Optimal. Leaf size=113 \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{9} x^{9/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{7} x^{7/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{11} c x^{11/2} (A c+2 b B)+\frac{2}{13} B c^2 x^{13/2} \]
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Rubi [A] time = 0.147911, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{2}{3} a^2 A x^{3/2}+\frac{2}{9} x^{9/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{7} x^{7/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{5} a x^{5/2} (a B+2 A b)+\frac{2}{11} c x^{11/2} (A c+2 b B)+\frac{2}{13} B c^2 x^{13/2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(A + B*x)*(a + b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 19.6446, size = 124, normalized size = 1.1 \[ \frac{2 A a^{2} x^{\frac{3}{2}}}{3} + \frac{2 B c^{2} x^{\frac{13}{2}}}{13} + \frac{2 a x^{\frac{5}{2}} \left (2 A b + B a\right )}{5} + \frac{2 c x^{\frac{11}{2}} \left (A c + 2 B b\right )}{11} + x^{\frac{9}{2}} \left (\frac{4 A b c}{9} + \frac{4 B a c}{9} + \frac{2 B b^{2}}{9}\right ) + x^{\frac{7}{2}} \left (\frac{4 A a c}{7} + \frac{2 A b^{2}}{7} + \frac{4 B a b}{7}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)**2*x**(1/2),x)
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Mathematica [A] time = 0.0968873, size = 93, normalized size = 0.82 \[ \frac{2 x^{3/2} \left (15015 a^2 A+5005 x^3 \left (2 a B c+2 A b c+b^2 B\right )+6435 x^2 \left (A \left (2 a c+b^2\right )+2 a b B\right )+9009 a x (a B+2 A b)+4095 c x^4 (A c+2 b B)+3465 B c^2 x^5\right )}{45045} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*(A + B*x)*(a + b*x + c*x^2)^2,x]
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Maple [A] time = 0.009, size = 102, normalized size = 0.9 \[{\frac{6930\,B{c}^{2}{x}^{5}+8190\,A{c}^{2}{x}^{4}+16380\,B{x}^{4}bc+20020\,A{x}^{3}bc+20020\,aBc{x}^{3}+10010\,B{b}^{2}{x}^{3}+25740\,aAc{x}^{2}+12870\,A{b}^{2}{x}^{2}+25740\,B{x}^{2}ab+36036\,aAbx+18018\,{a}^{2}Bx+30030\,A{a}^{2}}{45045}{x}^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)^2*x^(1/2),x)
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Maxima [A] time = 0.710764, size = 126, normalized size = 1.12 \[ \frac{2}{13} \, B c^{2} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{\frac{9}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{2}} + \frac{2}{7} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)*sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.31119, size = 130, normalized size = 1.15 \[ \frac{2}{45045} \,{\left (3465 \, B c^{2} x^{6} + 4095 \,{\left (2 \, B b c + A c^{2}\right )} x^{5} + 5005 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{4} + 15015 \, A a^{2} x + 6435 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{3} + 9009 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)*sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.20251, size = 121, normalized size = 1.07 \[ \frac{2 A a^{2} x^{\frac{3}{2}}}{3} + \frac{2 B c^{2} x^{\frac{13}{2}}}{13} + \frac{2 x^{\frac{11}{2}} \left (A c^{2} + 2 B b c\right )}{11} + \frac{2 x^{\frac{9}{2}} \left (2 A b c + 2 B a c + B b^{2}\right )}{9} + \frac{2 x^{\frac{7}{2}} \left (2 A a c + A b^{2} + 2 B a b\right )}{7} + \frac{2 x^{\frac{5}{2}} \left (2 A a b + B a^{2}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)**2*x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.269911, size = 139, normalized size = 1.23 \[ \frac{2}{13} \, B c^{2} x^{\frac{13}{2}} + \frac{4}{11} \, B b c x^{\frac{11}{2}} + \frac{2}{11} \, A c^{2} x^{\frac{11}{2}} + \frac{2}{9} \, B b^{2} x^{\frac{9}{2}} + \frac{4}{9} \, B a c x^{\frac{9}{2}} + \frac{4}{9} \, A b c x^{\frac{9}{2}} + \frac{4}{7} \, B a b x^{\frac{7}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + \frac{4}{7} \, A a c x^{\frac{7}{2}} + \frac{2}{5} \, B a^{2} x^{\frac{5}{2}} + \frac{4}{5} \, A a b x^{\frac{5}{2}} + \frac{2}{3} \, A a^{2} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)*sqrt(x),x, algorithm="giac")
[Out]