3.612 \(\int (a+b x)^3 \log (x) \, dx\)

Optimal. Leaf size=67 \[ -\frac{a^4 \log (x)}{4 b}-a^3 x-\frac{3}{4} a^2 b x^2-\frac{1}{3} a b^2 x^3+\frac{\log (x) (a+b x)^4}{4 b}-\frac{b^3 x^4}{16} \]

[Out]

-(a^3*x) - (3*a^2*b*x^2)/4 - (a*b^2*x^3)/3 - (b^3*x^4)/16 - (a^4*Log[x])/(4*b) +
((a + b*x)^4*Log[x])/(4*b)

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Rubi [A]  time = 0.0664176, antiderivative size = 67, 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 \[ -\frac{a^4 \log (x)}{4 b}-a^3 x-\frac{3}{4} a^2 b x^2-\frac{1}{3} a b^2 x^3+\frac{\log (x) (a+b x)^4}{4 b}-\frac{b^3 x^4}{16} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*Log[x],x]

[Out]

-(a^3*x) - (3*a^2*b*x^2)/4 - (a*b^2*x^3)/3 - (b^3*x^4)/16 - (a^4*Log[x])/(4*b) +
((a + b*x)^4*Log[x])/(4*b)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{4} \log{\left (x \right )}}{4 b} - a^{3} x - \frac{3 a^{2} b \int x\, dx}{2} - \frac{a b^{2} x^{3}}{3} - \frac{b^{3} x^{4}}{16} + \frac{\left (a + b x\right )^{4} \log{\left (x \right )}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*ln(x),x)

[Out]

-a**4*log(x)/(4*b) - a**3*x - 3*a**2*b*Integral(x, x)/2 - a*b**2*x**3/3 - b**3*x
**4/16 + (a + b*x)**4*log(x)/(4*b)

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Mathematica [A]  time = 0.0205071, size = 68, normalized size = 1.01 \[ \frac{1}{48} x \left (-48 a^3-36 a^2 b x+12 \log (x) \left (4 a^3+6 a^2 b x+4 a b^2 x^2+b^3 x^3\right )-16 a b^2 x^2-3 b^3 x^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3*Log[x],x]

[Out]

(x*(-48*a^3 - 36*a^2*b*x - 16*a*b^2*x^2 - 3*b^3*x^3 + 12*(4*a^3 + 6*a^2*b*x + 4*
a*b^2*x^2 + b^3*x^3)*Log[x]))/48

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Maple [A]  time = 0.003, size = 72, normalized size = 1.1 \[{\frac{{b}^{3}{x}^{4}\ln \left ( x \right ) }{4}}-{\frac{{b}^{3}{x}^{4}}{16}}+a{b}^{2}{x}^{3}\ln \left ( x \right ) -{\frac{a{b}^{2}{x}^{3}}{3}}+{\frac{3\,{a}^{2}b{x}^{2}\ln \left ( x \right ) }{2}}-{\frac{3\,{a}^{2}b{x}^{2}}{4}}+\ln \left ( x \right ) x{a}^{3}-{a}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*ln(x),x)

[Out]

1/4*b^3*x^4*ln(x)-1/16*b^3*x^4+a*b^2*x^3*ln(x)-1/3*a*b^2*x^3+3/2*a^2*b*x^2*ln(x)
-3/4*a^2*b*x^2+ln(x)*x*a^3-a^3*x

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Maxima [A]  time = 1.35796, size = 93, normalized size = 1.39 \[ -\frac{1}{16} \, b^{3} x^{4} - \frac{1}{3} \, a b^{2} x^{3} - \frac{3}{4} \, a^{2} b x^{2} - a^{3} x + \frac{1}{4} \,{\left (b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*log(x),x, algorithm="maxima")

[Out]

-1/16*b^3*x^4 - 1/3*a*b^2*x^3 - 3/4*a^2*b*x^2 - a^3*x + 1/4*(b^3*x^4 + 4*a*b^2*x
^3 + 6*a^2*b*x^2 + 4*a^3*x)*log(x)

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Fricas [A]  time = 0.207007, size = 93, normalized size = 1.39 \[ -\frac{1}{16} \, b^{3} x^{4} - \frac{1}{3} \, a b^{2} x^{3} - \frac{3}{4} \, a^{2} b x^{2} - a^{3} x + \frac{1}{4} \,{\left (b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x\right )} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*log(x),x, algorithm="fricas")

[Out]

-1/16*b^3*x^4 - 1/3*a*b^2*x^3 - 3/4*a^2*b*x^2 - a^3*x + 1/4*(b^3*x^4 + 4*a*b^2*x
^3 + 6*a^2*b*x^2 + 4*a^3*x)*log(x)

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Sympy [A]  time = 0.168731, size = 71, normalized size = 1.06 \[ - a^{3} x - \frac{3 a^{2} b x^{2}}{4} - \frac{a b^{2} x^{3}}{3} - \frac{b^{3} x^{4}}{16} + \left (a^{3} x + \frac{3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac{b^{3} x^{4}}{4}\right ) \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*ln(x),x)

[Out]

-a**3*x - 3*a**2*b*x**2/4 - a*b**2*x**3/3 - b**3*x**4/16 + (a**3*x + 3*a**2*b*x*
*2/2 + a*b**2*x**3 + b**3*x**4/4)*log(x)

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GIAC/XCAS [A]  time = 0.231184, size = 96, normalized size = 1.43 \[ \frac{1}{4} \, b^{3} x^{4}{\rm ln}\left (x\right ) - \frac{1}{16} \, b^{3} x^{4} + a b^{2} x^{3}{\rm ln}\left (x\right ) - \frac{1}{3} \, a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2}{\rm ln}\left (x\right ) - \frac{3}{4} \, a^{2} b x^{2} + a^{3} x{\rm ln}\left (x\right ) - a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*log(x),x, algorithm="giac")

[Out]

1/4*b^3*x^4*ln(x) - 1/16*b^3*x^4 + a*b^2*x^3*ln(x) - 1/3*a*b^2*x^3 + 3/2*a^2*b*x
^2*ln(x) - 3/4*a^2*b*x^2 + a^3*x*ln(x) - a^3*x