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

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

[Out]

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

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Rubi [A]  time = 0.0891105, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{1}{2} A b^2 x^2+\frac{1}{4} c x^4 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{5} B c^2 x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A b^{2} \int x\, dx + \frac{B c^{2} x^{5}}{5} + \frac{b x^{3} \left (2 A c + B b\right )}{3} + \frac{c x^{4} \left (A c + 2 B b\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0235507, size = 49, normalized size = 0.89 \[ \frac{1}{60} x^2 \left (30 A b^2+15 c x^2 (A c+2 b B)+20 b x (2 A c+b B)+12 B c^2 x^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(30*A*b^2 + 20*b*(b*B + 2*A*c)*x + 15*c*(2*b*B + A*c)*x^2 + 12*B*c^2*x^3))/
60

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Maple [A]  time = 0.002, size = 52, normalized size = 1. \[{\frac{B{c}^{2}{x}^{5}}{5}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,Abc+{b}^{2}B \right ){x}^{3}}{3}}+{\frac{A{b}^{2}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2/x,x)

[Out]

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

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Maxima [A]  time = 0.698245, size = 69, normalized size = 1.25 \[ \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{2} \, A b^{2} x^{2} + \frac{1}{4} \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{2} + 2 \, A b c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.269407, size = 69, normalized size = 1.25 \[ \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{2} \, A b^{2} x^{2} + \frac{1}{4} \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b^{2} + 2 \, A b c\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.121529, size = 54, normalized size = 0.98 \[ \frac{A b^{2} x^{2}}{2} + \frac{B c^{2} x^{5}}{5} + x^{4} \left (\frac{A c^{2}}{4} + \frac{B b c}{2}\right ) + x^{3} \left (\frac{2 A b c}{3} + \frac{B b^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.265737, size = 72, normalized size = 1.31 \[ \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{2} \, B b c x^{4} + \frac{1}{4} \, A c^{2} x^{4} + \frac{1}{3} \, B b^{2} x^{3} + \frac{2}{3} \, A b c x^{3} + \frac{1}{2} \, A b^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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