3.102.84 \(\int \frac {-18+24 x+31 x^2+6 x^3+(45+10 x) \log (9+2 x)}{9+2 x} \, dx\)

Optimal. Leaf size=20 \[ -5-2 x+x \left (x+x^2+5 \log (9+2 x)\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 35, normalized size of antiderivative = 1.75, number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6742, 1850, 2389, 2295} \begin {gather*} x^3+x^2-2 x+\frac {5}{2} (2 x+9) \log (2 x+9)-\frac {45}{2} \log (2 x+9) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-18 + 24*x + 31*x^2 + 6*x^3 + (45 + 10*x)*Log[9 + 2*x])/(9 + 2*x),x]

[Out]

-2*x + x^2 + x^3 - (45*Log[9 + 2*x])/2 + (5*(9 + 2*x)*Log[9 + 2*x])/2

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 {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-18+24 x+31 x^2+6 x^3}{9+2 x}+5 \log (9+2 x)\right ) \, dx\\ &=5 \int \log (9+2 x) \, dx+\int \frac {-18+24 x+31 x^2+6 x^3}{9+2 x} \, dx\\ &=\frac {5}{2} \operatorname {Subst}(\int \log (x) \, dx,x,9+2 x)+\int \left (3+2 x+3 x^2-\frac {45}{9+2 x}\right ) \, dx\\ &=-2 x+x^2+x^3-\frac {45}{2} \log (9+2 x)+\frac {5}{2} (9+2 x) \log (9+2 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 19, normalized size = 0.95 \begin {gather*} -2 x+x^2+x^3+5 x \log (9+2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18 + 24*x + 31*x^2 + 6*x^3 + (45 + 10*x)*Log[9 + 2*x])/(9 + 2*x),x]

[Out]

-2*x + x^2 + x^3 + 5*x*Log[9 + 2*x]

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fricas [A]  time = 1.27, size = 19, normalized size = 0.95 \begin {gather*} x^{3} + x^{2} + 5 \, x \log \left (2 \, x + 9\right ) - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+45)*log(2*x+9)+6*x^3+31*x^2+24*x-18)/(2*x+9),x, algorithm="fricas")

[Out]

x^3 + x^2 + 5*x*log(2*x + 9) - 2*x

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giac [A]  time = 0.16, size = 19, normalized size = 0.95 \begin {gather*} x^{3} + x^{2} + 5 \, x \log \left (2 \, x + 9\right ) - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+45)*log(2*x+9)+6*x^3+31*x^2+24*x-18)/(2*x+9),x, algorithm="giac")

[Out]

x^3 + x^2 + 5*x*log(2*x + 9) - 2*x

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maple [A]  time = 0.24, size = 20, normalized size = 1.00

method	result	size
norman	\(x^{2}+x^{3}-2 x +5 \ln \left (2 x +9\right ) x\)	\(20\)
risch	\(x^{2}+x^{3}-2 x +5 \ln \left (2 x +9\right ) x\)	\(20\)
derivativedivides	\(\frac {\left (2 x +9\right )^{3}}{8}+\frac {5 \ln \left (2 x +9\right ) \left (2 x +9\right )}{2}+\frac {199 x}{4}+\frac {1791}{8}-\frac {25 \left (2 x +9\right )^{2}}{8}-\frac {45 \ln \left (2 x +9\right )}{2}\)	\(45\)
default	\(\frac {\left (2 x +9\right )^{3}}{8}+\frac {5 \ln \left (2 x +9\right ) \left (2 x +9\right )}{2}+\frac {199 x}{4}+\frac {1791}{8}-\frac {25 \left (2 x +9\right )^{2}}{8}-\frac {45 \ln \left (2 x +9\right )}{2}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x+45)*ln(2*x+9)+6*x^3+31*x^2+24*x-18)/(2*x+9),x,method=_RETURNVERBOSE)

[Out]

x^2+x^3-2*x+5*ln(2*x+9)*x

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maxima [A]  time = 0.36, size = 40, normalized size = 2.00 \begin {gather*} x^{3} + x^{2} + \frac {5}{2} \, {\left (2 \, x - 9 \, \log \left (2 \, x + 9\right )\right )} \log \left (2 \, x + 9\right ) + \frac {45}{2} \, \log \left (2 \, x + 9\right )^{2} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+45)*log(2*x+9)+6*x^3+31*x^2+24*x-18)/(2*x+9),x, algorithm="maxima")

[Out]

x^3 + x^2 + 5/2*(2*x - 9*log(2*x + 9))*log(2*x + 9) + 45/2*log(2*x + 9)^2 - 2*x

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mupad [B]  time = 0.14, size = 19, normalized size = 0.95 \begin {gather*} x\,\left (5\,\ln \left (2\,x+9\right )-2\right )+x^2+x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x + log(2*x + 9)*(10*x + 45) + 31*x^2 + 6*x^3 - 18)/(2*x + 9),x)

[Out]

x*(5*log(2*x + 9) - 2) + x^2 + x^3

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sympy [A]  time = 0.11, size = 19, normalized size = 0.95 \begin {gather*} x^{3} + x^{2} + 5 x \log {\left (2 x + 9 \right )} - 2 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+45)*ln(2*x+9)+6*x**3+31*x**2+24*x-18)/(2*x+9),x)

[Out]

x**3 + x**2 + 5*x*log(2*x + 9) - 2*x

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