3.27.79 \(\int \frac {25+305 x+191 x^2-71 x^3-52 x^4-6 x^5}{25 x+10 x^2+x^3} \, dx\)

Optimal. Leaf size=28 \[ -5+x+\log \left (e^{4+\frac {2 (3-x) x (3+x)^2}{5+x}} x\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 23, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {1594, 27, 1620} \begin {gather*} -2 x^3+4 x^2-x-\frac {320}{x+5}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(25 + 305*x + 191*x^2 - 71*x^3 - 52*x^4 - 6*x^5)/(25*x + 10*x^2 + x^3),x]

[Out]

-x + 4*x^2 - 2*x^3 - 320/(5 + x) + Log[x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25+305 x+191 x^2-71 x^3-52 x^4-6 x^5}{x \left (25+10 x+x^2\right )} \, dx\\ &=\int \frac {25+305 x+191 x^2-71 x^3-52 x^4-6 x^5}{x (5+x)^2} \, dx\\ &=\int \left (-1+\frac {1}{x}+8 x-6 x^2+\frac {320}{(5+x)^2}\right ) \, dx\\ &=-x+4 x^2-2 x^3-\frac {320}{5+x}+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.82 \begin {gather*} -x+4 x^2-2 x^3-\frac {320}{5+x}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 + 305*x + 191*x^2 - 71*x^3 - 52*x^4 - 6*x^5)/(25*x + 10*x^2 + x^3),x]

[Out]

-x + 4*x^2 - 2*x^3 - 320/(5 + x) + Log[x]

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fricas [A]  time = 0.53, size = 34, normalized size = 1.21 \begin {gather*} -\frac {2 \, x^{4} + 6 \, x^{3} - 19 \, x^{2} - {\left (x + 5\right )} \log \relax (x) + 5 \, x + 320}{x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^5-52*x^4-71*x^3+191*x^2+305*x+25)/(x^3+10*x^2+25*x),x, algorithm="fricas")

[Out]

-(2*x^4 + 6*x^3 - 19*x^2 - (x + 5)*log(x) + 5*x + 320)/(x + 5)

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giac [A]  time = 0.24, size = 24, normalized size = 0.86 \begin {gather*} -2 \, x^{3} + 4 \, x^{2} - x - \frac {320}{x + 5} + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^5-52*x^4-71*x^3+191*x^2+305*x+25)/(x^3+10*x^2+25*x),x, algorithm="giac")

[Out]

-2*x^3 + 4*x^2 - x - 320/(x + 5) + log(abs(x))

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maple [A]  time = 0.02, size = 24, normalized size = 0.86

method	result	size
default	\(-2 x^{3}+4 x^{2}-x -\frac {320}{5+x}+\ln \relax (x )\)	\(24\)
risch	\(-2 x^{3}+4 x^{2}-x -\frac {320}{5+x}+\ln \relax (x )\)	\(24\)
norman	\(\frac {-2 x^{4}-6 x^{3}+19 x^{2}-295}{5+x}+\ln \relax (x )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^5-52*x^4-71*x^3+191*x^2+305*x+25)/(x^3+10*x^2+25*x),x,method=_RETURNVERBOSE)

[Out]

-2*x^3+4*x^2-x-320/(5+x)+ln(x)

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maxima [A]  time = 0.34, size = 23, normalized size = 0.82 \begin {gather*} -2 \, x^{3} + 4 \, x^{2} - x - \frac {320}{x + 5} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^5-52*x^4-71*x^3+191*x^2+305*x+25)/(x^3+10*x^2+25*x),x, algorithm="maxima")

[Out]

-2*x^3 + 4*x^2 - x - 320/(x + 5) + log(x)

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mupad [B]  time = 0.04, size = 23, normalized size = 0.82 \begin {gather*} \ln \relax (x)-x-\frac {320}{x+5}+4\,x^2-2\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((305*x + 191*x^2 - 71*x^3 - 52*x^4 - 6*x^5 + 25)/(25*x + 10*x^2 + x^3),x)

[Out]

log(x) - x - 320/(x + 5) + 4*x^2 - 2*x^3

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sympy [A]  time = 0.09, size = 19, normalized size = 0.68 \begin {gather*} - 2 x^{3} + 4 x^{2} - x + \log {\relax (x )} - \frac {320}{x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**5-52*x**4-71*x**3+191*x**2+305*x+25)/(x**3+10*x**2+25*x),x)

[Out]

-2*x**3 + 4*x**2 - x + log(x) - 320/(x + 5)

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