3.149 \(\int x^{7/2} (A+B x) \left (b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{15} b x^{15/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]

[Out]

(2*A*b^2*x^(13/2))/13 + (2*b*(b*B + 2*A*c)*x^(15/2))/15 + (2*c*(2*b*B + A*c)*x^(
17/2))/17 + (2*B*c^2*x^(19/2))/19

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Rubi [A]  time = 0.0982191, 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 \[ \frac{2}{13} A b^2 x^{13/2}+\frac{2}{17} c x^{17/2} (A c+2 b B)+\frac{2}{15} b x^{15/2} (2 A c+b B)+\frac{2}{19} B c^2 x^{19/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(2*A*b^2*x^(13/2))/13 + (2*b*(b*B + 2*A*c)*x^(15/2))/15 + (2*c*(2*b*B + A*c)*x^(
17/2))/17 + (2*B*c^2*x^(19/2))/19

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Rubi in Sympy [A]  time = 10.3204, size = 63, normalized size = 1. \[ \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} + \frac{2 b x^{\frac{15}{2}} \left (2 A c + B b\right )}{15} + \frac{2 c x^{\frac{17}{2}} \left (A c + 2 B b\right )}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(B*x+A)*(c*x**2+b*x)**2,x)

[Out]

2*A*b**2*x**(13/2)/13 + 2*B*c**2*x**(19/2)/19 + 2*b*x**(15/2)*(2*A*c + B*b)/15 +
2*c*x**(17/2)*(A*c + 2*B*b)/17

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Mathematica [A]  time = 0.0361024, size = 51, normalized size = 0.81 \[ \frac{2 x^{13/2} \left (4845 A b^2+3705 c x^2 (A c+2 b B)+4199 b x (2 A c+b B)+3315 B c^2 x^3\right )}{62985} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

(2*x^(13/2)*(4845*A*b^2 + 4199*b*(b*B + 2*A*c)*x + 3705*c*(2*b*B + A*c)*x^2 + 33
15*B*c^2*x^3))/62985

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Maple [A]  time = 0.007, size = 52, normalized size = 0.8 \[{\frac{6630\,B{c}^{2}{x}^{3}+7410\,A{c}^{2}{x}^{2}+14820\,B{x}^{2}bc+16796\,Abcx+8398\,{b}^{2}Bx+9690\,{b}^{2}A}{62985}{x}^{{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(B*x+A)*(c*x^2+b*x)^2,x)

[Out]

2/62985*x^(13/2)*(3315*B*c^2*x^3+3705*A*c^2*x^2+7410*B*b*c*x^2+8398*A*b*c*x+4199
*B*b^2*x+4845*A*b^2)

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Maxima [A]  time = 0.679714, size = 69, normalized size = 1.1 \[ \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} + \frac{2}{17} \,{\left (2 \, B b c + A c^{2}\right )} x^{\frac{17}{2}} + \frac{2}{15} \,{\left (B b^{2} + 2 \, A b c\right )} x^{\frac{15}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*x^(7/2),x, algorithm="maxima")

[Out]

2/19*B*c^2*x^(19/2) + 2/13*A*b^2*x^(13/2) + 2/17*(2*B*b*c + A*c^2)*x^(17/2) + 2/
15*(B*b^2 + 2*A*b*c)*x^(15/2)

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Fricas [A]  time = 0.302097, size = 76, normalized size = 1.21 \[ \frac{2}{62985} \,{\left (3315 \, B c^{2} x^{9} + 4845 \, A b^{2} x^{6} + 3705 \,{\left (2 \, B b c + A c^{2}\right )} x^{8} + 4199 \,{\left (B b^{2} + 2 \, A b c\right )} x^{7}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*x^(7/2),x, algorithm="fricas")

[Out]

2/62985*(3315*B*c^2*x^9 + 4845*A*b^2*x^6 + 3705*(2*B*b*c + A*c^2)*x^8 + 4199*(B*
b^2 + 2*A*b*c)*x^7)*sqrt(x)

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Sympy [A]  time = 43.5575, size = 80, normalized size = 1.27 \[ \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{4 A b c x^{\frac{15}{2}}}{15} + \frac{2 A c^{2} x^{\frac{17}{2}}}{17} + \frac{2 B b^{2} x^{\frac{15}{2}}}{15} + \frac{4 B b c x^{\frac{17}{2}}}{17} + \frac{2 B c^{2} x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)*(B*x+A)*(c*x**2+b*x)**2,x)

[Out]

2*A*b**2*x**(13/2)/13 + 4*A*b*c*x**(15/2)/15 + 2*A*c**2*x**(17/2)/17 + 2*B*b**2*
x**(15/2)/15 + 4*B*b*c*x**(17/2)/17 + 2*B*c**2*x**(19/2)/19

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GIAC/XCAS [A]  time = 0.26891, size = 72, normalized size = 1.14 \[ \frac{2}{19} \, B c^{2} x^{\frac{19}{2}} + \frac{4}{17} \, B b c x^{\frac{17}{2}} + \frac{2}{17} \, A c^{2} x^{\frac{17}{2}} + \frac{2}{15} \, B b^{2} x^{\frac{15}{2}} + \frac{4}{15} \, A b c x^{\frac{15}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)*x^(7/2),x, algorithm="giac")

[Out]

2/19*B*c^2*x^(19/2) + 4/17*B*b*c*x^(17/2) + 2/17*A*c^2*x^(17/2) + 2/15*B*b^2*x^(
15/2) + 4/15*A*b*c*x^(15/2) + 2/13*A*b^2*x^(13/2)