Integrand size = 17, antiderivative size = 98 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^3} \, dx=\frac {x (d+e x)^3}{4 a \left (a+c x^2\right )^2}-\frac {3 d (a e-c d x) (d+e x)}{8 a^2 c \left (a+c x^2\right )}+\frac {3 d \left (c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{3/2}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {743, 737, 211} \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^3} \, dx=\frac {3 d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (a e^2+c d^2\right )}{8 a^{5/2} c^{3/2}}-\frac {3 d (d+e x) (a e-c d x)}{8 a^2 c \left (a+c x^2\right )}+\frac {x (d+e x)^3}{4 a \left (a+c x^2\right )^2} \]
[In]
[Out]
Rule 211
Rule 737
Rule 743
Rubi steps \begin{align*} \text {integral}& = \frac {x (d+e x)^3}{4 a \left (a+c x^2\right )^2}+\frac {(3 d) \int \frac {(d+e x)^2}{\left (a+c x^2\right )^2} \, dx}{4 a} \\ & = \frac {x (d+e x)^3}{4 a \left (a+c x^2\right )^2}-\frac {3 d (a e-c d x) (d+e x)}{8 a^2 c \left (a+c x^2\right )}+\frac {\left (3 d \left (c d^2+a e^2\right )\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 c} \\ & = \frac {x (d+e x)^3}{4 a \left (a+c x^2\right )^2}-\frac {3 d (a e-c d x) (d+e x)}{8 a^2 c \left (a+c x^2\right )}+\frac {3 d \left (c d^2+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{3/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.30 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^3} \, dx=\frac {\frac {\sqrt {a} \left (-2 a^3 e^3+3 c^3 d^3 x^3+a c^2 d x \left (5 d^2+3 e^2 x^2\right )-a^2 c e \left (6 d^2+3 d e x+4 e^2 x^2\right )\right )}{\left (a+c x^2\right )^2}+3 \sqrt {c} d \left (c d^2+a e^2\right ) \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^2} \]
[In]
[Out]
Time = 2.17 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.21
method | result | size |
default | \(\frac {\frac {3 d \left (e^{2} a +c \,d^{2}\right ) x^{3}}{8 a^{2}}-\frac {e^{3} x^{2}}{2 c}-\frac {d \left (3 e^{2} a -5 c \,d^{2}\right ) x}{8 a c}-\frac {e \left (e^{2} a +3 c \,d^{2}\right )}{4 c^{2}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {3 d \left (e^{2} a +c \,d^{2}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 a^{2} c \sqrt {a c}}\) | \(119\) |
risch | \(\frac {\frac {3 d \left (e^{2} a +c \,d^{2}\right ) x^{3}}{8 a^{2}}-\frac {e^{3} x^{2}}{2 c}-\frac {d \left (3 e^{2} a -5 c \,d^{2}\right ) x}{8 a c}-\frac {e \left (e^{2} a +3 c \,d^{2}\right )}{4 c^{2}}}{\left (c \,x^{2}+a \right )^{2}}-\frac {3 d \ln \left (c x +\sqrt {-a c}\right ) e^{2}}{16 \sqrt {-a c}\, c a}-\frac {3 d^{3} \ln \left (c x +\sqrt {-a c}\right )}{16 \sqrt {-a c}\, a^{2}}+\frac {3 d \ln \left (-c x +\sqrt {-a c}\right ) e^{2}}{16 \sqrt {-a c}\, c a}+\frac {3 d^{3} \ln \left (-c x +\sqrt {-a c}\right )}{16 \sqrt {-a c}\, a^{2}}\) | \(195\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (84) = 168\).
Time = 0.47 (sec) , antiderivative size = 406, normalized size of antiderivative = 4.14 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^3} \, dx=\left [-\frac {8 \, a^{3} c e^{3} x^{2} + 12 \, a^{3} c d^{2} e + 4 \, a^{4} e^{3} - 6 \, {\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x^{3} + 3 \, {\left (a^{2} c d^{3} + a^{3} d e^{2} + {\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{4} + 2 \, {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (5 \, a^{2} c^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x}{16 \, {\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}, -\frac {4 \, a^{3} c e^{3} x^{2} + 6 \, a^{3} c d^{2} e + 2 \, a^{4} e^{3} - 3 \, {\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x^{3} - 3 \, {\left (a^{2} c d^{3} + a^{3} d e^{2} + {\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{4} + 2 \, {\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (5 \, a^{2} c^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x}{8 \, {\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (88) = 176\).
Time = 0.66 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.78 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^3} \, dx=- \frac {3 d \sqrt {- \frac {1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log {\left (- \frac {3 a^{3} c d \sqrt {- \frac {1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right )}{3 a d e^{2} + 3 c d^{3}} + x \right )}}{16} + \frac {3 d \sqrt {- \frac {1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log {\left (\frac {3 a^{3} c d \sqrt {- \frac {1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right )}{3 a d e^{2} + 3 c d^{3}} + x \right )}}{16} + \frac {- 2 a^{3} e^{3} - 6 a^{2} c d^{2} e - 4 a^{2} c e^{3} x^{2} + x^{3} \cdot \left (3 a c^{2} d e^{2} + 3 c^{3} d^{3}\right ) + x \left (- 3 a^{2} c d e^{2} + 5 a c^{2} d^{3}\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^3} \, dx=-\frac {4 \, a^{2} c e^{3} x^{2} + 6 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3} - 3 \, {\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{3} - {\left (5 \, a c^{2} d^{3} - 3 \, a^{2} c d e^{2}\right )} x}{8 \, {\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )}} + \frac {3 \, {\left (c d^{3} + a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^3} \, dx=\frac {3 \, {\left (c d^{3} + a d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c} + \frac {3 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d e^{2} x^{3} - 4 \, a^{2} c e^{3} x^{2} + 5 \, a c^{2} d^{3} x - 3 \, a^{2} c d e^{2} x - 6 \, a^{2} c d^{2} e - 2 \, a^{3} e^{3}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x)^3}{\left (a+c x^2\right )^3} \, dx=\frac {3\,d\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (c\,d^2+a\,e^2\right )}{8\,a^{5/2}\,c^{3/2}}-\frac {\frac {e^3\,x^2}{2\,c}+\frac {e\,\left (3\,c\,d^2+a\,e^2\right )}{4\,c^2}-\frac {3\,d\,x^3\,\left (c\,d^2+a\,e^2\right )}{8\,a^2}+\frac {d\,x\,\left (3\,a\,e^2-5\,c\,d^2\right )}{8\,a\,c}}{a^2+2\,a\,c\,x^2+c^2\,x^4} \]
[In]
[Out]