Optimal. Leaf size=63 \[ \frac {2}{9} A b^2 x^{9/2}+\frac {2}{17} c x^{17/2} (A c+2 b B)+\frac {2}{13} b x^{13/2} (2 A c+b B)+\frac {2}{21} B c^2 x^{21/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1584, 448} \[ \frac {2}{9} A b^2 x^{9/2}+\frac {2}{17} c x^{17/2} (A c+2 b B)+\frac {2}{13} b x^{13/2} (2 A c+b B)+\frac {2}{21} B c^2 x^{21/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 448
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{\sqrt {x}} \, dx &=\int x^{7/2} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^{7/2}+b (b B+2 A c) x^{11/2}+c (2 b B+A c) x^{15/2}+B c^2 x^{19/2}\right ) \, dx\\ &=\frac {2}{9} A b^2 x^{9/2}+\frac {2}{13} b (b B+2 A c) x^{13/2}+\frac {2}{17} c (2 b B+A c) x^{17/2}+\frac {2}{21} B c^2 x^{21/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 53, normalized size = 0.84 \[ \frac {2 x^{9/2} \left (1547 A b^2+819 c x^4 (A c+2 b B)+1071 b x^2 (2 A c+b B)+663 B c^2 x^6\right )}{13923} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.88, size = 56, normalized size = 0.89 \[ \frac {2}{13923} \, {\left (663 \, B c^{2} x^{10} + 819 \, {\left (2 \, B b c + A c^{2}\right )} x^{8} + 1547 \, A b^{2} x^{4} + 1071 \, {\left (B b^{2} + 2 \, A b c\right )} x^{6}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 53, normalized size = 0.84 \[ \frac {2}{21} \, B c^{2} x^{\frac {21}{2}} + \frac {4}{17} \, B b c x^{\frac {17}{2}} + \frac {2}{17} \, A c^{2} x^{\frac {17}{2}} + \frac {2}{13} \, B b^{2} x^{\frac {13}{2}} + \frac {4}{13} \, A b c x^{\frac {13}{2}} + \frac {2}{9} \, A b^{2} x^{\frac {9}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 56, normalized size = 0.89 \[ \frac {2 \left (663 B \,c^{2} x^{6}+819 A \,c^{2} x^{4}+1638 B b c \,x^{4}+2142 A b c \,x^{2}+1071 B \,b^{2} x^{2}+1547 b^{2} A \right ) x^{\frac {9}{2}}}{13923} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 51, normalized size = 0.81 \[ \frac {2}{21} \, B c^{2} x^{\frac {21}{2}} + \frac {2}{17} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {17}{2}} + \frac {2}{9} \, A b^{2} x^{\frac {9}{2}} + \frac {2}{13} \, {\left (B b^{2} + 2 \, A b c\right )} x^{\frac {13}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 51, normalized size = 0.81 \[ x^{13/2}\,\left (\frac {2\,B\,b^2}{13}+\frac {4\,A\,c\,b}{13}\right )+x^{17/2}\,\left (\frac {2\,A\,c^2}{17}+\frac {4\,B\,b\,c}{17}\right )+\frac {2\,A\,b^2\,x^{9/2}}{9}+\frac {2\,B\,c^2\,x^{21/2}}{21} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 9.94, size = 80, normalized size = 1.27 \[ \frac {2 A b^{2} x^{\frac {9}{2}}}{9} + \frac {4 A b c x^{\frac {13}{2}}}{13} + \frac {2 A c^{2} x^{\frac {17}{2}}}{17} + \frac {2 B b^{2} x^{\frac {13}{2}}}{13} + \frac {4 B b c x^{\frac {17}{2}}}{17} + \frac {2 B c^{2} x^{\frac {21}{2}}}{21} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________