Integrand size = 37, antiderivative size = 186 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{4 c^4 d^4}+\frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}-\frac {35 e^2 \left (c d^2-a e^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{9/2} d^{9/2}} \]
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Time = 0.08 (sec) , antiderivative size = 186, 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} \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {35 e^2 \left (c d^2-a e^2\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{9/2} d^{9/2}}+\frac {35 e^2 \sqrt {d+e x} \left (c d^2-a e^2\right )}{4 c^4 d^4}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3} \]
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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)^3} \, dx \\ & = -\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {(7 e) \int \frac {(d+e x)^{5/2}}{(a e+c d x)^2} \, dx}{4 c d} \\ & = -\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {\left (35 e^2\right ) \int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx}{8 c^2 d^2} \\ & = \frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {\left (35 e^2 \left (c d^2-a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{8 c^3 d^3} \\ & = \frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{4 c^4 d^4}+\frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {\left (35 e^2 \left (c d^2-a e^2\right )^2\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{8 c^4 d^4} \\ & = \frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{4 c^4 d^4}+\frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}+\frac {\left (35 e \left (c d^2-a e^2\right )^2\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 c^4 d^4} \\ & = \frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {d+e x}}{4 c^4 d^4}+\frac {35 e^2 (d+e x)^{3/2}}{12 c^3 d^3}-\frac {7 e (d+e x)^{5/2}}{4 c^2 d^2 (a e+c d x)}-\frac {(d+e x)^{7/2}}{2 c d (a e+c d x)^2}-\frac {35 e^2 \left (c d^2-a e^2\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{4 c^{9/2} d^{9/2}} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.11 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (105 a^3 e^6-35 a^2 c d e^4 (4 d-5 e x)+7 a c^2 d^2 e^2 \left (3 d^2-34 d e x+8 e^2 x^2\right )+c^3 d^3 \left (6 d^3+39 d^2 e x-80 d e^2 x^2-8 e^3 x^3\right )\right )}{12 c^4 d^4 (a e+c d x)^2}+\frac {35 \left (c d^2 e-a e^3\right )^2 \arctan \left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {-c d^2+a e^2}}\right )}{4 c^{9/2} d^{9/2} \sqrt {-c d^2+a e^2}} \]
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Time = 9.71 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-\frac {2 e^{2} \left (-x c d e +9 e^{2} a -10 c \,d^{2}\right ) \sqrt {e x +d}}{3 c^{4} d^{4}}+\frac {\left (2 a^{2} e^{4}-4 a c \,d^{2} e^{2}+2 c^{2} d^{4}\right ) e^{2} \left (\frac {-\frac {13 c d \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (-\frac {11 e^{2} a}{8}+\frac {11 c \,d^{2}}{8}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {35 \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{c^{4} d^{4}}\) | \(180\) |
pseudoelliptic | \(\frac {\frac {35 e^{2} \left (e^{2} a -c \,d^{2}\right )^{2} \left (c d x +a e \right )^{2} \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{4}-\frac {35 \left (\frac {2 \left (-\frac {4}{3} e^{3} x^{3}-\frac {40}{3} d \,e^{2} x^{2}+\frac {13}{2} d^{2} e x +d^{3}\right ) d^{3} c^{3}}{35}+\frac {e^{2} \left (\frac {8}{3} x^{2} e^{2}-\frac {34}{3} d e x +d^{2}\right ) d^{2} a \,c^{2}}{5}-\frac {4 e^{4} \left (-\frac {5 e x}{4}+d \right ) d \,a^{2} c}{3}+e^{6} a^{3}\right ) \sqrt {e x +d}\, \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}{4}}{c^{4} d^{4} \left (c d x +a e \right )^{2} \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\) | \(208\) |
derivativedivides | \(2 e^{2} \left (-\frac {-\frac {c d \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a \,e^{2} \sqrt {e x +d}-3 c \,d^{2} \sqrt {e x +d}}{c^{4} d^{4}}+\frac {\frac {\left (-\frac {13}{8} d \,e^{4} a^{2} c +\frac {13}{4} d^{3} e^{2} c^{2} a -\frac {13}{8} d^{5} c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} e^{6} a^{3}+\frac {33}{8} d^{2} e^{4} a^{2} c -\frac {33}{8} d^{4} e^{2} c^{2} a +\frac {11}{8} c^{3} d^{6}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {35 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{4} d^{4}}\right )\) | \(245\) |
default | \(2 e^{2} \left (-\frac {-\frac {c d \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a \,e^{2} \sqrt {e x +d}-3 c \,d^{2} \sqrt {e x +d}}{c^{4} d^{4}}+\frac {\frac {\left (-\frac {13}{8} d \,e^{4} a^{2} c +\frac {13}{4} d^{3} e^{2} c^{2} a -\frac {13}{8} d^{5} c^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {11}{8} e^{6} a^{3}+\frac {33}{8} d^{2} e^{4} a^{2} c -\frac {33}{8} d^{4} e^{2} c^{2} a +\frac {11}{8} c^{3} d^{6}\right ) \sqrt {e x +d}}{\left (c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {35 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c d \sqrt {e x +d}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}\right )}{8 \sqrt {\left (e^{2} a -c \,d^{2}\right ) c d}}}{c^{4} d^{4}}\right )\) | \(245\) |
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Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (154) = 308\).
