Optimal. Leaf size=85 \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{5} a^3 A x^{5/2}+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0403044, 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 = {448} \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{5} a^3 A x^{5/2}+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 448
Rubi steps
\begin{align*} \int x^{3/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx &=\int \left (a^3 A x^{3/2}+a^2 (3 A b+a B) x^{7/2}+3 a b (A b+a B) x^{11/2}+b^2 (A b+3 a B) x^{15/2}+b^3 B x^{19/2}\right ) \, dx\\ &=\frac{2}{5} a^3 A x^{5/2}+\frac{2}{9} a^2 (3 A b+a B) x^{9/2}+\frac{6}{13} a b (A b+a B) x^{13/2}+\frac{2}{17} b^2 (A b+3 a B) x^{17/2}+\frac{2}{21} b^3 B x^{21/2}\\ \end{align*}
Mathematica [A] time = 0.0360561, size = 85, normalized size = 1. \[ \frac{2}{9} a^2 x^{9/2} (a B+3 A b)+\frac{2}{5} a^3 A x^{5/2}+\frac{2}{17} b^2 x^{17/2} (3 a B+A b)+\frac{6}{13} a b x^{13/2} (a B+A b)+\frac{2}{21} b^3 B x^{21/2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 80, normalized size = 0.9 \begin{align*}{\frac{6630\,B{b}^{3}{x}^{8}+8190\,{x}^{6}A{b}^{3}+24570\,{x}^{6}Ba{b}^{2}+32130\,{x}^{4}Aa{b}^{2}+32130\,{x}^{4}B{a}^{2}b+46410\,{x}^{2}A{a}^{2}b+15470\,{x}^{2}B{a}^{3}+27846\,A{a}^{3}}{69615}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.1162, size = 99, normalized size = 1.16 \begin{align*} \frac{2}{21} \, B b^{3} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{17}{2}} + \frac{6}{13} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{13}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} + \frac{2}{9} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.82957, size = 198, normalized size = 2.33 \begin{align*} \frac{2}{69615} \,{\left (3315 \, B b^{3} x^{10} + 4095 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{8} + 16065 \,{\left (B a^{2} b + A a b^{2}\right )} x^{6} + 13923 \, A a^{3} x^{2} + 7735 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 11.3913, size = 114, normalized size = 1.34 \begin{align*} \frac{2 A a^{3} x^{\frac{5}{2}}}{5} + \frac{2 A a^{2} b x^{\frac{9}{2}}}{3} + \frac{6 A a b^{2} x^{\frac{13}{2}}}{13} + \frac{2 A b^{3} x^{\frac{17}{2}}}{17} + \frac{2 B a^{3} x^{\frac{9}{2}}}{9} + \frac{6 B a^{2} b x^{\frac{13}{2}}}{13} + \frac{6 B a b^{2} x^{\frac{17}{2}}}{17} + \frac{2 B b^{3} x^{\frac{21}{2}}}{21} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13626, size = 104, normalized size = 1.22 \begin{align*} \frac{2}{21} \, B b^{3} x^{\frac{21}{2}} + \frac{6}{17} \, B a b^{2} x^{\frac{17}{2}} + \frac{2}{17} \, A b^{3} x^{\frac{17}{2}} + \frac{6}{13} \, B a^{2} b x^{\frac{13}{2}} + \frac{6}{13} \, A a b^{2} x^{\frac{13}{2}} + \frac{2}{9} \, B a^{3} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} b x^{\frac{9}{2}} + \frac{2}{5} \, A a^{3} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]