Optimal. Leaf size=63 \[ \frac{2}{9} A b^2 x^{9/2}+\frac{2}{13} c x^{13/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{15} B c^2 x^{15/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0293865, antiderivative size = 63, 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} \[ \frac{2}{9} A b^2 x^{9/2}+\frac{2}{13} c x^{13/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{15} B c^2 x^{15/2} \]
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 )^2 \, dx &=\int \left (A b^2 x^{7/2}+b (b B+2 A c) x^{9/2}+c (2 b B+A c) x^{11/2}+B c^2 x^{13/2}\right ) \, dx\\ &=\frac{2}{9} A b^2 x^{9/2}+\frac{2}{11} b (b B+2 A c) x^{11/2}+\frac{2}{13} c (2 b B+A c) x^{13/2}+\frac{2}{15} B c^2 x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.0165733, size = 55, normalized size = 0.87 \[ \frac{2 x^{9/2} \left (5 A \left (143 b^2+234 b c x+99 c^2 x^2\right )+3 B x \left (195 b^2+330 b c x+143 c^2 x^2\right )\right )}{6435} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 52, normalized size = 0.8 \begin{align*}{\frac{858\,B{c}^{2}{x}^{3}+990\,A{c}^{2}{x}^{2}+1980\,B{x}^{2}bc+2340\,Abcx+1170\,{b}^{2}Bx+1430\,A{b}^{2}}{6435}{x}^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.09531, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{11}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.80242, size = 140, normalized size = 2.22 \begin{align*} \frac{2}{6435} \,{\left (429 \, B c^{2} x^{7} + 715 \, A b^{2} x^{4} + 495 \,{\left (2 \, B b c + A c^{2}\right )} x^{6} + 585 \,{\left (B b^{2} + 2 \, A b c\right )} x^{5}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 4.71182, size = 80, normalized size = 1.27 \begin{align*} \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{4 A b c x^{\frac{11}{2}}}{11} + \frac{2 A c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} + \frac{4 B b c x^{\frac{13}{2}}}{13} + \frac{2 B c^{2} x^{\frac{15}{2}}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10803, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{4}{13} \, B b c x^{\frac{13}{2}} + \frac{2}{13} \, A c^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A b c x^{\frac{11}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]