Optimal. Leaf size=85 \[ \frac{2}{13} b^2 x^{13/2} (3 A c+b B)+\frac{2}{11} A b^3 x^{11/2}+\frac{2}{17} c^2 x^{17/2} (A c+3 b B)+\frac{2}{5} b c x^{15/2} (A c+b B)+\frac{2}{19} B c^3 x^{19/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0418129, antiderivative size = 85, normalized size of antiderivative = 1., 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} \[ \frac{2}{13} b^2 x^{13/2} (3 A c+b B)+\frac{2}{11} A b^3 x^{11/2}+\frac{2}{17} c^2 x^{17/2} (A c+3 b B)+\frac{2}{5} b c x^{15/2} (A c+b B)+\frac{2}{19} B c^3 x^{19/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 765
Rubi steps
\begin{align*} \int x^{3/2} (A+B x) \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^{9/2}+b^2 (b B+3 A c) x^{11/2}+3 b c (b B+A c) x^{13/2}+c^2 (3 b B+A c) x^{15/2}+B c^3 x^{17/2}\right ) \, dx\\ &=\frac{2}{11} A b^3 x^{11/2}+\frac{2}{13} b^2 (b B+3 A c) x^{13/2}+\frac{2}{5} b c (b B+A c) x^{15/2}+\frac{2}{17} c^2 (3 b B+A c) x^{17/2}+\frac{2}{19} B c^3 x^{19/2}\\ \end{align*}
Mathematica [A] time = 0.0442834, size = 70, normalized size = 0.82 \[ \frac{2 \left (B x^{11/2} (b+c x)^4-\frac{x^{11/2} \left (2805 b^2 c x+1105 b^3+2431 b c^2 x^2+715 c^3 x^3\right ) (11 b B-19 A c)}{12155}\right )}{19 c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 76, normalized size = 0.9 \begin{align*}{\frac{24310\,B{c}^{3}{x}^{4}+27170\,A{x}^{3}{c}^{3}+81510\,B{x}^{3}b{c}^{2}+92378\,A{x}^{2}b{c}^{2}+92378\,B{x}^{2}{b}^{2}c+106590\,A{b}^{2}cx+35530\,{b}^{3}Bx+41990\,A{b}^{3}}{230945}{x}^{{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.16816, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{19} \, B c^{3} x^{\frac{19}{2}} + \frac{2}{11} \, A b^{3} x^{\frac{11}{2}} + \frac{2}{17} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac{17}{2}} + \frac{2}{5} \,{\left (B b^{2} c + A b c^{2}\right )} x^{\frac{15}{2}} + \frac{2}{13} \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac{13}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.8556, size = 203, normalized size = 2.39 \begin{align*} \frac{2}{230945} \,{\left (12155 \, B c^{3} x^{9} + 20995 \, A b^{3} x^{5} + 13585 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{8} + 46189 \,{\left (B b^{2} c + A b c^{2}\right )} x^{7} + 17765 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 9.6158, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A b^{3} x^{\frac{11}{2}}}{11} + \frac{6 A b^{2} c x^{\frac{13}{2}}}{13} + \frac{2 A b c^{2} x^{\frac{15}{2}}}{5} + \frac{2 A c^{3} x^{\frac{17}{2}}}{17} + \frac{2 B b^{3} x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} c x^{\frac{15}{2}}}{5} + \frac{6 B b c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B c^{3} x^{\frac{19}{2}}}{19} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15209, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{19} \, B c^{3} x^{\frac{19}{2}} + \frac{6}{17} \, B b c^{2} x^{\frac{17}{2}} + \frac{2}{17} \, A c^{3} x^{\frac{17}{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}{13} \, B b^{3} x^{\frac{13}{2}} + \frac{6}{13} \, A b^{2} c x^{\frac{13}{2}} + \frac{2}{11} \, A b^{3} x^{\frac{11}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]