3.2378 \(\int \frac{(5-x) (3+2 x)^4}{2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=43 \[ -\frac{4 x^4}{3}+\frac{32 x^3}{27}+\frac{1156 x^2}{27}+\frac{11576 x}{81}-6 \log (x+1)+\frac{10625}{243} \log (3 x+2) \]

[Out]

(11576*x)/81 + (1156*x^2)/27 + (32*x^3)/27 - (4*x^4)/3 - 6*Log[1 + x] + (10625*Log[2 + 3*x])/243

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Rubi [A]  time = 0.0244776, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {800, 632, 31} \[ -\frac{4 x^4}{3}+\frac{32 x^3}{27}+\frac{1156 x^2}{27}+\frac{11576 x}{81}-6 \log (x+1)+\frac{10625}{243} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[((5 - x)*(3 + 2*x)^4)/(2 + 5*x + 3*x^2),x]

[Out]

(11576*x)/81 + (1156*x^2)/27 + (32*x^3)/27 - (4*x^4)/3 - 6*Log[1 + x] + (10625*Log[2 + 3*x])/243

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{(5-x) (3+2 x)^4}{2+5 x+3 x^2} \, dx &=\int \left (\frac{11576}{81}+\frac{2312 x}{27}+\frac{32 x^2}{9}-\frac{16 x^3}{3}+\frac{9653+9167 x}{81 \left (2+5 x+3 x^2\right )}\right ) \, dx\\ &=\frac{11576 x}{81}+\frac{1156 x^2}{27}+\frac{32 x^3}{27}-\frac{4 x^4}{3}+\frac{1}{81} \int \frac{9653+9167 x}{2+5 x+3 x^2} \, dx\\ &=\frac{11576 x}{81}+\frac{1156 x^2}{27}+\frac{32 x^3}{27}-\frac{4 x^4}{3}-18 \int \frac{1}{3+3 x} \, dx+\frac{10625}{81} \int \frac{1}{2+3 x} \, dx\\ &=\frac{11576 x}{81}+\frac{1156 x^2}{27}+\frac{32 x^3}{27}-\frac{4 x^4}{3}-6 \log (1+x)+\frac{10625}{243} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0239609, size = 43, normalized size = 1. \[ \frac{1}{972} \left (42500 \log (-6 x-4)-3 \left (432 x^4-384 x^3-13872 x^2-46304 x+1944 \log (-2 (x+1))-41727\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((5 - x)*(3 + 2*x)^4)/(2 + 5*x + 3*x^2),x]

[Out]

(42500*Log[-4 - 6*x] - 3*(-41727 - 46304*x - 13872*x^2 - 384*x^3 + 432*x^4 + 1944*Log[-2*(1 + x)]))/972

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Maple [A]  time = 0.007, size = 34, normalized size = 0.8 \begin{align*}{\frac{11576\,x}{81}}+{\frac{1156\,{x}^{2}}{27}}+{\frac{32\,{x}^{3}}{27}}-{\frac{4\,{x}^{4}}{3}}-6\,\ln \left ( 1+x \right ) +{\frac{10625\,\ln \left ( 2+3\,x \right ) }{243}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5-x)*(3+2*x)^4/(3*x^2+5*x+2),x)

[Out]

11576/81*x+1156/27*x^2+32/27*x^3-4/3*x^4-6*ln(1+x)+10625/243*ln(2+3*x)

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Maxima [A]  time = 1.18946, size = 45, normalized size = 1.05 \begin{align*} -\frac{4}{3} \, x^{4} + \frac{32}{27} \, x^{3} + \frac{1156}{27} \, x^{2} + \frac{11576}{81} \, x + \frac{10625}{243} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^4/(3*x^2+5*x+2),x, algorithm="maxima")

[Out]

-4/3*x^4 + 32/27*x^3 + 1156/27*x^2 + 11576/81*x + 10625/243*log(3*x + 2) - 6*log(x + 1)

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Fricas [A]  time = 2.10573, size = 120, normalized size = 2.79 \begin{align*} -\frac{4}{3} \, x^{4} + \frac{32}{27} \, x^{3} + \frac{1156}{27} \, x^{2} + \frac{11576}{81} \, x + \frac{10625}{243} \, \log \left (3 \, x + 2\right ) - 6 \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^4/(3*x^2+5*x+2),x, algorithm="fricas")

[Out]

-4/3*x^4 + 32/27*x^3 + 1156/27*x^2 + 11576/81*x + 10625/243*log(3*x + 2) - 6*log(x + 1)

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Sympy [A]  time = 0.129265, size = 41, normalized size = 0.95 \begin{align*} - \frac{4 x^{4}}{3} + \frac{32 x^{3}}{27} + \frac{1156 x^{2}}{27} + \frac{11576 x}{81} + \frac{10625 \log{\left (x + \frac{2}{3} \right )}}{243} - 6 \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)**4/(3*x**2+5*x+2),x)

[Out]

-4*x**4/3 + 32*x**3/27 + 1156*x**2/27 + 11576*x/81 + 10625*log(x + 2/3)/243 - 6*log(x + 1)

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Giac [A]  time = 1.12103, size = 47, normalized size = 1.09 \begin{align*} -\frac{4}{3} \, x^{4} + \frac{32}{27} \, x^{3} + \frac{1156}{27} \, x^{2} + \frac{11576}{81} \, x + \frac{10625}{243} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 6 \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5-x)*(3+2*x)^4/(3*x^2+5*x+2),x, algorithm="giac")

[Out]

-4/3*x^4 + 32/27*x^3 + 1156/27*x^2 + 11576/81*x + 10625/243*log(abs(3*x + 2)) - 6*log(abs(x + 1))