Integrand size = 17, antiderivative size = 123 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac {2 d e^3 x^2}{c}+\frac {e^4 x^3}{3 c}+\frac {\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{5/2}}+\frac {2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {716, 649, 211, 266} \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )}{\sqrt {a} c^{5/2}}+\frac {2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2}+\frac {e^2 x \left (6 c d^2-a e^2\right )}{c^2}+\frac {2 d e^3 x^2}{c}+\frac {e^4 x^3}{3 c} \]
[In]
[Out]
Rule 211
Rule 266
Rule 649
Rule 716
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e^2 \left (6 c d^2-a e^2\right )}{c^2}+\frac {4 d e^3 x}{c}+\frac {e^4 x^2}{c}+\frac {c^2 d^4-6 a c d^2 e^2+a^2 e^4+4 c d e \left (c d^2-a e^2\right ) x}{c^2 \left (a+c x^2\right )}\right ) \, dx \\ & = \frac {e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac {2 d e^3 x^2}{c}+\frac {e^4 x^3}{3 c}+\frac {\int \frac {c^2 d^4-6 a c d^2 e^2+a^2 e^4+4 c d e \left (c d^2-a e^2\right ) x}{a+c x^2} \, dx}{c^2} \\ & = \frac {e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac {2 d e^3 x^2}{c}+\frac {e^4 x^3}{3 c}+\frac {\left (4 d e \left (c d^2-a e^2\right )\right ) \int \frac {x}{a+c x^2} \, dx}{c}+\frac {\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \int \frac {1}{a+c x^2} \, dx}{c^2} \\ & = \frac {e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac {2 d e^3 x^2}{c}+\frac {e^4 x^3}{3 c}+\frac {\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{5/2}}+\frac {2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {\left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} c^{5/2}}+\frac {e \left (-3 a e^3 x+c e x \left (18 d^2+6 d e x+e^2 x^2\right )+6 \left (c d^3-a d e^2\right ) \log \left (a+c x^2\right )\right )}{3 c^2} \]
[In]
[Out]
Time = 2.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {e^{2} \left (-\frac {1}{3} c \,e^{2} x^{3}-2 c d e \,x^{2}+a \,e^{2} x -6 c \,d^{2} x \right )}{c^{2}}+\frac {\frac {\left (-4 d \,e^{3} a c +4 d^{3} e \,c^{2}\right ) \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{c^{2}}\) | \(117\) |
risch | \(\frac {e^{4} x^{3}}{3 c}+\frac {2 d \,e^{3} x^{2}}{c}-\frac {e^{4} a x}{c^{2}}+\frac {6 e^{2} d^{2} x}{c}-\frac {2 a \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}+\frac {2 \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d^{3} e}{c}+\frac {\ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}}{2 c^{3} a}-\frac {2 a \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}+\frac {2 \ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) d^{3} e}{c}-\frac {\ln \left (e^{4} a^{3}-6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right )^{2}}}{2 c^{3} a}\) | \(554\) |
[In]
[Out]
none
Time = 0.40 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.30 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\left [\frac {2 \, a c^{2} e^{4} x^{3} + 12 \, a c^{2} d e^{3} x^{2} - 3 \, {\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 6 \, {\left (6 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x + 12 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} \log \left (c x^{2} + a\right )}{6 \, a c^{3}}, \frac {a c^{2} e^{4} x^{3} + 6 \, a c^{2} d e^{3} x^{2} + 3 \, {\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + 3 \, {\left (6 \, a c^{2} d^{2} e^{2} - a^{2} c e^{4}\right )} x + 6 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} \log \left (c x^{2} + a\right )}{3 \, a c^{3}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (117) = 234\).
Time = 0.53 (sec) , antiderivative size = 401, normalized size of antiderivative = 3.26 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=x \left (- \frac {a e^{4}}{c^{2}} + \frac {6 d^{2} e^{2}}{c}\right ) + \left (- \frac {2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} - \frac {\sqrt {- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) \log {\left (x + \frac {4 a^{2} d e^{3} + 2 a c^{2} \left (- \frac {2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} - \frac {\sqrt {- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) - 4 a c d^{3} e}{a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}} \right )} + \left (- \frac {2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} + \frac {\sqrt {- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) \log {\left (x + \frac {4 a^{2} d e^{3} + 2 a c^{2} \left (- \frac {2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} + \frac {\sqrt {- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) - 4 a c d^{3} e}{a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}} \right )} + \frac {2 d e^{3} x^{2}}{c} + \frac {e^{4} x^{3}}{3 c} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.93 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {2 \, {\left (c d^{3} e - a d e^{3}\right )} \log \left (c x^{2} + a\right )}{c^{2}} + \frac {{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c^{2}} + \frac {c e^{4} x^{3} + 6 \, c d e^{3} x^{2} + 3 \, {\left (6 \, c d^{2} e^{2} - a e^{4}\right )} x}{3 \, c^{2}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {2 \, {\left (c d^{3} e - a d e^{3}\right )} \log \left (c x^{2} + a\right )}{c^{2}} + \frac {{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c} c^{2}} + \frac {c^{2} e^{4} x^{3} + 6 \, c^{2} d e^{3} x^{2} + 18 \, c^{2} d^{2} e^{2} x - 3 \, a c e^{4} x}{3 \, c^{3}} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.03 \[ \int \frac {(d+e x)^4}{a+c x^2} \, dx=\frac {e^4\,x^3}{3\,c}-x\,\left (\frac {a\,e^4}{c^2}-\frac {6\,d^2\,e^2}{c}\right )+\frac {2\,d\,e^3\,x^2}{c}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (a^2\,e^4-6\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{\sqrt {a}\,c^{5/2}}-\frac {\ln \left (c\,x^2+a\right )\,\left (16\,a^2\,c^3\,d\,e^3-16\,a\,c^4\,d^3\,e\right )}{8\,a\,c^5} \]
[In]
[Out]