7 c^2 d^2 e}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}+\frac {7 c^{5/2} d^{5/2} e \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}} \]
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Time = 0.07 (sec) , antiderivative size = 192, 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, 44, 53, 65, 214} \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {7 c^{5/2} d^{5/2} e \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{9/2}}-\frac {7 c^2 d^2 e}{\sqrt {d+e x} \left (c d^2-a e^2\right )^4}-\frac {7 c d e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}-\frac {1}{(d+e x)^{5/2} \left (c d^2-a e^2\right ) (a e+c d x)}-\frac {7 e}{5 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2} \]
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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)^2 (d+e x)^{7/2}} \, dx \\ & = -\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {(7 e) \int \frac {1}{(a e+c d x) (d+e x)^{7/2}} \, dx}{2 \left (c d^2-a e^2\right )} \\ & = -\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {(7 c d e) \int \frac {1}{(a e+c d x) (d+e x)^{5/2}} \, dx}{2 \left (c d^2-a e^2\right )^2} \\ & = -\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {7 c d e}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {\left (7 c^2 d^2 e\right ) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{2 \left (c d^2-a e^2\right )^3} \\ & = -\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {7 c d e}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {7 c^2 d^2 e}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}-\frac {\left (7 c^3 d^3 e\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 \left (c d^2-a e^2\right )^4} \\ & = -\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {7 c d e}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {7 c^2 d^2 e}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}-\frac {\left (7 c^3 d^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 )}{\left (c d^2-a e^2\right )^4} \\ & = -\frac {7 e}{5 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}-\frac {1}{\left (c d^2-a e^2\right ) (a e+c d x) (d+e x)^{5/2}}-\frac {7 c d e}{3 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}-\frac {7 c^2 d^2 e}{\left (c d^2-a e^2\right )^4 \sqrt {d+e x}}+\frac {7 c^{5/2} d^{5/2} e \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 = 0.40 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\frac {-6 a^3 e^6+2 a^2 c d e^4 (16 d+7 e x)-2 a c^2 d^2 e^2 \left (58 d^2+84 d e x+35 e^2 x^2\right )-c^3 d^3 \left (15 d^3+161 d^2 e x+245 d e^2 x^2+105 e^3 x^3\right )}{15 \left (c d^2-a e^2\right )^4 (a e+c d x) (d+e x)^{5/2}}-\frac {7 c^{5/2} d^{5/2} e \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{\left (-c d^2+a e^2\right )^{9/2}} \]
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Time = 2.82 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.95
method | result | size |
derivativedivides | \(2 e \left (-\frac {1}{5 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {3 c^{2} d^{2}}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {e x +d}}+\frac {2 c d}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {c^{3} d^{3} \left (\frac {\sqrt {e x +d}}{2 c d \left (e x +d \right )+2 e^{2} a -2 c \,d^{2}}+\frac {7 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}\right )\) | \(183\) |
default | \(2 e \left (-\frac {1}{5 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {5}{2}}}-\frac {3 c^{2} d^{2}}{\left (e^{2} a -c \,d^{2}\right )^{4} \sqrt {e x +d}}+\frac {2 c d}{3 \left (e^{2} a -c \,d^{2}\right )^{3} \left (e x +d \right )^{\frac {3}{2}}}-\frac {c^{3} d^{3} \left (\frac {\sqrt {e x +d}}{2 c d \left (e x +d \right )+2 e^{2} a -2 c \,d^{2}}+\frac {7 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{\left (e^{2} a -c \,d^{2}\right )^{4}}\right )\) | \(183\) |
pseudoelliptic | \(-\frac {2 \left (\frac {35 c^{3} d^{3} e \left (e x +d \right )^{\frac {5}{2}} \left (c d x +a e \right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{2}+\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\, \left (\frac {5 \left (7 e^{3} x^{3}+\frac {49}{3} d \,e^{2} x^{2}+\frac {161}{15} d^{2} e x +d^{3}\right ) d^{3} c^{3}}{2}+\frac {58 \left (\frac {35}{58} x^{2} e^{2}+\frac {42}{29} d e x +d^{2}\right ) e^{2} d^{2} a \,c^{2}}{3}-\frac {16 \left (\frac {7 e x}{16}+d \right ) e^{4} d \,a^{2} c}{3}+e^{6} a^{3}\right )\right )}{5 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}\, \left (e^{2} a -c \,d^{2}\right )^{4} \left (c d x +a e \right )}\) | \(211\) |
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Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (166) = 332\).
Time = 0.46 (sec) , antiderivative size = 1331, normalized size of antiderivative = 6.93 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {7}{2}} \left (a e + c d x\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (166) = 332\).
Time = 0.29 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.77 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {7 \, c^{3} d^{3} e \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt {-c^{2} d^{3} + a c d e^{2}}} - \frac {\sqrt {e x + d} c^{3} d^{3} e}{{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}} - \frac {2 \, {\left (45 \, {\left (e x + d\right )}^{2} c^{2} d^{2} e + 10 \, {\left (e x + d\right )} c^{2} d^{3} e + 3 \, c^{2} d^{4} e - 10 \, {\left (e x + d\right )} a c d e^{3} - 6 \, a c d^{2} e^{3} + 3 \, a^{2} e^{5}\right )}}{15 \, {\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} {\left (e x + d\right )}^{\frac {5}{2}}} \]
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Time = 10.13 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx=-\frac {\frac {2\,e}{5\,\left (a\,e^2-c\,d^2\right )}-\frac {14\,c\,d\,e\,\left (d+e\,x\right )}{15\,{\left (a\,e^2-c\,d^2\right )}^2}+\frac {14\,c^2\,d^2\,e\,{\left (d+e\,x\right )}^2}{3\,{\left (a\,e^2-c\,d^2\right )}^3}+\frac {7\,c^3\,d^3\,e\,{\left (d+e\,x\right )}^3}{{\left (a\,e^2-c\,d^2\right )}^4}}{\left (a\,e^2-c\,d^2\right )\,{\left (d+e\,x\right )}^{5/2}+c\,d\,{\left (d+e\,x\right )}^{7/2}}-\frac {7\,c^{5/2}\,d^{5/2}\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\left (a^4\,e^8-4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4-4\,a\,c^3\,d^6\,e^2+c^4\,d^8\right )}{{\left (a\,e^2-c\,d^2\right )}^{9/2}}\right )}{{\left (a\,e^2-c\,d^2\right )}^{9/2}} \]
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