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

Optimal. Leaf size=40 \[ A b^2 \log (x)+2 A b c x+\frac{1}{2} A c^2 x^2+\frac{B (b+c x)^3}{3 c} \]

[Out]

2*A*b*c*x + (A*c^2*x^2)/2 + (B*(b + c*x)^3)/(3*c) + A*b^2*Log[x]

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Rubi [A]  time = 0.0536276, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ A b^2 \log (x)+2 A b c x+\frac{1}{2} A c^2 x^2+\frac{B (b+c x)^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

2*A*b*c*x + (A*c^2*x^2)/2 + (B*(b + c*x)^3)/(3*c) + A*b^2*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0268239, size = 43, normalized size = 1.08 \[ A b^2 \log (x)+b c x (2 A+B x)+\frac{1}{6} c^2 x^2 (3 A+2 B x)+b^2 B x \]

Antiderivative was successfully verified.

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

[Out]

b^2*B*x + b*c*x*(2*A + B*x) + (c^2*x^2*(3*A + 2*B*x))/6 + A*b^2*Log[x]

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Maple [A]  time = 0.003, size = 46, normalized size = 1.2 \[{\frac{B{c}^{2}{x}^{3}}{3}}+{\frac{A{c}^{2}{x}^{2}}{2}}+B{x}^{2}bc+2\,Abcx+{b}^{2}Bx+A{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.69917, size = 62, normalized size = 1.55 \[ \frac{1}{3} \, B c^{2} x^{3} + A b^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B b c + A c^{2}\right )} x^{2} +{\left (B b^{2} + 2 \, A b c\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.270684, size = 62, normalized size = 1.55 \[ \frac{1}{3} \, B c^{2} x^{3} + A b^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B b c + A c^{2}\right )} x^{2} +{\left (B b^{2} + 2 \, A b c\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.21555, size = 46, normalized size = 1.15 \[ A b^{2} \log{\left (x \right )} + \frac{B c^{2} x^{3}}{3} + x^{2} \left (\frac{A c^{2}}{2} + B b c\right ) + x \left (2 A b c + B b^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.266865, size = 62, normalized size = 1.55 \[ \frac{1}{3} \, B c^{2} x^{3} + B b c x^{2} + \frac{1}{2} \, A c^{2} x^{2} + B b^{2} x + 2 \, A b c x + A b^{2}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*B*c^2*x^3 + B*b*c*x^2 + 1/2*A*c^2*x^2 + B*b^2*x + 2*A*b*c*x + A*b^2*ln(abs(x
))