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

Optimal. Leaf size=35 \[ \frac {(a+b x)^3 \log (a+b x)}{3 b}-\frac {(a+b x)^3}{9 b} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2390, 2304} \[ \frac {(a+b x)^3 \log (a+b x)}{3 b}-\frac {(a+b x)^3}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*Log[a + b*x],x]

[Out]

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2390

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 44, normalized size = 1.26 \[ \frac {(a+b x)^3 \log (a+b x)}{3 b}-\frac {1}{9} x \left (3 a^2+3 a b x+b^2 x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2*Log[a + b*x],x]

[Out]

-1/9*(x*(3*a^2 + 3*a*b*x + b^2*x^2)) + ((a + b*x)^3*Log[a + b*x])/(3*b)

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fricas [B]  time = 0.46, size = 64, normalized size = 1.83 \[ -\frac {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x - 3 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right )}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.22, size = 31, normalized size = 0.89 \[ \frac {{\left (b x + a\right )}^{3} \log \left (b x + a\right )}{3 \, b} - \frac {{\left (b x + a\right )}^{3}}{9 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.07, size = 82, normalized size = 2.34 \[ \frac {b^{2} x^{3} \ln \left (b x +a \right )}{3}+a b \,x^{2} \ln \left (b x +a \right )-\frac {b^{2} x^{3}}{9}+a^{2} x \ln \left (b x +a \right )-\frac {a b \,x^{2}}{3}+\frac {a^{3} \ln \left (b x +a \right )}{3 b}-\frac {a^{2} x}{3}-\frac {a^{3}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.60, size = 74, normalized size = 2.11 \[ \frac {1}{9} \, {\left (\frac {3 \, a^{3} \log \left (b x + a\right )}{b^{2}} - \frac {b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x}{b}\right )} b + \frac {1}{3} \, {\left (b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x\right )} \log \left (b x + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.46, size = 73, normalized size = 2.09 \[ \frac {a^3\,\ln \left (a+b\,x\right )}{3\,b}-\frac {b^2\,x^3}{9}-\frac {a^2\,x}{3}+\frac {b^2\,x^3\,\ln \left (a+b\,x\right )}{3}-\frac {a\,b\,x^2}{3}+a^2\,x\,\ln \left (a+b\,x\right )+a\,b\,x^2\,\ln \left (a+b\,x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(a + b*x)*(a + b*x)^2,x)

[Out]

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

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sympy [B]  time = 0.23, size = 63, normalized size = 1.80 \[ \frac {a^{3} \log {\left (a + b x \right )}}{3 b} - \frac {a^{2} x}{3} - \frac {a b x^{2}}{3} - \frac {b^{2} x^{3}}{9} + \left (a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}\right ) \log {\left (a + b x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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