Integrand size = 22, antiderivative size = 85 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{7} a^3 A x^{7/2}+\frac {2}{11} a^2 (3 A b+a B) x^{11/2}+\frac {2}{5} a b (A b+a B) x^{15/2}+\frac {2}{19} b^2 (A b+3 a B) x^{19/2}+\frac {2}{23} b^3 B x^{23/2} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459} \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{7} a^3 A x^{7/2}+\frac {2}{11} a^2 x^{11/2} (a B+3 A b)+\frac {2}{19} b^2 x^{19/2} (3 a B+A b)+\frac {2}{5} a b x^{15/2} (a B+A b)+\frac {2}{23} b^3 B x^{23/2} \]
[In]
[Out]
Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 A x^{5/2}+a^2 (3 A b+a B) x^{9/2}+3 a b (A b+a B) x^{13/2}+b^2 (A b+3 a B) x^{17/2}+b^3 B x^{21/2}\right ) \, dx \\ & = \frac {2}{7} a^3 A x^{7/2}+\frac {2}{11} a^2 (3 A b+a B) x^{11/2}+\frac {2}{5} a b (A b+a B) x^{15/2}+\frac {2}{19} b^2 (A b+3 a B) x^{19/2}+\frac {2}{23} b^3 B x^{23/2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.95 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2 x^{7/2} \left (2185 a^3 \left (11 A+7 B x^2\right )+3059 a^2 b x^2 \left (15 A+11 B x^2\right )+1771 a b^2 x^4 \left (19 A+15 B x^2\right )+385 b^3 x^6 \left (23 A+19 B x^2\right )\right )}{168245} \]
[In]
[Out]
Time = 2.73 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 b^{3} B \,x^{\frac {23}{2}}}{23}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{3} A \,x^{\frac {7}{2}}}{7}\) | \(76\) |
default | \(\frac {2 b^{3} B \,x^{\frac {23}{2}}}{23}+\frac {2 \left (b^{3} A +3 a \,b^{2} B \right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 a \,b^{2} A +3 a^{2} b B \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 a^{2} b A +a^{3} B \right ) x^{\frac {11}{2}}}{11}+\frac {2 a^{3} A \,x^{\frac {7}{2}}}{7}\) | \(76\) |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (7315 b^{3} B \,x^{8}+8855 x^{6} b^{3} A +26565 x^{6} a \,b^{2} B +33649 A a \,b^{2} x^{4}+33649 B \,a^{2} b \,x^{4}+45885 x^{2} a^{2} b A +15295 B \,a^{3} x^{2}+24035 a^{3} A \right )}{168245}\) | \(80\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (7315 b^{3} B \,x^{8}+8855 x^{6} b^{3} A +26565 x^{6} a \,b^{2} B +33649 A a \,b^{2} x^{4}+33649 B \,a^{2} b \,x^{4}+45885 x^{2} a^{2} b A +15295 B \,a^{3} x^{2}+24035 a^{3} A \right )}{168245}\) | \(80\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (7315 b^{3} B \,x^{8}+8855 x^{6} b^{3} A +26565 x^{6} a \,b^{2} B +33649 A a \,b^{2} x^{4}+33649 B \,a^{2} b \,x^{4}+45885 x^{2} a^{2} b A +15295 B \,a^{3} x^{2}+24035 a^{3} A \right )}{168245}\) | \(80\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{168245} \, {\left (7315 \, B b^{3} x^{11} + 8855 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 33649 \, {\left (B a^{2} b + A a b^{2}\right )} x^{7} + 24035 \, A a^{3} x^{3} + 15295 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} \sqrt {x} \]
[In]
[Out]
Time = 0.93 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2 A a^{3} x^{\frac {7}{2}}}{7} + \frac {6 A a^{2} b x^{\frac {11}{2}}}{11} + \frac {2 A a b^{2} x^{\frac {15}{2}}}{5} + \frac {2 A b^{3} x^{\frac {19}{2}}}{19} + \frac {2 B a^{3} x^{\frac {11}{2}}}{11} + \frac {2 B a^{2} b x^{\frac {15}{2}}}{5} + \frac {6 B a b^{2} x^{\frac {19}{2}}}{19} + \frac {2 B b^{3} x^{\frac {23}{2}}}{23} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{23} \, B b^{3} x^{\frac {23}{2}} + \frac {2}{19} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {19}{2}} + \frac {2}{5} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {15}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} + \frac {2}{11} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {11}{2}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=\frac {2}{23} \, B b^{3} x^{\frac {23}{2}} + \frac {6}{19} \, B a b^{2} x^{\frac {19}{2}} + \frac {2}{19} \, A b^{3} x^{\frac {19}{2}} + \frac {2}{5} \, B a^{2} b x^{\frac {15}{2}} + \frac {2}{5} \, A a b^{2} x^{\frac {15}{2}} + \frac {2}{11} \, B a^{3} x^{\frac {11}{2}} + \frac {6}{11} \, A a^{2} b x^{\frac {11}{2}} + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx=x^{11/2}\,\left (\frac {2\,B\,a^3}{11}+\frac {6\,A\,b\,a^2}{11}\right )+x^{19/2}\,\left (\frac {2\,A\,b^3}{19}+\frac {6\,B\,a\,b^2}{19}\right )+\frac {2\,A\,a^3\,x^{7/2}}{7}+\frac {2\,B\,b^3\,x^{23/2}}{23}+\frac {2\,a\,b\,x^{15/2}\,\left (A\,b+B\,a\right )}{5} \]
[In]
[Out]