3.1118 \(\int (A+B x) \left (b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{3} A b^2 x^3+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{4} b x^4 (2 A c+b B)+\frac{1}{6} B c^2 x^6 \]

[Out]

(A*b^2*x^3)/3 + (b*(b*B + 2*A*c)*x^4)/4 + (c*(2*b*B + A*c)*x^5)/5 + (B*c^2*x^6)/
6

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Rubi [A]  time = 0.12268, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{1}{3} A b^2 x^3+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{4} b x^4 (2 A c+b B)+\frac{1}{6} B c^2 x^6 \]

Antiderivative was successfully verified.

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

[Out]

(A*b^2*x^3)/3 + (b*(b*B + 2*A*c)*x^4)/4 + (c*(2*b*B + A*c)*x^5)/5 + (B*c^2*x^6)/
6

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Rubi in Sympy [A]  time = 13.6129, size = 49, normalized size = 0.89 \[ \frac{A b^{2} x^{3}}{3} + \frac{B c^{2} x^{6}}{6} + \frac{b x^{4} \left (2 A c + B b\right )}{4} + \frac{c x^{5} \left (A c + 2 B b\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b**2*x**3/3 + B*c**2*x**6/6 + b*x**4*(2*A*c + B*b)/4 + c*x**5*(A*c + 2*B*b)/5

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Mathematica [A]  time = 0.0190358, size = 49, normalized size = 0.89 \[ \frac{1}{60} x^3 \left (20 A b^2+12 c x^2 (A c+2 b B)+15 b x (2 A c+b B)+10 B c^2 x^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(20*A*b^2 + 15*b*(b*B + 2*A*c)*x + 12*c*(2*b*B + A*c)*x^2 + 10*B*c^2*x^3))/
60

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Maple [A]  time = 0.001, size = 52, normalized size = 1. \[{\frac{B{c}^{2}{x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,Abc+{b}^{2}B \right ){x}^{4}}{4}}+{\frac{A{b}^{2}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2,x)

[Out]

1/6*B*c^2*x^6+1/5*(A*c^2+2*B*b*c)*x^5+1/4*(2*A*b*c+B*b^2)*x^4+1/3*A*b^2*x^3

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Maxima [A]  time = 0.69235, size = 69, normalized size = 1.25 \[ \frac{1}{6} \, B c^{2} x^{6} + \frac{1}{3} \, A b^{2} x^{3} + \frac{1}{5} \,{\left (2 \, B b c + A c^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B b^{2} + 2 \, A b c\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*c^2*x^6 + 1/3*A*b^2*x^3 + 1/5*(2*B*b*c + A*c^2)*x^5 + 1/4*(B*b^2 + 2*A*b*c
)*x^4

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Fricas [A]  time = 0.274969, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} c^{2} B + \frac{2}{5} x^{5} c b B + \frac{1}{5} x^{5} c^{2} A + \frac{1}{4} x^{4} b^{2} B + \frac{1}{2} x^{4} c b A + \frac{1}{3} x^{3} b^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^6*c^2*B + 2/5*x^5*c*b*B + 1/5*x^5*c^2*A + 1/4*x^4*b^2*B + 1/2*x^4*c*b*A +
1/3*x^3*b^2*A

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Sympy [A]  time = 0.115798, size = 54, normalized size = 0.98 \[ \frac{A b^{2} x^{3}}{3} + \frac{B c^{2} x^{6}}{6} + x^{5} \left (\frac{A c^{2}}{5} + \frac{2 B b c}{5}\right ) + x^{4} \left (\frac{A b c}{2} + \frac{B b^{2}}{4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*b**2*x**3/3 + B*c**2*x**6/6 + x**5*(A*c**2/5 + 2*B*b*c/5) + x**4*(A*b*c/2 + B*
b**2/4)

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GIAC/XCAS [A]  time = 0.277479, size = 72, normalized size = 1.31 \[ \frac{1}{6} \, B c^{2} x^{6} + \frac{2}{5} \, B b c x^{5} + \frac{1}{5} \, A c^{2} x^{5} + \frac{1}{4} \, B b^{2} x^{4} + \frac{1}{2} \, A b c x^{4} + \frac{1}{3} \, A b^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*B*c^2*x^6 + 2/5*B*b*c*x^5 + 1/5*A*c^2*x^5 + 1/4*B*b^2*x^4 + 1/2*A*b*c*x^4 +
1/3*A*b^2*x^3