Integrand size = 37, antiderivative size = 111 \[ \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^2}+\frac {4 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right )^2 (d+e x)} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {672, 664} \[ \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {4 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x) \left (c d^2-a e^2\right )^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 (d+e x)^2 \left (c d^2-a e^2\right )} \]
[In]
[Out]
Rule 664
Rule 672
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^2}+\frac {(2 c d) \int \frac {1}{(d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 \left (c d^2-a e^2\right )} \\ & = \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right ) (d+e x)^2}+\frac {4 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 \left (c d^2-a e^2\right )^2 (d+e x)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-a e^2+c d (3 d+2 e x)\right )}{3 \left (c d^2-a e^2\right )^2 (d+e x)^2} \]
[In]
[Out]
Time = 2.78 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.73
method | result | size |
trager | \(-\frac {2 \left (-2 x c d e +e^{2} a -3 c \,d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (e x +d \right )^{2}}\) | \(81\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-2 x c d e +e^{2} a -3 c \,d^{2}\right )}{3 \left (e x +d \right ) \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(89\) |
default | \(\frac {-\frac {2 \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{3 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {4 c d e \sqrt {c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}}{3 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )}}{e^{2}}\) | \(131\) |
[In]
[Out]
none
Time = 0.46 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.26 \[ \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + 3 \, c d^{2} - a e^{2}\right )}}{3 \, {\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} + {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )}} \]
[In]
[Out]
\[ \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {1}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{2}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {4 \, \sqrt {c d e} c d \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{3 \, {\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + \frac {2 \, {\left (3 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c d e - {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}}\right )}}{3 \, {\left (c d^{2} e^{2} - a e^{4}\right )} {\left (c d^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - a e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )}} \]
[In]
[Out]
Time = 10.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2\,\left (3\,c\,d^2+2\,c\,x\,d\,e-a\,e^2\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{3\,{\left (a\,e^2-c\,d^2\right )}^2\,{\left (d+e\,x\right )}^2} \]
[In]
[Out]