Time = 0.34 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.43 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\left [\frac {105 \, {\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} + {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, {\left (8 \, c^{3} d^{3} e^{3} x^{3} - 6 \, c^{3} d^{6} - 21 \, a c^{2} d^{4} e^{2} + 140 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 8 \, {\left (10 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} - {\left (39 \, c^{3} d^{5} e - 238 \, a c^{2} d^{3} e^{3} + 175 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}}, -\frac {105 \, {\left (a^{2} c d^{2} e^{4} - a^{3} e^{6} + {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (8 \, c^{3} d^{3} e^{3} x^{3} - 6 \, c^{3} d^{6} - 21 \, a c^{2} d^{4} e^{2} + 140 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 8 \, {\left (10 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} - {\left (39 \, c^{3} d^{5} e - 238 \, a c^{2} d^{3} e^{3} + 175 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{12 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.63 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {35 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \arctan \left (\frac {\sqrt {e x + d} c d}{\sqrt {-c^{2} d^{3} + a c d e^{2}}}\right )}{4 \, \sqrt {-c^{2} d^{3} + a c d e^{2}} c^{4} d^{4}} - \frac {13 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{5} e^{2} - 11 \, \sqrt {e x + d} c^{3} d^{6} e^{2} - 26 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{2} d^{3} e^{4} + 33 \, \sqrt {e x + d} a c^{2} d^{4} e^{4} + 13 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} c d e^{6} - 33 \, \sqrt {e x + d} a^{2} c d^{2} e^{6} + 11 \, \sqrt {e x + d} a^{3} e^{8}}{4 \, {\left ({\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}^{2} c^{4} d^{4}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c^{6} d^{6} e^{2} + 9 \, \sqrt {e x + d} c^{6} d^{7} e^{2} - 9 \, \sqrt {e x + d} a c^{5} d^{5} e^{4}\right )}}{3 \, c^{9} d^{9}} \]
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Time = 10.00 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.72 \[ \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx=\frac {2\,e^2\,{\left (d+e\,x\right )}^{3/2}}{3\,c^3\,d^3}-\frac {{\left (d+e\,x\right )}^{3/2}\,\left (\frac {13\,a^2\,c\,d\,e^6}{4}-\frac {13\,a\,c^2\,d^3\,e^4}{2}+\frac {13\,c^3\,d^5\,e^2}{4}\right )+\sqrt {d+e\,x}\,\left (\frac {11\,a^3\,e^8}{4}-\frac {33\,a^2\,c\,d^2\,e^6}{4}+\frac {33\,a\,c^2\,d^4\,e^4}{4}-\frac {11\,c^3\,d^6\,e^2}{4}\right )}{c^6\,d^8-\left (2\,c^6\,d^7-2\,a\,c^5\,d^5\,e^2\right )\,\left (d+e\,x\right )+c^6\,d^6\,{\left (d+e\,x\right )}^2-2\,a\,c^5\,d^6\,e^2+a^2\,c^4\,d^4\,e^4}+\frac {2\,e^2\,\left (3\,c^3\,d^4-3\,a\,c^2\,d^2\,e^2\right )\,\sqrt {d+e\,x}}{c^6\,d^6}+\frac {35\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e^2\,{\left (a\,e^2-c\,d^2\right )}^{3/2}\,\sqrt {d+e\,x}}{a^2\,e^6-2\,a\,c\,d^2\,e^4+c^2\,d^4\,e^2}\right )\,{\left (a\,e^2-c\,d^2\right )}^{3/2}}{4\,c^{9/2}\,d^{9/2}} \]
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