Optimal. Leaf size=113 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{13} x^{13/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{11} x^{11/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{17} B c^2 x^{17/2} \]
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Rubi [A] time = 0.149938, 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 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{13} x^{13/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{11} x^{11/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{17} B c^2 x^{17/2} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)*(A + B*x)*(a + b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 19.6744, size = 124, normalized size = 1.1 \[ \frac{2 A a^{2} x^{\frac{7}{2}}}{7} + \frac{2 B c^{2} x^{\frac{17}{2}}}{17} + \frac{2 a x^{\frac{9}{2}} \left (2 A b + B a\right )}{9} + \frac{2 c x^{\frac{15}{2}} \left (A c + 2 B b\right )}{15} + x^{\frac{13}{2}} \left (\frac{4 A b c}{13} + \frac{4 B a c}{13} + \frac{2 B b^{2}}{13}\right ) + x^{\frac{11}{2}} \left (\frac{4 A a c}{11} + \frac{2 A b^{2}}{11} + \frac{4 B a b}{11}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)*(B*x+A)*(c*x**2+b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.0813931, size = 113, normalized size = 1. \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{13} x^{13/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac{2}{11} x^{11/2} \left (2 a A c+2 a b B+A b^2\right )+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{17} B c^2 x^{17/2} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)*(A + B*x)*(a + b*x + c*x^2)^2,x]
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Maple [A] time = 0.01, size = 102, normalized size = 0.9 \[{\frac{90090\,B{c}^{2}{x}^{5}+102102\,A{c}^{2}{x}^{4}+204204\,B{x}^{4}bc+235620\,A{x}^{3}bc+235620\,aBc{x}^{3}+117810\,B{b}^{2}{x}^{3}+278460\,aAc{x}^{2}+139230\,A{b}^{2}{x}^{2}+278460\,B{x}^{2}ab+340340\,aAbx+170170\,{a}^{2}Bx+218790\,A{a}^{2}}{765765}{x}^{{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)*(B*x+A)*(c*x^2+b*x+a)^2,x)
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Maxima [A] time = 0.711498, size = 126, normalized size = 1.12 \[ \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{2}{15} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{15}{2}} + \frac{2}{13} \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{\frac{13}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{9}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)*x^(5/2),x, algorithm="maxima")
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Fricas [A] time = 0.317244, size = 132, normalized size = 1.17 \[ \frac{2}{765765} \,{\left (45045 \, B c^{2} x^{8} + 51051 \,{\left (2 \, B b c + A c^{2}\right )} x^{7} + 58905 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{6} + 109395 \, A a^{2} x^{3} + 69615 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{5} + 85085 \,{\left (B a^{2} + 2 \, A a b\right )} x^{4}\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)*x^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 23.4037, size = 162, normalized size = 1.43 \[ \frac{2 A a^{2} x^{\frac{7}{2}}}{7} + \frac{4 A a b x^{\frac{9}{2}}}{9} + \frac{4 A a c x^{\frac{11}{2}}}{11} + \frac{2 A b^{2} x^{\frac{11}{2}}}{11} + \frac{4 A b c x^{\frac{13}{2}}}{13} + \frac{2 A c^{2} x^{\frac{15}{2}}}{15} + \frac{2 B a^{2} x^{\frac{9}{2}}}{9} + \frac{4 B a b x^{\frac{11}{2}}}{11} + \frac{4 B a c x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} + \frac{4 B b c x^{\frac{15}{2}}}{15} + \frac{2 B c^{2} x^{\frac{17}{2}}}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(5/2)*(B*x+A)*(c*x**2+b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.271013, size = 139, normalized size = 1.23 \[ \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{4}{15} \, B b c x^{\frac{15}{2}} + \frac{2}{15} \, A c^{2} x^{\frac{15}{2}} + \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{13} \, B a c x^{\frac{13}{2}} + \frac{4}{13} \, A b c x^{\frac{13}{2}} + \frac{4}{11} \, B a b x^{\frac{11}{2}} + \frac{2}{11} \, A b^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A a c x^{\frac{11}{2}} + \frac{2}{9} \, B a^{2} x^{\frac{9}{2}} + \frac{4}{9} \, A a b x^{\frac{9}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2*(B*x + A)*x^(5/2),x, algorithm="giac")
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