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

Optimal. Leaf size=59 \[ -\frac{b^2 x}{3 a^2}+\frac{b^3 \log (a x+b)}{3 a^3}+\frac{b x^2}{6 a}+\frac{1}{3} x^3 \log (a x+b)-\frac{x^3}{9} \]

[Out]

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

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Rubi [A]  time = 0.0316671, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2395, 43} \[ -\frac{b^2 x}{3 a^2}+\frac{b^3 \log (a x+b)}{3 a^3}+\frac{b x^2}{6 a}+\frac{1}{3} x^3 \log (a x+b)-\frac{x^3}{9} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Log[b + a*x],x]

[Out]

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

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^2 \log (b+a x) \, dx &=\frac{1}{3} x^3 \log (b+a x)-\frac{1}{3} a \int \frac{x^3}{b+a x} \, dx\\ &=\frac{1}{3} x^3 \log (b+a x)-\frac{1}{3} a \int \left (\frac{b^2}{a^3}-\frac{b x}{a^2}+\frac{x^2}{a}-\frac{b^3}{a^3 (b+a x)}\right ) \, dx\\ &=-\frac{b^2 x}{3 a^2}+\frac{b x^2}{6 a}-\frac{x^3}{9}+\frac{b^3 \log (b+a x)}{3 a^3}+\frac{1}{3} x^3 \log (b+a x)\\ \end{align*}

Mathematica [A]  time = 0.0172276, size = 59, normalized size = 1. \[ -\frac{b^2 x}{3 a^2}+\frac{b^3 \log (a x+b)}{3 a^3}+\frac{b x^2}{6 a}+\frac{1}{3} x^3 \log (a x+b)-\frac{x^3}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Log[b + a*x],x]

[Out]

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

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Maple [A]  time = 0.003, size = 58, normalized size = 1. \begin{align*}{\frac{{x}^{3}\ln \left ( ax+b \right ) }{3}}+{\frac{{b}^{3}\ln \left ( ax+b \right ) }{3\,{a}^{3}}}-{\frac{{x}^{3}}{9}}+{\frac{b{x}^{2}}{6\,a}}-{\frac{{b}^{2}x}{3\,{a}^{2}}}-{\frac{11\,{b}^{3}}{18\,{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*ln(a*x+b),x)

[Out]

1/3*x^3*ln(a*x+b)+1/3*b^3*ln(a*x+b)/a^3-1/9*x^3+1/6*b*x^2/a-1/3*b^2*x/a^2-11/18*b^3/a^3

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Maxima [A]  time = 0.945145, size = 77, normalized size = 1.31 \begin{align*} \frac{1}{3} \, x^{3} \log \left (a x + b\right ) + \frac{1}{18} \, a{\left (\frac{6 \, b^{3} \log \left (a x + b\right )}{a^{4}} - \frac{2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*log(a*x + b) + 1/18*a*(6*b^3*log(a*x + b)/a^4 - (2*a^2*x^3 - 3*a*b*x^2 + 6*b^2*x)/a^3)

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Fricas [A]  time = 1.96776, size = 111, normalized size = 1.88 \begin{align*} -\frac{2 \, a^{3} x^{3} - 3 \, a^{2} b x^{2} + 6 \, a b^{2} x - 6 \,{\left (a^{3} x^{3} + b^{3}\right )} \log \left (a x + b\right )}{18 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(2*a^3*x^3 - 3*a^2*b*x^2 + 6*a*b^2*x - 6*(a^3*x^3 + b^3)*log(a*x + b))/a^3

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Sympy [A]  time = 0.330298, size = 54, normalized size = 0.92 \begin{align*} - a \left (\frac{x^{3}}{9 a} - \frac{b x^{2}}{6 a^{2}} + \frac{b^{2} x}{3 a^{3}} - \frac{b^{3} \log{\left (a x + b \right )}}{3 a^{4}}\right ) + \frac{x^{3} \log{\left (a x + b \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*ln(a*x+b),x)

[Out]

-a*(x**3/(9*a) - b*x**2/(6*a**2) + b**2*x/(3*a**3) - b**3*log(a*x + b)/(3*a**4)) + x**3*log(a*x + b)/3

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Giac [A]  time = 1.07197, size = 127, normalized size = 2.15 \begin{align*} \frac{{\left (a x + b\right )}^{3} \log \left (a x + b\right )}{3 \, a^{3}} - \frac{{\left (a x + b\right )}^{2} b \log \left (a x + b\right )}{a^{3}} + \frac{{\left (a x + b\right )} b^{2} \log \left (a x + b\right )}{a^{3}} - \frac{{\left (a x + b\right )}^{3}}{9 \, a^{3}} + \frac{{\left (a x + b\right )}^{2} b}{2 \, a^{3}} - \frac{{\left (a x + b\right )} b^{2}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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