Integrand size = 20, antiderivative size = 71 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {(b c-a d)^2 \left (a+b x^2\right )^3}{6 b^3}+\frac {d (b c-a d) \left (a+b x^2\right )^4}{4 b^3}+\frac {d^2 \left (a+b x^2\right )^5}{10 b^3} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {455, 45} \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {d \left (a+b x^2\right )^4 (b c-a d)}{4 b^3}+\frac {\left (a+b x^2\right )^3 (b c-a d)^2}{6 b^3}+\frac {d^2 \left (a+b x^2\right )^5}{10 b^3} \]
[In]
[Out]
Rule 45
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int (a+b x)^2 (c+d x)^2 \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {(b c-a d)^2 (a+b x)^2}{b^2}+\frac {2 d (b c-a d) (a+b x)^3}{b^2}+\frac {d^2 (a+b x)^4}{b^2}\right ) \, dx,x,x^2\right ) \\ & = \frac {(b c-a d)^2 \left (a+b x^2\right )^3}{6 b^3}+\frac {d (b c-a d) \left (a+b x^2\right )^4}{4 b^3}+\frac {d^2 \left (a+b x^2\right )^5}{10 b^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.14 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{60} x^2 \left (30 a^2 c^2+30 a c (b c+a d) x^2+10 \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^4+15 b d (b c+a d) x^6+6 b^2 d^2 x^8\right ) \]
[In]
[Out]
Time = 2.63 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.25
method | result | size |
norman | \(\frac {b^{2} d^{2} x^{10}}{10}+\left (\frac {1}{4} a b \,d^{2}+\frac {1}{4} b^{2} c d \right ) x^{8}+\left (\frac {1}{6} a^{2} d^{2}+\frac {2}{3} a b c d +\frac {1}{6} b^{2} c^{2}\right ) x^{6}+\left (\frac {1}{2} a^{2} c d +\frac {1}{2} b \,c^{2} a \right ) x^{4}+\frac {a^{2} c^{2} x^{2}}{2}\) | \(89\) |
default | \(\frac {b^{2} d^{2} x^{10}}{10}+\frac {\left (2 a b \,d^{2}+2 b^{2} c d \right ) x^{8}}{8}+\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{6}}{6}+\frac {\left (2 a^{2} c d +2 b \,c^{2} a \right ) x^{4}}{4}+\frac {a^{2} c^{2} x^{2}}{2}\) | \(90\) |
gosper | \(\frac {1}{10} b^{2} d^{2} x^{10}+\frac {1}{4} x^{8} a b \,d^{2}+\frac {1}{4} x^{8} b^{2} c d +\frac {1}{6} x^{6} a^{2} d^{2}+\frac {2}{3} x^{6} a b c d +\frac {1}{6} x^{6} b^{2} c^{2}+\frac {1}{2} x^{4} a^{2} c d +\frac {1}{2} x^{4} b \,c^{2} a +\frac {1}{2} a^{2} c^{2} x^{2}\) | \(95\) |
risch | \(\frac {1}{10} b^{2} d^{2} x^{10}+\frac {1}{4} x^{8} a b \,d^{2}+\frac {1}{4} x^{8} b^{2} c d +\frac {1}{6} x^{6} a^{2} d^{2}+\frac {2}{3} x^{6} a b c d +\frac {1}{6} x^{6} b^{2} c^{2}+\frac {1}{2} x^{4} a^{2} c d +\frac {1}{2} x^{4} b \,c^{2} a +\frac {1}{2} a^{2} c^{2} x^{2}\) | \(95\) |
parallelrisch | \(\frac {1}{10} b^{2} d^{2} x^{10}+\frac {1}{4} x^{8} a b \,d^{2}+\frac {1}{4} x^{8} b^{2} c d +\frac {1}{6} x^{6} a^{2} d^{2}+\frac {2}{3} x^{6} a b c d +\frac {1}{6} x^{6} b^{2} c^{2}+\frac {1}{2} x^{4} a^{2} c d +\frac {1}{2} x^{4} b \,c^{2} a +\frac {1}{2} a^{2} c^{2} x^{2}\) | \(95\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.20 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{10} \, b^{2} d^{2} x^{10} + \frac {1}{4} \, {\left (b^{2} c d + a b d^{2}\right )} x^{8} + \frac {1}{6} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{6} + \frac {1}{2} \, a^{2} c^{2} x^{2} + \frac {1}{2} \, {\left (a b c^{2} + a^{2} c d\right )} x^{4} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.32 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {a^{2} c^{2} x^{2}}{2} + \frac {b^{2} d^{2} x^{10}}{10} + x^{8} \left (\frac {a b d^{2}}{4} + \frac {b^{2} c d}{4}\right ) + x^{6} \left (\frac {a^{2} d^{2}}{6} + \frac {2 a b c d}{3} + \frac {b^{2} c^{2}}{6}\right ) + x^{4} \left (\frac {a^{2} c d}{2} + \frac {a b c^{2}}{2}\right ) \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.20 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{10} \, b^{2} d^{2} x^{10} + \frac {1}{4} \, {\left (b^{2} c d + a b d^{2}\right )} x^{8} + \frac {1}{6} \, {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{6} + \frac {1}{2} \, a^{2} c^{2} x^{2} + \frac {1}{2} \, {\left (a b c^{2} + a^{2} c d\right )} x^{4} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.32 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=\frac {1}{10} \, b^{2} d^{2} x^{10} + \frac {1}{4} \, b^{2} c d x^{8} + \frac {1}{4} \, a b d^{2} x^{8} + \frac {1}{6} \, b^{2} c^{2} x^{6} + \frac {2}{3} \, a b c d x^{6} + \frac {1}{6} \, a^{2} d^{2} x^{6} + \frac {1}{2} \, a b c^{2} x^{4} + \frac {1}{2} \, a^{2} c d x^{4} + \frac {1}{2} \, a^{2} c^{2} x^{2} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.10 \[ \int x \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx=x^6\,\left (\frac {a^2\,d^2}{6}+\frac {2\,a\,b\,c\,d}{3}+\frac {b^2\,c^2}{6}\right )+\frac {a^2\,c^2\,x^2}{2}+\frac {b^2\,d^2\,x^{10}}{10}+\frac {a\,c\,x^4\,\left (a\,d+b\,c\right )}{2}+\frac {b\,d\,x^8\,\left (a\,d+b\,c\right )}{4} \]
[In]
[Out]