3.66 \(\int (b+a x)^2 \log (x) \, dx\)

Optimal. Leaf size=54 \[ -\frac{a^2 x^3}{9}-\frac{b^3 \log (x)}{3 a}-\frac{1}{2} a b x^2+\frac{\log (x) (a x+b)^3}{3 a}-b^2 x \]

[Out]

-(b^2*x) - (a*b*x^2)/2 - (a^2*x^3)/9 - (b^3*Log[x])/(3*a) + ((b + a*x)^3*Log[x])
/(3*a)

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Rubi [A]  time = 0.0522001, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{a^2 x^3}{9}-\frac{b^3 \log (x)}{3 a}-\frac{1}{2} a b x^2+\frac{\log (x) (a x+b)^3}{3 a}-b^2 x \]

Antiderivative was successfully verified.

[In]  Int[(b + a*x)^2*Log[x],x]

[Out]

-(b^2*x) - (a*b*x^2)/2 - (a^2*x^3)/9 - (b^3*Log[x])/(3*a) + ((b + a*x)^3*Log[x])
/(3*a)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2} x^{3}}{9} - a b \int x\, dx - b^{2} x - \frac{b^{3} \log{\left (x \right )}}{3 a} + \frac{\left (a x + b\right )^{3} \log{\left (x \right )}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x+b)**2*ln(x),x)

[Out]

-a**2*x**3/9 - a*b*Integral(x, x) - b**2*x - b**3*log(x)/(3*a) + (a*x + b)**3*lo
g(x)/(3*a)

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Mathematica [A]  time = 0.0143132, size = 46, normalized size = 0.85 \[ \frac{1}{18} x \left (6 \log (x) \left (a^2 x^2+3 a b x+3 b^2\right )-2 a^2 x^2-9 a b x-18 b^2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b + a*x)^2*Log[x],x]

[Out]

(x*(-18*b^2 - 9*a*b*x - 2*a^2*x^2 + 6*(3*b^2 + 3*a*b*x + a^2*x^2)*Log[x]))/18

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Maple [A]  time = 0.001, size = 48, normalized size = 0.9 \[{\frac{{a}^{2}{x}^{3}\ln \left ( x \right ) }{3}}-{\frac{{a}^{2}{x}^{3}}{9}}+ab{x}^{2}\ln \left ( x \right ) -{\frac{ab{x}^{2}}{2}}+\ln \left ( x \right ) x{b}^{2}-{b}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x+b)^2*ln(x),x)

[Out]

1/3*a^2*x^3*ln(x)-1/9*a^2*x^3+a*b*x^2*ln(x)-1/2*a*b*x^2+ln(x)*x*b^2-b^2*x

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Maxima [A]  time = 1.42692, size = 63, normalized size = 1.17 \[ -\frac{1}{9} \, a^{2} x^{3} - \frac{1}{2} \, a b x^{2} - b^{2} x + \frac{1}{3} \,{\left (a^{2} x^{3} + 3 \, a b x^{2} + 3 \, b^{2} x\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*a^2*x^3 - 1/2*a*b*x^2 - b^2*x + 1/3*(a^2*x^3 + 3*a*b*x^2 + 3*b^2*x)*log(x)

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Fricas [A]  time = 0.204077, size = 63, normalized size = 1.17 \[ -\frac{1}{9} \, a^{2} x^{3} - \frac{1}{2} \, a b x^{2} - b^{2} x + \frac{1}{3} \,{\left (a^{2} x^{3} + 3 \, a b x^{2} + 3 \, b^{2} x\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*a^2*x^3 - 1/2*a*b*x^2 - b^2*x + 1/3*(a^2*x^3 + 3*a*b*x^2 + 3*b^2*x)*log(x)

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Sympy [A]  time = 0.132458, size = 44, normalized size = 0.81 \[ - \frac{a^{2} x^{3}}{9} - \frac{a b x^{2}}{2} - b^{2} x + \left (\frac{a^{2} x^{3}}{3} + a b x^{2} + b^{2} x\right ) \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x+b)**2*ln(x),x)

[Out]

-a**2*x**3/9 - a*b*x**2/2 - b**2*x + (a**2*x**3/3 + a*b*x**2 + b**2*x)*log(x)

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GIAC/XCAS [A]  time = 0.199139, size = 63, normalized size = 1.17 \[ \frac{1}{3} \, a^{2} x^{3}{\rm ln}\left (x\right ) - \frac{1}{9} \, a^{2} x^{3} + a b x^{2}{\rm ln}\left (x\right ) - \frac{1}{2} \, a b x^{2} + b^{2} x{\rm ln}\left (x\right ) - b^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^2*x^3*ln(x) - 1/9*a^2*x^3 + a*b*x^2*ln(x) - 1/2*a*b*x^2 + b^2*x*ln(x) - b^
2*x