Integrand size = 37, antiderivative size = 274 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {3 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 c^{5/2} d^{5/2} e^{7/2}} \]
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Time = 0.11 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {678, 626, 635, 212} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=-\frac {3 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{5/2} d^{5/2} e^{7/2}}+\frac {3 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^2 d^2 e^3}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{5 e}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (a e^2+c d^2+2 c d e x\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} \]
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Rule 212
Rule 626
Rule 635
Rule 678
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (2 c d^2 e-e \left (c d^2+a e^2\right )\right ) \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2} \, dx}{2 e^2} \\ & = \frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}+\frac {\left (3 \left (c d^2-a e^2\right )^3\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32 c d e^2} \\ & = \frac {3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (3 \left (c d^2-a e^2\right )^5\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 c^2 d^2 e^3} \\ & = \frac {3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {\left (3 \left (c d^2-a e^2\right )^5\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{128 c^2 d^2 e^3} \\ & = \frac {3 \left (c d^2-a e^2\right )^3 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{128 c^2 d^2 e^3}+\frac {1}{16} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{5 e}-\frac {3 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{256 c^{5/2} d^{5/2} e^{7/2}} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {((a e+c d x) (d+e x))^{3/2} \left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \left (-15 a^4 e^8+10 a^3 c d e^6 (7 d+e x)+2 a^2 c^2 d^2 e^4 \left (64 d^2+233 d e x+124 e^2 x^2\right )+2 a c^3 d^3 e^2 \left (-35 d^3+23 d^2 e x+256 d e^2 x^2+168 e^3 x^3\right )+c^4 d^4 \left (15 d^4-10 d^3 e x+8 d^2 e^2 x^2+176 d e^3 x^3+128 e^4 x^4\right )\right )}{(a e+c d x) (d+e x)}-\frac {15 \left (c d^2-a e^2\right )^5 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{(a e+c d x)^{3/2} (d+e x)^{3/2}}\right )}{640 c^{5/2} d^{5/2} e^{7/2}} \]
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Time = 2.84 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 c d e}-\frac {3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (e^{2} a -c \,d^{2}\right )^{2} \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{16 c d e}\right )}{2}}{e}\) | \(327\) |
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Time = 0.33 (sec) , antiderivative size = 844, normalized size of antiderivative = 3.08 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\left [\frac {15 \, {\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 d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (128 \, c^{5} d^{5} e^{5} x^{4} + 15 \, c^{5} d^{9} e - 70 \, a c^{4} d^{7} e^{3} + 128 \, a^{2} c^{3} d^{5} e^{5} + 70 \, a^{3} c^{2} d^{3} e^{7} - 15 \, a^{4} c d e^{9} + 16 \, {\left (11 \, c^{5} d^{6} e^{4} + 21 \, a c^{4} d^{4} e^{6}\right )} x^{3} + 8 \, {\left (c^{5} d^{7} e^{3} + 64 \, a c^{4} d^{5} e^{5} + 31 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} - 2 \, {\left (5 \, c^{5} d^{8} e^{2} - 23 \, a c^{4} d^{6} e^{4} - 233 \, a^{2} c^{3} d^{4} e^{6} - 5 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2560 \, c^{3} d^{3} e^{4}}, \frac {15 \, {\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 d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (128 \, c^{5} d^{5} e^{5} x^{4} + 15 \, c^{5} d^{9} e - 70 \, a c^{4} d^{7} e^{3} + 128 \, a^{2} c^{3} d^{5} e^{5} + 70 \, a^{3} c^{2} d^{3} e^{7} - 15 \, a^{4} c d e^{9} + 16 \, {\left (11 \, c^{5} d^{6} e^{4} + 21 \, a c^{4} d^{4} e^{6}\right )} x^{3} + 8 \, {\left (c^{5} d^{7} e^{3} + 64 \, a c^{4} d^{5} e^{5} + 31 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} - 2 \, {\left (5 \, c^{5} d^{8} e^{2} - 23 \, a c^{4} d^{6} e^{4} - 233 \, a^{2} c^{3} d^{4} e^{6} - 5 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{1280 \, c^{3} d^{3} e^{4}}\right ] \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.39 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\frac {1}{640} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, c^{2} d^{2} e x + \frac {11 \, c^{6} d^{7} e^{4} + 21 \, a c^{5} d^{5} e^{6}}{c^{4} d^{4} e^{4}}\right )} x + \frac {c^{6} d^{8} e^{3} + 64 \, a c^{5} d^{6} e^{5} + 31 \, a^{2} c^{4} d^{4} e^{7}}{c^{4} d^{4} e^{4}}\right )} x - \frac {5 \, c^{6} d^{9} e^{2} - 23 \, a c^{5} d^{7} e^{4} - 233 \, a^{2} c^{4} d^{5} e^{6} - 5 \, a^{3} c^{3} d^{3} e^{8}}{c^{4} d^{4} e^{4}}\right )} x + \frac {15 \, c^{6} d^{10} e - 70 \, a c^{5} d^{8} e^{3} + 128 \, a^{2} c^{4} d^{6} e^{5} + 70 \, a^{3} c^{3} d^{4} e^{7} - 15 \, a^{4} c^{2} d^{2} e^{9}}{c^{4} d^{4} e^{4}}\right )} + \frac {3 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{256 \, \sqrt {c d e} c^{2} d^{2} e^{3}} \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{d+e\,x} \,d x \]
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