3.151 \(\int x^{3/2} (A+B x) \left (b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{9} A b^2 x^{9/2}+\frac{2}{13} c x^{13/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{15} B c^2 x^{15/2} \]

[Out]

(2*A*b^2*x^(9/2))/9 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(13
/2))/13 + (2*B*c^2*x^(15/2))/15

_______________________________________________________________________________________

Rubi [A]  time = 0.0950487, 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 \[ \frac{2}{9} A b^2 x^{9/2}+\frac{2}{13} c x^{13/2} (A c+2 b B)+\frac{2}{11} b x^{11/2} (2 A c+b B)+\frac{2}{15} B c^2 x^{15/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(3/2)*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(2*A*b^2*x^(9/2))/9 + (2*b*(b*B + 2*A*c)*x^(11/2))/11 + (2*c*(2*b*B + A*c)*x^(13
/2))/13 + (2*B*c^2*x^(15/2))/15

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.3122, size = 63, normalized size = 1. \[ \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{2 B c^{2} x^{\frac{15}{2}}}{15} + \frac{2 b x^{\frac{11}{2}} \left (2 A c + B b\right )}{11} + \frac{2 c x^{\frac{13}{2}} \left (A c + 2 B b\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(3/2)*(B*x+A)*(c*x**2+b*x)**2,x)

[Out]

2*A*b**2*x**(9/2)/9 + 2*B*c**2*x**(15/2)/15 + 2*b*x**(11/2)*(2*A*c + B*b)/11 + 2
*c*x**(13/2)*(A*c + 2*B*b)/13

_______________________________________________________________________________________

Mathematica [A]  time = 0.0328309, size = 51, normalized size = 0.81 \[ \frac{2 x^{9/2} \left (715 A b^2+495 c x^2 (A c+2 b B)+585 b x (2 A c+b B)+429 B c^2 x^3\right )}{6435} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(3/2)*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(2*x^(9/2)*(715*A*b^2 + 585*b*(b*B + 2*A*c)*x + 495*c*(2*b*B + A*c)*x^2 + 429*B*
c^2*x^3))/6435

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 52, normalized size = 0.8 \[{\frac{858\,B{c}^{2}{x}^{3}+990\,A{c}^{2}{x}^{2}+1980\,B{x}^{2}bc+2340\,Abcx+1170\,{b}^{2}Bx+1430\,{b}^{2}A}{6435}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(3/2)*(B*x+A)*(c*x^2+b*x)^2,x)

[Out]

2/6435*x^(9/2)*(429*B*c^2*x^3+495*A*c^2*x^2+990*B*b*c*x^2+1170*A*b*c*x+585*B*b^2
*x+715*A*b^2)

_______________________________________________________________________________________

Maxima [A]  time = 0.682235, size = 69, normalized size = 1.1 \[ \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{13}{2}} + \frac{2}{11} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*x^(3/2),x, algorithm="maxima")

[Out]

2/15*B*c^2*x^(15/2) + 2/9*A*b^2*x^(9/2) + 2/13*(2*B*b*c + A*c^2)*x^(13/2) + 2/11
*(B*b^2 + 2*A*b*c)*x^(11/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.285079, size = 76, normalized size = 1.21 \[ \frac{2}{6435} \,{\left (429 \, B c^{2} x^{7} + 715 \, A b^{2} x^{4} + 495 \,{\left (2 \, B b c + A c^{2}\right )} x^{6} + 585 \,{\left (B b^{2} + 2 \, A b c\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*x^(3/2),x, algorithm="fricas")

[Out]

2/6435*(429*B*c^2*x^7 + 715*A*b^2*x^4 + 495*(2*B*b*c + A*c^2)*x^6 + 585*(B*b^2 +
2*A*b*c)*x^5)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 10.1276, size = 80, normalized size = 1.27 \[ \frac{2 A b^{2} x^{\frac{9}{2}}}{9} + \frac{4 A b c x^{\frac{11}{2}}}{11} + \frac{2 A c^{2} x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} x^{\frac{11}{2}}}{11} + \frac{4 B b c x^{\frac{13}{2}}}{13} + \frac{2 B c^{2} x^{\frac{15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(3/2)*(B*x+A)*(c*x**2+b*x)**2,x)

[Out]

2*A*b**2*x**(9/2)/9 + 4*A*b*c*x**(11/2)/11 + 2*A*c**2*x**(13/2)/13 + 2*B*b**2*x*
*(11/2)/11 + 4*B*b*c*x**(13/2)/13 + 2*B*c**2*x**(15/2)/15

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.267125, size = 72, normalized size = 1.14 \[ \frac{2}{15} \, B c^{2} x^{\frac{15}{2}} + \frac{4}{13} \, B b c x^{\frac{13}{2}} + \frac{2}{13} \, A c^{2} x^{\frac{13}{2}} + \frac{2}{11} \, B b^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A b c x^{\frac{11}{2}} + \frac{2}{9} \, A b^{2} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*x^(3/2),x, algorithm="giac")

[Out]

2/15*B*c^2*x^(15/2) + 4/13*B*b*c*x^(13/2) + 2/13*A*c^2*x^(13/2) + 2/11*B*b^2*x^(
11/2) + 4/11*A*b*c*x^(11/2) + 2/9*A*b^2*x^(9/2)