3.65 \(\int \frac{(a+b x)^2 (A+B x)}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0372604, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ a^2 A \log (x)+2 a A b x+\frac{B (a+b x)^3}{3 b}+\frac{1}{2} A b^2 x^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0252649, size = 43, normalized size = 1.08 \[ a^2 A \log (x)+a^2 B x+a b x (2 A+B x)+\frac{1}{6} b^2 x^2 (3 A+2 B x) \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(B*x+A)/x,x)

[Out]

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

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Maxima [A]  time = 1.33049, size = 62, normalized size = 1.55 \[ \frac{1}{3} \, B b^{2} x^{3} + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B a b + A b^{2}\right )} x^{2} +{\left (B a^{2} + 2 \, A a b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.200406, size = 62, normalized size = 1.55 \[ \frac{1}{3} \, B b^{2} x^{3} + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, B a b + A b^{2}\right )} x^{2} +{\left (B a^{2} + 2 \, A a b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.23805, size = 46, normalized size = 1.15 \[ A a^{2} \log{\left (x \right )} + \frac{B b^{2} x^{3}}{3} + x^{2} \left (\frac{A b^{2}}{2} + B a b\right ) + x \left (2 A a b + B a^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.253532, size = 62, normalized size = 1.55 \[ \frac{1}{3} \, B b^{2} x^{3} + B a b x^{2} + \frac{1}{2} \, A b^{2} x^{2} + B a^{2} x + 2 \, A a b x + A a^{2}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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