Integrand size = 26, antiderivative size = 85 \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2}{7} A b^3 x^{7/2}+\frac {2}{11} b^2 (b B+3 A c) x^{11/2}+\frac {2}{5} b c (b B+A c) x^{15/2}+\frac {2}{19} c^2 (3 b B+A c) x^{19/2}+\frac {2}{23} B c^3 x^{23/2} \]
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Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1598, 459} \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2}{7} A b^3 x^{7/2}+\frac {2}{11} b^2 x^{11/2} (3 A c+b B)+\frac {2}{19} c^2 x^{19/2} (A c+3 b B)+\frac {2}{5} b c x^{15/2} (A c+b B)+\frac {2}{23} B c^3 x^{23/2} \]
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Rule 459
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int x^{5/2} \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx \\ & = \int \left (A b^3 x^{5/2}+b^2 (b B+3 A c) x^{9/2}+3 b c (b B+A c) x^{13/2}+c^2 (3 b B+A c) x^{17/2}+B c^3 x^{21/2}\right ) \, dx \\ & = \frac {2}{7} A b^3 x^{7/2}+\frac {2}{11} b^2 (b B+3 A c) x^{11/2}+\frac {2}{5} b c (b B+A c) x^{15/2}+\frac {2}{19} c^2 (3 b B+A c) x^{19/2}+\frac {2}{23} B c^3 x^{23/2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.98 \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2 x^{7/2} \left (23 A \left (1045 b^3+1995 b^2 c x^2+1463 b c^2 x^4+385 c^3 x^6\right )+7 B x^2 \left (2185 b^3+4807 b^2 c x^2+3795 b c^2 x^4+1045 c^3 x^6\right )\right )}{168245} \]
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Time = 1.77 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {2 B \,c^{3} x^{\frac {23}{2}}}{23}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 b^{2} A c +B \,b^{3}\right ) x^{\frac {11}{2}}}{11}+\frac {2 A \,b^{3} x^{\frac {7}{2}}}{7}\) | \(76\) |
default | \(\frac {2 B \,c^{3} x^{\frac {23}{2}}}{23}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {19}{2}}}{19}+\frac {2 \left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{\frac {15}{2}}}{15}+\frac {2 \left (3 b^{2} A c +B \,b^{3}\right ) x^{\frac {11}{2}}}{11}+\frac {2 A \,b^{3} x^{\frac {7}{2}}}{7}\) | \(76\) |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (7315 B \,c^{3} x^{8}+8855 A \,c^{3} x^{6}+26565 x^{6} B b \,c^{2}+33649 A b \,c^{2} x^{4}+33649 x^{4} B \,b^{2} c +45885 A \,b^{2} c \,x^{2}+15295 b^{3} B \,x^{2}+24035 b^{3} A \right )}{168245}\) | \(80\) |
trager | \(\frac {2 x^{\frac {7}{2}} \left (7315 B \,c^{3} x^{8}+8855 A \,c^{3} x^{6}+26565 x^{6} B b \,c^{2}+33649 A b \,c^{2} x^{4}+33649 x^{4} B \,b^{2} c +45885 A \,b^{2} c \,x^{2}+15295 b^{3} B \,x^{2}+24035 b^{3} A \right )}{168245}\) | \(80\) |
risch | \(\frac {2 x^{\frac {7}{2}} \left (7315 B \,c^{3} x^{8}+8855 A \,c^{3} x^{6}+26565 x^{6} B b \,c^{2}+33649 A b \,c^{2} x^{4}+33649 x^{4} B \,b^{2} c +45885 A \,b^{2} c \,x^{2}+15295 b^{3} B \,x^{2}+24035 b^{3} A \right )}{168245}\) | \(80\) |
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Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.92 \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2}{168245} \, {\left (7315 \, B c^{3} x^{11} + 8855 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{9} + 33649 \, {\left (B b^{2} c + A b c^{2}\right )} x^{7} + 24035 \, A b^{3} x^{3} + 15295 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{5}\right )} \sqrt {x} \]
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Time = 1.80 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34 \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2 A b^{3} x^{\frac {7}{2}}}{7} + \frac {6 A b^{2} c x^{\frac {11}{2}}}{11} + \frac {2 A b c^{2} x^{\frac {15}{2}}}{5} + \frac {2 A c^{3} x^{\frac {19}{2}}}{19} + \frac {2 B b^{3} x^{\frac {11}{2}}}{11} + \frac {2 B b^{2} c x^{\frac {15}{2}}}{5} + \frac {6 B b c^{2} x^{\frac {19}{2}}}{19} + \frac {2 B c^{3} x^{\frac {23}{2}}}{23} \]
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Time = 0.22 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.86 \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {2}{19} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {19}{2}} + \frac {2}{5} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {15}{2}} + \frac {2}{7} \, A b^{3} x^{\frac {7}{2}} + \frac {2}{11} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {11}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.91 \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{7/2}} \, dx=\frac {2}{23} \, B c^{3} x^{\frac {23}{2}} + \frac {6}{19} \, B b c^{2} x^{\frac {19}{2}} + \frac {2}{19} \, A c^{3} x^{\frac {19}{2}} + \frac {2}{5} \, B b^{2} c x^{\frac {15}{2}} + \frac {2}{5} \, A b c^{2} x^{\frac {15}{2}} + \frac {2}{11} \, B b^{3} x^{\frac {11}{2}} + \frac {6}{11} \, A b^{2} c x^{\frac {11}{2}} + \frac {2}{7} \, A b^{3} x^{\frac {7}{2}} \]
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Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{7/2}} \, dx=x^{11/2}\,\left (\frac {2\,B\,b^3}{11}+\frac {6\,A\,c\,b^2}{11}\right )+x^{19/2}\,\left (\frac {2\,A\,c^3}{19}+\frac {6\,B\,b\,c^2}{19}\right )+\frac {2\,A\,b^3\,x^{7/2}}{7}+\frac {2\,B\,c^3\,x^{23/2}}{23}+\frac {2\,b\,c\,x^{15/2}\,\left (A\,c+B\,b\right )}{5} \]
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