3.150 \(\int x^{5/2} (A+B x) \left (b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{11} A b^2 x^{11/2}+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{13} b x^{13/2} (2 A c+b B)+\frac{2}{17} B c^2 x^{17/2} \]

[Out]

(2*A*b^2*x^(11/2))/11 + (2*b*(b*B + 2*A*c)*x^(13/2))/13 + (2*c*(2*b*B + A*c)*x^(
15/2))/15 + (2*B*c^2*x^(17/2))/17

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Rubi [A]  time = 0.0957146, 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 \[ \frac{2}{11} A b^2 x^{11/2}+\frac{2}{15} c x^{15/2} (A c+2 b B)+\frac{2}{13} b x^{13/2} (2 A c+b B)+\frac{2}{17} B c^2 x^{17/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(2*A*b^2*x^(11/2))/11 + (2*b*(b*B + 2*A*c)*x^(13/2))/13 + (2*c*(2*b*B + A*c)*x^(
15/2))/15 + (2*B*c^2*x^(17/2))/17

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Rubi in Sympy [A]  time = 10.257, size = 63, normalized size = 1. \[ \frac{2 A b^{2} x^{\frac{11}{2}}}{11} + \frac{2 B c^{2} x^{\frac{17}{2}}}{17} + \frac{2 b x^{\frac{13}{2}} \left (2 A c + B b\right )}{13} + \frac{2 c x^{\frac{15}{2}} \left (A c + 2 B b\right )}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(B*x+A)*(c*x**2+b*x)**2,x)

[Out]

2*A*b**2*x**(11/2)/11 + 2*B*c**2*x**(17/2)/17 + 2*b*x**(13/2)*(2*A*c + B*b)/13 +
2*c*x**(15/2)*(A*c + 2*B*b)/15

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Mathematica [A]  time = 0.0331458, size = 51, normalized size = 0.81 \[ \frac{2 x^{11/2} \left (3315 A b^2+2431 c x^2 (A c+2 b B)+2805 b x (2 A c+b B)+2145 B c^2 x^3\right )}{36465} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(2*x^(11/2)*(3315*A*b^2 + 2805*b*(b*B + 2*A*c)*x + 2431*c*(2*b*B + A*c)*x^2 + 21
45*B*c^2*x^3))/36465

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Maple [A]  time = 0.01, size = 52, normalized size = 0.8 \[{\frac{4290\,B{c}^{2}{x}^{3}+4862\,A{c}^{2}{x}^{2}+9724\,B{x}^{2}bc+11220\,Abcx+5610\,{b}^{2}Bx+6630\,{b}^{2}A}{36465}{x}^{{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)*(c*x^2+b*x)^2,x)

[Out]

2/36465*x^(11/2)*(2145*B*c^2*x^3+2431*A*c^2*x^2+4862*B*b*c*x^2+5610*A*b*c*x+2805
*B*b^2*x+3315*A*b^2)

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Maxima [A]  time = 0.675935, size = 69, normalized size = 1.1 \[ \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{2}{11} \, A b^{2} x^{\frac{11}{2}} + \frac{2}{15} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{15}{2}} + \frac{2}{13} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*x^(5/2),x, algorithm="maxima")

[Out]

2/17*B*c^2*x^(17/2) + 2/11*A*b^2*x^(11/2) + 2/15*(2*B*b*c + A*c^2)*x^(15/2) + 2/
13*(B*b^2 + 2*A*b*c)*x^(13/2)

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Fricas [A]  time = 0.286772, size = 76, normalized size = 1.21 \[ \frac{2}{36465} \,{\left (2145 \, B c^{2} x^{8} + 3315 \, A b^{2} x^{5} + 2431 \,{\left (2 \, B b c + A c^{2}\right )} x^{7} + 2805 \,{\left (B b^{2} + 2 \, A b c\right )} x^{6}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*x^(5/2),x, algorithm="fricas")

[Out]

2/36465*(2145*B*c^2*x^8 + 3315*A*b^2*x^5 + 2431*(2*B*b*c + A*c^2)*x^7 + 2805*(B*
b^2 + 2*A*b*c)*x^6)*sqrt(x)

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Sympy [A]  time = 21.6007, size = 80, normalized size = 1.27 \[ \frac{2 A b^{2} x^{\frac{11}{2}}}{11} + \frac{4 A b c x^{\frac{13}{2}}}{13} + \frac{2 A c^{2} x^{\frac{15}{2}}}{15} + \frac{2 B b^{2} x^{\frac{13}{2}}}{13} + \frac{4 B b c x^{\frac{15}{2}}}{15} + \frac{2 B c^{2} x^{\frac{17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(B*x+A)*(c*x**2+b*x)**2,x)

[Out]

2*A*b**2*x**(11/2)/11 + 4*A*b*c*x**(13/2)/13 + 2*A*c**2*x**(15/2)/15 + 2*B*b**2*
x**(13/2)/13 + 4*B*b*c*x**(15/2)/15 + 2*B*c**2*x**(17/2)/17

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GIAC/XCAS [A]  time = 0.269156, size = 72, normalized size = 1.14 \[ \frac{2}{17} \, B c^{2} x^{\frac{17}{2}} + \frac{4}{15} \, B b c x^{\frac{15}{2}} + \frac{2}{15} \, A c^{2} x^{\frac{15}{2}} + \frac{2}{13} \, B b^{2} x^{\frac{13}{2}} + \frac{4}{13} \, A b c x^{\frac{13}{2}} + \frac{2}{11} \, A b^{2} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*x^(5/2),x, algorithm="giac")

[Out]

2/17*B*c^2*x^(17/2) + 4/15*B*b*c*x^(15/2) + 2/15*A*c^2*x^(15/2) + 2/13*B*b^2*x^(
13/2) + 4/13*A*b*c*x^(13/2) + 2/11*A*b^2*x^(11/2)