Integrand size = 37, antiderivative size = 201 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {1}{8} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e}+\frac {\left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 c^{3/2} d^{3/2} e^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {678, 626, 635, 212} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 c^{3/2} d^{3/2} e^{5/2}}+\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 e}+\frac {1}{8} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \]
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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 )^{3/2}}{3 e}-\frac {\left (2 c d^2 e-e \left (c d^2+a e^2\right )\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{2 e^2} \\ & = \frac {1}{8} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \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}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e}+\frac {\left (c d^2-a e^2\right )^3 \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 c d e^2} \\ & = \frac {1}{8} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \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}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e}+\frac {\left (c d^2-a e^2\right )^3 \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 )}{8 c d e^2} \\ & = \frac {1}{8} \left (\frac {a}{c d}-\frac {d}{e^2}\right ) \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}+\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 e}+\frac {\left (c d^2-a e^2\right )^3 \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 )}{16 c^{3/2} d^{3/2} e^{5/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\sqrt {c} \sqrt {d} \sqrt {e} \left (3 a^2 e^4+2 a c d e^2 (4 d+7 e x)+c^2 d^2 \left (-3 d^2+2 d e x+8 e^2 x^2\right )\right )+\frac {3 \left (c d^2-a e^2\right )^3 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{24 c^{3/2} d^{3/2} e^{5/2}} \]
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Time = 0.59 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {\frac {\left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3}+\frac {\left (e^{2} a -c \,d^{2}\right ) \left (\frac {\left (2 c d e \left (x +\frac {d}{e}\right )+e^{2} a -c \,d^{2}\right ) \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{4 c d e}-\frac {\left (e^{2} a -c \,d^{2}\right )^{2} \ln \left (\frac {\frac {e^{2} a}{2}-\frac {c \,d^{2}}{2}+c d e \left (x +\frac {d}{e}\right )}{\sqrt {c d e}}+\sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{8 c d e \sqrt {c d e}}\right )}{2}}{e}\) | \(230\) |
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Time = 0.33 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.65 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\left [-\frac {3 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} - 3 \, c^{3} d^{5} e + 8 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5} + 2 \, {\left (c^{3} d^{4} e^{2} + 7 \, a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{96 \, c^{2} d^{2} e^{3}}, -\frac {3 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) - 2 \, {\left (8 \, c^{3} d^{3} e^{3} x^{2} - 3 \, c^{3} d^{5} e + 8 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5} + 2 \, {\left (c^{3} d^{4} e^{2} + 7 \, a c^{2} d^{2} e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, c^{2} d^{2} e^{3}}\right ] \]
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Time = 3.61 (sec) , antiderivative size = 751, normalized size of antiderivative = 3.74 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=a e \left (\begin {cases} \left (\frac {x}{2} + \frac {\frac {a e^{2}}{4} + \frac {c d^{2}}{4}}{c d e}\right ) \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} + \left (\frac {a d e}{2} - \frac {\left (\frac {a e^{2}}{4} + \frac {c d^{2}}{4}\right ) \left (a e^{2} + c d^{2}\right )}{2 c d e}\right ) \left (\begin {cases} \frac {\log {\left (a e^{2} + c d^{2} + 2 c d e x + 2 \sqrt {c d e} \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \right )}}{\sqrt {c d e}} & \text {for}\: a d e - \frac {\left (a e^{2} + c d^{2}\right )^{2}}{4 c d e} \neq 0 \\\frac {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right ) \log {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e} \right )}}{\sqrt {c d e \left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c d e \neq 0 \\\frac {2 \left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {3}{2}}}{3 \left (a e^{2} + c d^{2}\right )} & \text {for}\: a e^{2} + c d^{2} \neq 0 \\x \sqrt {a d e} & \text {otherwise} \end {cases}\right ) + c d \left (\begin {cases} \left (- \frac {a \left (\frac {a e^{2}}{6} + \frac {c d^{2}}{6}\right )}{2 c} - \frac {\left (a e^{2} + c d^{2}\right ) \left (\frac {a d e}{3} - \frac {\left (\frac {a e^{2}}{6} + \frac {c d^{2}}{6}\right ) \left (\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}\right )}{2 c d e}\right )}{2 c d e}\right ) \left (\begin {cases} \frac {\log {\left (a e^{2} + c d^{2} + 2 c d e x + 2 \sqrt {c d e} \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \right )}}{\sqrt {c d e}} & \text {for}\: a d e - \frac {\left (a e^{2} + c d^{2}\right )^{2}}{4 c d e} \neq 0 \\\frac {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right ) \log {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e} \right )}}{\sqrt {c d e \left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \left (\frac {x^{2}}{3} + \frac {x \left (\frac {a e^{2}}{6} + \frac {c d^{2}}{6}\right )}{2 c d e} + \frac {\frac {a d e}{3} - \frac {\left (\frac {a e^{2}}{6} + \frac {c d^{2}}{6}\right ) \left (\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}\right )}{2 c d e}}{c d e}\right ) \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} & \text {for}\: c d e \neq 0 \\\frac {2 \left (- \frac {a d e \left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {3}{2}}}{3} + \frac {\left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {5}{2}}}{5}\right )}{\left (a e^{2} + c d^{2}\right )^{2}} & \text {for}\: a e^{2} + c d^{2} \neq 0 \\\frac {x^{2} \sqrt {a d e}}{2} & \text {otherwise} \end {cases}\right ) \]
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Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.35 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\frac {1}{24} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (2 \, {\left (4 \, c d x + \frac {c^{3} d^{4} e + 7 \, a c^{2} d^{2} e^{3}}{c^{2} d^{2} e^{2}}\right )} x - \frac {3 \, c^{3} d^{5} - 8 \, a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4}}{c^{2} d^{2} e^{2}}\right )} - \frac {{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{16 \, \sqrt {c d e} c d e^{2}} \]
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Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{d+e\,x} \,d x \]
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