3.2396 \(\int \frac{(5-x) (3+2 x)^5}{(2+5 x+3 x^2)^3} \, dx\)

Optimal. Leaf size=64 \[ \frac{502254 x+398585}{486 \left (3 x^2+5 x+2\right )}-\frac{57499 x+56041}{486 \left (3 x^2+5 x+2\right )^2}-\frac{32 x}{27}-1085 \log (x+1)+\frac{29375}{27} \log (3 x+2) \]

[Out]

(-32*x)/27 - (56041 + 57499*x)/(486*(2 + 5*x + 3*x^2)^2) + (398585 + 502254*x)/(486*(2 + 5*x + 3*x^2)) - 1085*
Log[1 + x] + (29375*Log[2 + 3*x])/27

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Rubi [A]  time = 0.0800288, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {816, 1660, 1657, 632, 31} \[ \frac{502254 x+398585}{486 \left (3 x^2+5 x+2\right )}-\frac{57499 x+56041}{486 \left (3 x^2+5 x+2\right )^2}-\frac{32 x}{27}-1085 \log (x+1)+\frac{29375}{27} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-32*x)/27 - (56041 + 57499*x)/(486*(2 + 5*x + 3*x^2)^2) + (398585 + 502254*x)/(486*(2 + 5*x + 3*x^2)) - 1085*
Log[1 + x] + (29375*Log[2 + 3*x])/27

Rule 816

Int[((d_) + (e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[(a
+ b*x + c*x^2)^p*ExpandIntegrand[(d + e*x)^m*(f + g*x), 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] && ILtQ[p, -1] && IGtQ[m, 0] && RationalQ[a, b, c, d, e, f, g]

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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)^5}{\left (2+5 x+3 x^2\right )^3} \, dx &=\int \frac{\frac{13}{2} (3+2 x)^5-\frac{1}{2} (3+2 x)^6}{\left (2+5 x+3 x^2\right )^3} \, dx\\ &=-\frac{56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}-\frac{1}{2} \int \frac{-\frac{72539}{243}-\frac{87920 x}{81}-\frac{9824 x^2}{27}+\frac{160 x^3}{9}+\frac{64 x^4}{3}}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac{56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac{398585+502254 x}{486 \left (2+5 x+3 x^2\right )}+\frac{1}{2} \int \frac{\frac{58942}{27}+\frac{160 x}{27}-\frac{64 x^2}{9}}{2+5 x+3 x^2} \, dx\\ &=-\frac{56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac{398585+502254 x}{486 \left (2+5 x+3 x^2\right )}+\frac{1}{2} \int \left (-\frac{64}{27}+\frac{10 (1969+16 x)}{9 \left (2+5 x+3 x^2\right )}\right ) \, dx\\ &=-\frac{32 x}{27}-\frac{56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac{398585+502254 x}{486 \left (2+5 x+3 x^2\right )}+\frac{5}{9} \int \frac{1969+16 x}{2+5 x+3 x^2} \, dx\\ &=-\frac{32 x}{27}-\frac{56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac{398585+502254 x}{486 \left (2+5 x+3 x^2\right )}-3255 \int \frac{1}{3+3 x} \, dx+\frac{29375}{9} \int \frac{1}{2+3 x} \, dx\\ &=-\frac{32 x}{27}-\frac{56041+57499 x}{486 \left (2+5 x+3 x^2\right )^2}+\frac{398585+502254 x}{486 \left (2+5 x+3 x^2\right )}-1085 \log (1+x)+\frac{29375}{27} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0357525, size = 81, normalized size = 1.27 \[ \frac{-1728 x^5-8352 x^4+486510 x^3+1221179 x^2+176250 \left (3 x^2+5 x+2\right )^2 \log (-6 x-4)-175770 \left (3 x^2+5 x+2\right )^2 \log (-2 (x+1))+973450 x+245891}{162 \left (3 x^2+5 x+2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(245891 + 973450*x + 1221179*x^2 + 486510*x^3 - 8352*x^4 - 1728*x^5 + 176250*(2 + 5*x + 3*x^2)^2*Log[-4 - 6*x]
- 175770*(2 + 5*x + 3*x^2)^2*Log[-2*(1 + x)])/(162*(2 + 5*x + 3*x^2)^2)

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Maple [A]  time = 0.01, size = 51, normalized size = 0.8 \begin{align*} -{\frac{32\,x}{27}}+3\, \left ( 1+x \right ) ^{-2}+113\, \left ( 1+x \right ) ^{-1}-1085\,\ln \left ( 1+x \right ) -{\frac{53125}{162\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{6250}{18+27\,x}}+{\frac{29375\,\ln \left ( 2+3\,x \right ) }{27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/27*x+3/(1+x)^2+113/(1+x)-1085*ln(1+x)-53125/162/(2+3*x)^2+6250/9/(2+3*x)+29375/27*ln(2+3*x)

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Maxima [A]  time = 0.995824, size = 77, normalized size = 1.2 \begin{align*} -\frac{32}{27} \, x + \frac{502254 \, x^{3} + 1235675 \, x^{2} + 979978 \, x + 247043}{162 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + \frac{29375}{27} \, \log \left (3 \, x + 2\right ) - 1085 \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/27*x + 1/162*(502254*x^3 + 1235675*x^2 + 979978*x + 247043)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4) + 29375/2
7*log(3*x + 2) - 1085*log(x + 1)

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Fricas [A]  time = 1.16202, size = 311, normalized size = 4.86 \begin{align*} -\frac{1728 \, x^{5} + 5760 \, x^{4} - 495150 \, x^{3} - 1231835 \, x^{2} - 176250 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 175770 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) - 979210 \, x - 247043}{162 \,{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/162*(1728*x^5 + 5760*x^4 - 495150*x^3 - 1231835*x^2 - 176250*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(3*x +
2) + 175770*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(x + 1) - 979210*x - 247043)/(9*x^4 + 30*x^3 + 37*x^2 + 2
0*x + 4)

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Sympy [A]  time = 0.203175, size = 56, normalized size = 0.88 \begin{align*} - \frac{32 x}{27} + \frac{502254 x^{3} + 1235675 x^{2} + 979978 x + 247043}{1458 x^{4} + 4860 x^{3} + 5994 x^{2} + 3240 x + 648} + \frac{29375 \log{\left (x + \frac{2}{3} \right )}}{27} - 1085 \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32*x/27 + (502254*x**3 + 1235675*x**2 + 979978*x + 247043)/(1458*x**4 + 4860*x**3 + 5994*x**2 + 3240*x + 648)
+ 29375*log(x + 2/3)/27 - 1085*log(x + 1)

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Giac [A]  time = 1.11713, size = 66, normalized size = 1.03 \begin{align*} -\frac{32}{27} \, x + \frac{502254 \, x^{3} + 1235675 \, x^{2} + 979978 \, x + 247043}{162 \,{\left (3 \, x + 2\right )}^{2}{\left (x + 1\right )}^{2}} + \frac{29375}{27} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 1085 \, \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)^5/(3*x^2+5*x+2)^3,x, algorithm="giac")

[Out]

-32/27*x + 1/162*(502254*x^3 + 1235675*x^2 + 979978*x + 247043)/((3*x + 2)^2*(x + 1)^2) + 29375/27*log(abs(3*x
+ 2)) - 1085*log(abs(x + 1))