Optimal. Leaf size=63 \[ \frac {2}{7} A b^2 x^{7/2}+\frac {2}{11} c x^{11/2} (A c+2 b B)+\frac {2}{9} b x^{9/2} (2 A c+b B)+\frac {2}{13} B c^2 x^{13/2} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {765} \begin {gather*} \frac {2}{7} A b^2 x^{7/2}+\frac {2}{11} c x^{11/2} (A c+2 b B)+\frac {2}{9} b x^{9/2} (2 A c+b B)+\frac {2}{13} B c^2 x^{13/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin {align*} \int \sqrt {x} (A+B x) \left (b x+c x^2\right )^2 \, dx &=\int \left (A b^2 x^{5/2}+b (b B+2 A c) x^{7/2}+c (2 b B+A c) x^{9/2}+B c^2 x^{11/2}\right ) \, dx\\ &=\frac {2}{7} A b^2 x^{7/2}+\frac {2}{9} b (b B+2 A c) x^{9/2}+\frac {2}{11} c (2 b B+A c) x^{11/2}+\frac {2}{13} B c^2 x^{13/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 55, normalized size = 0.87 \begin {gather*} \frac {2 x^{7/2} \left (13 A \left (99 b^2+154 b c x+63 c^2 x^2\right )+7 B x \left (143 b^2+234 b c x+99 c^2 x^2\right )\right )}{9009} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.03, size = 69, normalized size = 1.10 \begin {gather*} \frac {2 \left (1287 A b^2 x^{7/2}+2002 A b c x^{9/2}+819 A c^2 x^{11/2}+1001 b^2 B x^{9/2}+1638 b B c x^{11/2}+693 B c^2 x^{13/2}\right )}{9009} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.39, size = 56, normalized size = 0.89 \begin {gather*} \frac {2}{9009} \, {\left (693 \, B c^{2} x^{6} + 1287 \, A b^{2} x^{3} + 819 \, {\left (2 \, B b c + A c^{2}\right )} x^{5} + 1001 \, {\left (B b^{2} + 2 \, A b c\right )} x^{4}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 53, normalized size = 0.84 \begin {gather*} \frac {2}{13} \, B c^{2} x^{\frac {13}{2}} + \frac {4}{11} \, B b c x^{\frac {11}{2}} + \frac {2}{11} \, A c^{2} x^{\frac {11}{2}} + \frac {2}{9} \, B b^{2} x^{\frac {9}{2}} + \frac {4}{9} \, A b c x^{\frac {9}{2}} + \frac {2}{7} \, A b^{2} x^{\frac {7}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 52, normalized size = 0.83 \begin {gather*} \frac {2 \left (693 B \,c^{2} x^{3}+819 A \,c^{2} x^{2}+1638 B b c \,x^{2}+2002 A b c x +1001 B \,b^{2} x +1287 A \,b^{2}\right ) x^{\frac {7}{2}}}{9009} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.53, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{13} \, B c^{2} x^{\frac {13}{2}} + \frac {2}{7} \, A b^{2} x^{\frac {7}{2}} + \frac {2}{11} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (B b^{2} + 2 \, A b c\right )} x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 51, normalized size = 0.81 \begin {gather*} x^{9/2}\,\left (\frac {2\,B\,b^2}{9}+\frac {4\,A\,c\,b}{9}\right )+x^{11/2}\,\left (\frac {2\,A\,c^2}{11}+\frac {4\,B\,b\,c}{11}\right )+\frac {2\,A\,b^2\,x^{7/2}}{7}+\frac {2\,B\,c^2\,x^{13/2}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.24, size = 66, normalized size = 1.05 \begin {gather*} \frac {2 A b^{2} x^{\frac {7}{2}}}{7} + \frac {2 B c^{2} x^{\frac {13}{2}}}{13} + \frac {2 x^{\frac {11}{2}} \left (A c^{2} + 2 B b c\right )}{11} + \frac {2 x^{\frac {9}{2}} \left (2 A b c + B b^{2}\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________