Integrand size = 18, antiderivative size = 45 \[ \int x^m \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {a A x^{1+m}}{1+m}+\frac {(A b+a B) x^{3+m}}{3+m}+\frac {b B x^{5+m}}{5+m} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {459} \[ \int x^m \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {x^{m+3} (a B+A b)}{m+3}+\frac {a A x^{m+1}}{m+1}+\frac {b B x^{m+5}}{m+5} \]
[In]
[Out]
Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a A x^m+(A b+a B) x^{2+m}+b B x^{4+m}\right ) \, dx \\ & = \frac {a A x^{1+m}}{1+m}+\frac {(A b+a B) x^{3+m}}{3+m}+\frac {b B x^{5+m}}{5+m} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int x^m \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=x^{1+m} \left (\frac {a A}{1+m}+\frac {(A b+a B) x^2}{3+m}+\frac {b B x^4}{5+m}\right ) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.18
method | result | size |
norman | \(\frac {\left (A b +B a \right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {A a x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {B b \,x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}\) | \(53\) |
risch | \(\frac {x \left (B b \,m^{2} x^{4}+4 B b m \,x^{4}+A b \,m^{2} x^{2}+B a \,m^{2} x^{2}+3 b B \,x^{4}+6 A b m \,x^{2}+6 B a m \,x^{2}+A a \,m^{2}+5 A b \,x^{2}+5 B a \,x^{2}+8 A a m +15 A a \right ) x^{m}}{\left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(109\) |
gosper | \(\frac {x^{1+m} \left (B b \,m^{2} x^{4}+4 B b m \,x^{4}+A b \,m^{2} x^{2}+B a \,m^{2} x^{2}+3 b B \,x^{4}+6 A b m \,x^{2}+6 B a m \,x^{2}+A a \,m^{2}+5 A b \,x^{2}+5 B a \,x^{2}+8 A a m +15 A a \right )}{\left (1+m \right ) \left (3+m \right ) \left (5+m \right )}\) | \(110\) |
parallelrisch | \(\frac {B \,x^{5} x^{m} b \,m^{2}+4 B \,x^{5} x^{m} b m +A \,x^{3} x^{m} b \,m^{2}+3 B \,x^{5} x^{m} b +B \,x^{3} x^{m} a \,m^{2}+6 A \,x^{3} x^{m} b m +6 B \,x^{3} x^{m} a m +5 A \,x^{3} x^{m} b +A x \,x^{m} a \,m^{2}+5 B \,x^{3} x^{m} a +8 A x \,x^{m} a m +15 A x \,x^{m} a}{\left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(144\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (45) = 90\).
Time = 0.25 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.04 \[ \int x^m \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {{\left ({\left (B b m^{2} + 4 \, B b m + 3 \, B b\right )} x^{5} + {\left ({\left (B a + A b\right )} m^{2} + 5 \, B a + 5 \, A b + 6 \, {\left (B a + A b\right )} m\right )} x^{3} + {\left (A a m^{2} + 8 \, A a m + 15 \, A a\right )} x\right )} x^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (37) = 74\).
Time = 0.32 (sec) , antiderivative size = 410, normalized size of antiderivative = 9.11 \[ \int x^m \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\begin {cases} - \frac {A a}{4 x^{4}} - \frac {A b}{2 x^{2}} - \frac {B a}{2 x^{2}} + B b \log {\left (x \right )} & \text {for}\: m = -5 \\- \frac {A a}{2 x^{2}} + A b \log {\left (x \right )} + B a \log {\left (x \right )} + \frac {B b x^{2}}{2} & \text {for}\: m = -3 \\A a \log {\left (x \right )} + \frac {A b x^{2}}{2} + \frac {B a x^{2}}{2} + \frac {B b x^{4}}{4} & \text {for}\: m = -1 \\\frac {A a m^{2} x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {8 A a m x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {15 A a x x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {A b m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {6 A b m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {5 A b x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {B a m^{2} x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {6 B a m x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {5 B a x^{3} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {B b m^{2} x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {4 B b m x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} + \frac {3 B b x^{5} x^{m}}{m^{3} + 9 m^{2} + 23 m + 15} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.18 \[ \int x^m \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {B b x^{m + 5}}{m + 5} + \frac {B a x^{m + 3}}{m + 3} + \frac {A b x^{m + 3}}{m + 3} + \frac {A a x^{m + 1}}{m + 1} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (45) = 90\).
Time = 0.33 (sec) , antiderivative size = 143, normalized size of antiderivative = 3.18 \[ \int x^m \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=\frac {B b m^{2} x^{5} x^{m} + 4 \, B b m x^{5} x^{m} + B a m^{2} x^{3} x^{m} + A b m^{2} x^{3} x^{m} + 3 \, B b x^{5} x^{m} + 6 \, B a m x^{3} x^{m} + 6 \, A b m x^{3} x^{m} + A a m^{2} x x^{m} + 5 \, B a x^{3} x^{m} + 5 \, A b x^{3} x^{m} + 8 \, A a m x x^{m} + 15 \, A a x x^{m}}{m^{3} + 9 \, m^{2} + 23 \, m + 15} \]
[In]
[Out]
Time = 5.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.11 \[ \int x^m \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx=x^m\,\left (\frac {x^3\,\left (A\,b+B\,a\right )\,\left (m^2+6\,m+5\right )}{m^3+9\,m^2+23\,m+15}+\frac {B\,b\,x^5\,\left (m^2+4\,m+3\right )}{m^3+9\,m^2+23\,m+15}+\frac {A\,a\,x\,\left (m^2+8\,m+15\right )}{m^3+9\,m^2+23\,m+15}\right ) \]
[In]
[Out]