3.152 \(\int \sqrt{x} (A+B x) \left (b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{7} A b^2 x^{7/2}+\frac{2}{11} c x^{11/2} (A c+2 b B)+\frac{2}{9} b x^{9/2} (2 A c+b B)+\frac{2}{13} B c^2 x^{13/2} \]

[Out]

(2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*c)*x^(9/2))/9 + (2*c*(2*b*B + A*c)*x^(11/2
))/11 + (2*B*c^2*x^(13/2))/13

_______________________________________________________________________________________

Rubi [A]  time = 0.0915379, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2}{7} A b^2 x^{7/2}+\frac{2}{11} c x^{11/2} (A c+2 b B)+\frac{2}{9} b x^{9/2} (2 A c+b B)+\frac{2}{13} B c^2 x^{13/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(2*A*b^2*x^(7/2))/7 + (2*b*(b*B + 2*A*c)*x^(9/2))/9 + (2*c*(2*b*B + A*c)*x^(11/2
))/11 + (2*B*c^2*x^(13/2))/13

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.3028, size = 63, normalized size = 1. \[ \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + \frac{2 B c^{2} x^{\frac{13}{2}}}{13} + \frac{2 b x^{\frac{9}{2}} \left (2 A c + B b\right )}{9} + \frac{2 c x^{\frac{11}{2}} \left (A c + 2 B b\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**2*x**(1/2),x)

[Out]

2*A*b**2*x**(7/2)/7 + 2*B*c**2*x**(13/2)/13 + 2*b*x**(9/2)*(2*A*c + B*b)/9 + 2*c
*x**(11/2)*(A*c + 2*B*b)/11

_______________________________________________________________________________________

Mathematica [A]  time = 0.0348884, size = 51, normalized size = 0.81 \[ \frac{2 x^{7/2} \left (1287 A b^2+819 c x^2 (A c+2 b B)+1001 b x (2 A c+b B)+693 B c^2 x^3\right )}{9009} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(2*x^(7/2)*(1287*A*b^2 + 1001*b*(b*B + 2*A*c)*x + 819*c*(2*b*B + A*c)*x^2 + 693*
B*c^2*x^3))/9009

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 52, normalized size = 0.8 \[{\frac{1386\,B{c}^{2}{x}^{3}+1638\,A{c}^{2}{x}^{2}+3276\,B{x}^{2}bc+4004\,Abcx+2002\,{b}^{2}Bx+2574\,{b}^{2}A}{9009}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2*x^(1/2),x)

[Out]

2/9009*x^(7/2)*(693*B*c^2*x^3+819*A*c^2*x^2+1638*B*b*c*x^2+2002*A*b*c*x+1001*B*b
^2*x+1287*A*b^2)

_______________________________________________________________________________________

Maxima [A]  time = 0.680598, size = 69, normalized size = 1.1 \[ \frac{2}{13} \, B c^{2} x^{\frac{13}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{11}{2}} + \frac{2}{9} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*sqrt(x),x, algorithm="maxima")

[Out]

2/13*B*c^2*x^(13/2) + 2/7*A*b^2*x^(7/2) + 2/11*(2*B*b*c + A*c^2)*x^(11/2) + 2/9*
(B*b^2 + 2*A*b*c)*x^(9/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.300561, size = 76, normalized size = 1.21 \[ \frac{2}{9009} \,{\left (693 \, B c^{2} x^{6} + 1287 \, A b^{2} x^{3} + 819 \,{\left (2 \, B b c + A c^{2}\right )} x^{5} + 1001 \,{\left (B b^{2} + 2 \, A b c\right )} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*sqrt(x),x, algorithm="fricas")

[Out]

2/9009*(693*B*c^2*x^6 + 1287*A*b^2*x^3 + 819*(2*B*b*c + A*c^2)*x^5 + 1001*(B*b^2
+ 2*A*b*c)*x^4)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 3.78393, size = 66, normalized size = 1.05 \[ \frac{2 A b^{2} x^{\frac{7}{2}}}{7} + \frac{2 B c^{2} x^{\frac{13}{2}}}{13} + \frac{2 x^{\frac{11}{2}} \left (A c^{2} + 2 B b c\right )}{11} + \frac{2 x^{\frac{9}{2}} \left (2 A b c + B b^{2}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**2*x**(1/2),x)

[Out]

2*A*b**2*x**(7/2)/7 + 2*B*c**2*x**(13/2)/13 + 2*x**(11/2)*(A*c**2 + 2*B*b*c)/11
+ 2*x**(9/2)*(2*A*b*c + B*b**2)/9

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270332, size = 72, normalized size = 1.14 \[ \frac{2}{13} \, B c^{2} x^{\frac{13}{2}} + \frac{4}{11} \, B b c x^{\frac{11}{2}} + \frac{2}{11} \, A c^{2} x^{\frac{11}{2}} + \frac{2}{9} \, B b^{2} x^{\frac{9}{2}} + \frac{4}{9} \, A b c x^{\frac{9}{2}} + \frac{2}{7} \, A b^{2} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*sqrt(x),x, algorithm="giac")

[Out]

2/13*B*c^2*x^(13/2) + 4/11*B*b*c*x^(11/2) + 2/11*A*c^2*x^(11/2) + 2/9*B*b^2*x^(9
/2) + 4/9*A*b*c*x^(9/2) + 2/7*A*b^2*x^(7/2)