3.69.18 $\int \frac{45+26x+112x^2+18x^3+(-36-6x)\log (6+x)}{6+x}dx$

Optimal. Leaf size=27 $$2x-x^2-3(1+2x)(-1-x^2+\log (6+x))$$

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Rubi [A]  time = 0.10, antiderivative size = 29, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 4, integrand size = 31, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.129, Rules used = {6742, 1850, 2389, 2295} $$\begin{array}{c}6x^3+2x^2+8x-6(x+6)\log (x+6)+33\log (x+6)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(45 + 26*x + 112*x^2 + 18*x^3 + (-36 - 6*x)*Log[6 + x])/(6 + x),x]

[Out]

8*x + 2*x^2 + 6*x^3 + 33*Log[6 + x] - 6*(6 + x)*Log[6 + x]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (\frac{45+26x+112x^2+18x^3}{6+x}-6\log (6+x))dx\\ & =-(6\int \log (6+x)dx)+\int \frac{45+26x+112x^2+18x^3}{6+x}dx\\ & =-(6\mathrm{Subst}(\int \log (x)dx,x,6+x))+\int (2+4x+18x^2+\frac{33}{6+x})dx\\ & =8x+2x^2+6x^3+33\log (6+x)-6(6+x)\log (6+x)\end{array}\end{array}$$

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Mathematica [A]  time = 0.03, size = 27, normalized size = 1.00 $$\begin{array}{c}8x+2x^2+6x^3-3\log (6+x)-6x\log (6+x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(45 + 26*x + 112*x^2 + 18*x^3 + (-36 - 6*x)*Log[6 + x])/(6 + x),x]

[Out]

8*x + 2*x^2 + 6*x^3 - 3*Log[6 + x] - 6*x*Log[6 + x]

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fricas [A]  time = 0.48, size = 25, normalized size = 0.93 $$\begin{array}{c}6x^3+2x^2-3(2x+1)\log (x+6)+8x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x-36)*log(x+6)+18*x^3+112*x^2+26*x+45)/(x+6),x, algorithm="fricas")

[Out]

6*x^3 + 2*x^2 - 3*(2*x + 1)*log(x + 6) + 8*x

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giac [A]  time = 0.12, size = 27, normalized size = 1.00 $$\begin{array}{c}6x^3+2x^2-6x\log (x+6)+8x-3\log (x+6)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x-36)*log(x+6)+18*x^3+112*x^2+26*x+45)/(x+6),x, algorithm="giac")

[Out]

6*x^3 + 2*x^2 - 6*x*log(x + 6) + 8*x - 3*log(x + 6)

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maple [A]  time = 0.06, size = 28, normalized size = 1.04

method	result	size
norman	$-3\ln (x+6)+8x+2x^2+6x^3-6x\ln (x+6)$	$28$
risch	$-3\ln (x+6)+8x+2x^2+6x^3-6x\ln (x+6)$	$28$
derivativedivides	$6{(x+6)}^3-6(x+6)\ln (x+6)+632x+3792-106{(x+6)}^2+33\ln (x+6)$	$35$
default	$6{(x+6)}^3-6(x+6)\ln (x+6)+632x+3792-106{(x+6)}^2+33\ln (x+6)$	$35$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x-36)*ln(x+6)+18*x^3+112*x^2+26*x+45)/(x+6),x,method=_RETURNVERBOSE)

[Out]

-3*ln(x+6)+8*x+2*x^2+6*x^3-6*x*ln(x+6)

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maxima [A]  time = 0.42, size = 42, normalized size = 1.56 $$\begin{array}{c}6x^3+2x^2-6(x-6\log (x+6))\log (x+6)-36\log {(x+6)}^2+8x-3\log (x+6)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x-36)*log(x+6)+18*x^3+112*x^2+26*x+45)/(x+6),x, algorithm="maxima")

[Out]

6*x^3 + 2*x^2 - 6*(x - 6*log(x + 6))*log(x + 6) - 36*log(x + 6)^2 + 8*x - 3*log(x + 6)

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mupad [B]  time = 0.10, size = 28, normalized size = 1.04 $$\begin{array}{c}2x^2-3\ln (x+6)+6x^3-x(6\ln (x+6)-8)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((26*x + 112*x^2 + 18*x^3 - log(x + 6)*(6*x + 36) + 45)/(x + 6),x)

[Out]

2*x^2 - 3*log(x + 6) + 6*x^3 - x*(6*log(x + 6) - 8)

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sympy [A]  time = 0.12, size = 27, normalized size = 1.00 $$\begin{array}{c}6x^3+2x^2-6x\log (x+6)+8x-3\log (x+6)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x-36)*ln(x+6)+18*x**3+112*x**2+26*x+45)/(x+6),x)

[Out]

6*x**3 + 2*x**2 - 6*x*log(x + 6) + 8*x - 3*log(x + 6)

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