Optimal. Leaf size=159 \[ \frac{2}{3} a^6 A x^{3/2}+\frac{2}{5} a^5 x^{5/2} (a B+6 A b)+\frac{6}{7} a^4 b x^{7/2} (2 a B+5 A b)+\frac{10}{9} a^3 b^2 x^{9/2} (3 a B+4 A b)+\frac{10}{11} a^2 b^3 x^{11/2} (4 a B+3 A b)+\frac{2}{15} b^5 x^{15/2} (6 a B+A b)+\frac{6}{13} a b^4 x^{13/2} (5 a B+2 A b)+\frac{2}{17} b^6 B x^{17/2} \]
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Rubi [A] time = 0.193634, 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 \[ \frac{2}{3} a^6 A x^{3/2}+\frac{2}{5} a^5 x^{5/2} (a B+6 A b)+\frac{6}{7} a^4 b x^{7/2} (2 a B+5 A b)+\frac{10}{9} a^3 b^2 x^{9/2} (3 a B+4 A b)+\frac{10}{11} a^2 b^3 x^{11/2} (4 a B+3 A b)+\frac{2}{15} b^5 x^{15/2} (6 a B+A b)+\frac{6}{13} a b^4 x^{13/2} (5 a B+2 A b)+\frac{2}{17} b^6 B x^{17/2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
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Rubi in Sympy [A] time = 42.1793, size = 165, normalized size = 1.04 \[ \frac{2 A a^{6} x^{\frac{3}{2}}}{3} + \frac{2 B b^{6} x^{\frac{17}{2}}}{17} + \frac{2 a^{5} x^{\frac{5}{2}} \left (6 A b + B a\right )}{5} + \frac{6 a^{4} b x^{\frac{7}{2}} \left (5 A b + 2 B a\right )}{7} + \frac{10 a^{3} b^{2} x^{\frac{9}{2}} \left (4 A b + 3 B a\right )}{9} + \frac{10 a^{2} b^{3} x^{\frac{11}{2}} \left (3 A b + 4 B a\right )}{11} + \frac{6 a b^{4} x^{\frac{13}{2}} \left (2 A b + 5 B a\right )}{13} + \frac{2 b^{5} x^{\frac{15}{2}} \left (A b + 6 B a\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3*x**(1/2),x)
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Mathematica [A] time = 0.0636334, size = 159, normalized size = 1. \[ \frac{2}{3} a^6 A x^{3/2}+\frac{2}{5} a^5 x^{5/2} (a B+6 A b)+\frac{6}{7} a^4 b x^{7/2} (2 a B+5 A b)+\frac{10}{9} a^3 b^2 x^{9/2} (3 a B+4 A b)+\frac{10}{11} a^2 b^3 x^{11/2} (4 a B+3 A b)+\frac{2}{15} b^5 x^{15/2} (6 a B+A b)+\frac{6}{13} a b^4 x^{13/2} (5 a B+2 A b)+\frac{2}{17} b^6 B x^{17/2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
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Maple [A] time = 0.011, size = 148, normalized size = 0.9 \[{\frac{90090\,B{b}^{6}{x}^{7}+102102\,A{b}^{6}{x}^{6}+612612\,B{x}^{6}a{b}^{5}+706860\,aA{b}^{5}{x}^{5}+1767150\,B{x}^{5}{a}^{2}{b}^{4}+2088450\,{a}^{2}A{b}^{4}{x}^{4}+2784600\,B{x}^{4}{a}^{3}{b}^{3}+3403400\,{a}^{3}A{b}^{3}{x}^{3}+2552550\,B{x}^{3}{a}^{4}{b}^{2}+3281850\,{a}^{4}A{b}^{2}{x}^{2}+1312740\,B{x}^{2}{a}^{5}b+1837836\,{a}^{5}Abx+306306\,B{a}^{6}x+510510\,A{a}^{6}}{765765}{x}^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3*x^(1/2),x)
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Maxima [A] time = 0.685035, size = 198, normalized size = 1.25 \[ \frac{2}{17} \, B b^{6} x^{\frac{17}{2}} + \frac{2}{3} \, A a^{6} x^{\frac{3}{2}} + \frac{2}{15} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{15}{2}} + \frac{6}{13} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac{13}{2}} + \frac{10}{11} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac{11}{2}} + \frac{10}{9} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{\frac{9}{2}} + \frac{6}{7} \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{\frac{5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)*sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.305812, size = 203, normalized size = 1.28 \[ \frac{2}{765765} \,{\left (45045 \, B b^{6} x^{8} + 255255 \, A a^{6} x + 51051 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{7} + 176715 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{6} + 348075 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{5} + 425425 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{4} + 328185 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{3} + 153153 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{2}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)*sqrt(x),x, algorithm="fricas")
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Sympy [A] time = 6.54274, size = 182, normalized size = 1.14 \[ \frac{2 A a^{6} x^{\frac{3}{2}}}{3} + \frac{2 B b^{6} x^{\frac{17}{2}}}{17} + \frac{2 x^{\frac{15}{2}} \left (A b^{6} + 6 B a b^{5}\right )}{15} + \frac{2 x^{\frac{13}{2}} \left (6 A a b^{5} + 15 B a^{2} b^{4}\right )}{13} + \frac{2 x^{\frac{11}{2}} \left (15 A a^{2} b^{4} + 20 B a^{3} b^{3}\right )}{11} + \frac{2 x^{\frac{9}{2}} \left (20 A a^{3} b^{3} + 15 B a^{4} b^{2}\right )}{9} + \frac{2 x^{\frac{7}{2}} \left (15 A a^{4} b^{2} + 6 B a^{5} b\right )}{7} + \frac{2 x^{\frac{5}{2}} \left (6 A a^{5} b + B a^{6}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3*x**(1/2),x)
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GIAC/XCAS [A] time = 0.270207, size = 201, normalized size = 1.26 \[ \frac{2}{17} \, B b^{6} x^{\frac{17}{2}} + \frac{4}{5} \, B a b^{5} x^{\frac{15}{2}} + \frac{2}{15} \, A b^{6} x^{\frac{15}{2}} + \frac{30}{13} \, B a^{2} b^{4} x^{\frac{13}{2}} + \frac{12}{13} \, A a b^{5} x^{\frac{13}{2}} + \frac{40}{11} \, B a^{3} b^{3} x^{\frac{11}{2}} + \frac{30}{11} \, A a^{2} b^{4} x^{\frac{11}{2}} + \frac{10}{3} \, B a^{4} b^{2} x^{\frac{9}{2}} + \frac{40}{9} \, A a^{3} b^{3} x^{\frac{9}{2}} + \frac{12}{7} \, B a^{5} b x^{\frac{7}{2}} + \frac{30}{7} \, A a^{4} b^{2} x^{\frac{7}{2}} + \frac{2}{5} \, B a^{6} x^{\frac{5}{2}} + \frac{12}{5} \, A a^{5} b x^{\frac{5}{2}} + \frac{2}{3} \, A a^{6} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A)*sqrt(x),x, algorithm="giac")
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