Optimal. Leaf size=159 \[ \frac{10}{17} a^2 b^3 x^{17/2} (4 a B+3 A b)+\frac{2}{3} a^3 b^2 x^{15/2} (3 a B+4 A b)+\frac{6}{13} a^4 b x^{13/2} (2 a B+5 A b)+\frac{2}{11} a^5 x^{11/2} (a B+6 A b)+\frac{2}{9} a^6 A x^{9/2}+\frac{2}{21} b^5 x^{21/2} (6 a B+A b)+\frac{6}{19} a b^4 x^{19/2} (5 a B+2 A b)+\frac{2}{23} b^6 B x^{23/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0895278, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 76} \[ \frac{10}{17} a^2 b^3 x^{17/2} (4 a B+3 A b)+\frac{2}{3} a^3 b^2 x^{15/2} (3 a B+4 A b)+\frac{6}{13} a^4 b x^{13/2} (2 a B+5 A b)+\frac{2}{11} a^5 x^{11/2} (a B+6 A b)+\frac{2}{9} a^6 A x^{9/2}+\frac{2}{21} b^5 x^{21/2} (6 a B+A b)+\frac{6}{19} a b^4 x^{19/2} (5 a B+2 A b)+\frac{2}{23} b^6 B x^{23/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 76
Rubi steps
\begin{align*} \int x^{7/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int x^{7/2} (a+b x)^6 (A+B x) \, dx\\ &=\int \left (a^6 A x^{7/2}+a^5 (6 A b+a B) x^{9/2}+3 a^4 b (5 A b+2 a B) x^{11/2}+5 a^3 b^2 (4 A b+3 a B) x^{13/2}+5 a^2 b^3 (3 A b+4 a B) x^{15/2}+3 a b^4 (2 A b+5 a B) x^{17/2}+b^5 (A b+6 a B) x^{19/2}+b^6 B x^{21/2}\right ) \, dx\\ &=\frac{2}{9} a^6 A x^{9/2}+\frac{2}{11} a^5 (6 A b+a B) x^{11/2}+\frac{6}{13} a^4 b (5 A b+2 a B) x^{13/2}+\frac{2}{3} a^3 b^2 (4 A b+3 a B) x^{15/2}+\frac{10}{17} a^2 b^3 (3 A b+4 a B) x^{17/2}+\frac{6}{19} a b^4 (2 A b+5 a B) x^{19/2}+\frac{2}{21} b^5 (A b+6 a B) x^{21/2}+\frac{2}{23} b^6 B x^{23/2}\\ \end{align*}
Mathematica [A] time = 0.0971016, size = 131, normalized size = 0.82 \[ \frac{2 \left (\frac{1}{2} \left (\frac{30}{17} a^2 b^4 x^{17/2}+\frac{8}{3} a^3 b^3 x^{15/2}+\frac{30}{13} a^4 b^2 x^{13/2}+\frac{12}{11} a^5 b x^{11/2}+\frac{2}{9} a^6 x^{9/2}+\frac{12}{19} a b^5 x^{19/2}+\frac{2}{21} b^6 x^{21/2}\right ) (23 A b-9 a B)+B x^{9/2} (a+b x)^7\right )}{23 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 148, normalized size = 0.9 \begin{align*}{\frac{5819814\,B{b}^{6}{x}^{7}+6374082\,A{b}^{6}{x}^{6}+38244492\,B{x}^{6}a{b}^{5}+42270228\,aA{b}^{5}{x}^{5}+105675570\,B{x}^{5}{a}^{2}{b}^{4}+118107990\,{a}^{2}A{b}^{4}{x}^{4}+157477320\,B{x}^{4}{a}^{3}{b}^{3}+178474296\,{a}^{3}A{b}^{3}{x}^{3}+133855722\,B{x}^{3}{a}^{4}{b}^{2}+154448910\,{a}^{4}A{b}^{2}{x}^{2}+61779564\,B{x}^{2}{a}^{5}b+73012212\,{a}^{5}Abx+12168702\,B{a}^{6}x+14872858\,A{a}^{6}}{66927861}{x}^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.06476, size = 198, normalized size = 1.25 \begin{align*} \frac{2}{23} \, B b^{6} x^{\frac{23}{2}} + \frac{2}{9} \, A a^{6} x^{\frac{9}{2}} + \frac{2}{21} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{21}{2}} + \frac{6}{19} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac{19}{2}} + \frac{10}{17} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac{17}{2}} + \frac{2}{3} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac{15}{2}} + \frac{6}{13} \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{\frac{11}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.55913, size = 402, normalized size = 2.53 \begin{align*} \frac{2}{66927861} \,{\left (2909907 \, B b^{6} x^{11} + 7436429 \, A a^{6} x^{4} + 3187041 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{10} + 10567557 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{9} + 19684665 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{8} + 22309287 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{7} + 15444891 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{6} + 6084351 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{5}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 34.4588, size = 214, normalized size = 1.35 \begin{align*} \frac{2 A a^{6} x^{\frac{9}{2}}}{9} + \frac{12 A a^{5} b x^{\frac{11}{2}}}{11} + \frac{30 A a^{4} b^{2} x^{\frac{13}{2}}}{13} + \frac{8 A a^{3} b^{3} x^{\frac{15}{2}}}{3} + \frac{30 A a^{2} b^{4} x^{\frac{17}{2}}}{17} + \frac{12 A a b^{5} x^{\frac{19}{2}}}{19} + \frac{2 A b^{6} x^{\frac{21}{2}}}{21} + \frac{2 B a^{6} x^{\frac{11}{2}}}{11} + \frac{12 B a^{5} b x^{\frac{13}{2}}}{13} + 2 B a^{4} b^{2} x^{\frac{15}{2}} + \frac{40 B a^{3} b^{3} x^{\frac{17}{2}}}{17} + \frac{30 B a^{2} b^{4} x^{\frac{19}{2}}}{19} + \frac{4 B a b^{5} x^{\frac{21}{2}}}{7} + \frac{2 B b^{6} x^{\frac{23}{2}}}{23} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12174, size = 201, normalized size = 1.26 \begin{align*} \frac{2}{23} \, B b^{6} x^{\frac{23}{2}} + \frac{4}{7} \, B a b^{5} x^{\frac{21}{2}} + \frac{2}{21} \, A b^{6} x^{\frac{21}{2}} + \frac{30}{19} \, B a^{2} b^{4} x^{\frac{19}{2}} + \frac{12}{19} \, A a b^{5} x^{\frac{19}{2}} + \frac{40}{17} \, B a^{3} b^{3} x^{\frac{17}{2}} + \frac{30}{17} \, A a^{2} b^{4} x^{\frac{17}{2}} + 2 \, B a^{4} b^{2} x^{\frac{15}{2}} + \frac{8}{3} \, A a^{3} b^{3} x^{\frac{15}{2}} + \frac{12}{13} \, B a^{5} b x^{\frac{13}{2}} + \frac{30}{13} \, A a^{4} b^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B a^{6} x^{\frac{11}{2}} + \frac{12}{11} \, A a^{5} b x^{\frac{11}{2}} + \frac{2}{9} \, A a^{6} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]