3.86.75 \(\int \frac {15+30 x+5 x^2+8 x^3+x^4-2 x^5}{x^2+2 x^3+x^4} \, dx\)

Optimal. Leaf size=29 \[ 4+\left (1+e^4\right )^2-x^2+5 \left (-4+x-\frac {3}{x (1+x)}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 21, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {1594, 27, 1620} \begin {gather*} -x^2+5 x+\frac {15}{x+1}-\frac {15}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(15 + 30*x + 5*x^2 + 8*x^3 + x^4 - 2*x^5)/(x^2 + 2*x^3 + x^4),x]

[Out]

-15/x + 5*x - x^2 + 15/(1 + 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 {15+30 x+5 x^2+8 x^3+x^4-2 x^5}{x^2 \left (1+2 x+x^2\right )} \, dx\\ &=\int \frac {15+30 x+5 x^2+8 x^3+x^4-2 x^5}{x^2 (1+x)^2} \, dx\\ &=\int \left (5+\frac {15}{x^2}-2 x-\frac {15}{(1+x)^2}\right ) \, dx\\ &=-\frac {15}{x}+5 x-x^2+\frac {15}{1+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.72 \begin {gather*} -\frac {15}{x}+5 x-x^2+\frac {15}{1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15 + 30*x + 5*x^2 + 8*x^3 + x^4 - 2*x^5)/(x^2 + 2*x^3 + x^4),x]

[Out]

-15/x + 5*x - x^2 + 15/(1 + x)

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fricas [A]  time = 0.57, size = 24, normalized size = 0.83 \begin {gather*} -\frac {x^{4} - 4 \, x^{3} - 5 \, x^{2} + 15}{x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5+x^4+8*x^3+5*x^2+30*x+15)/(x^4+2*x^3+x^2),x, algorithm="fricas")

[Out]

-(x^4 - 4*x^3 - 5*x^2 + 15)/(x^2 + x)

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giac [A]  time = 0.22, size = 18, normalized size = 0.62 \begin {gather*} -x^{2} + 5 \, x - \frac {15}{x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5+x^4+8*x^3+5*x^2+30*x+15)/(x^4+2*x^3+x^2),x, algorithm="giac")

[Out]

-x^2 + 5*x - 15/(x^2 + x)

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maple [A]  time = 0.03, size = 20, normalized size = 0.69

method	result	size
risch	\(-x^{2}+5 x -\frac {15}{x \left (x +1\right )}\)	\(20\)
default	\(-x^{2}+5 x +\frac {15}{x +1}-\frac {15}{x}\)	\(22\)
gosper	\(-\frac {x^{4}-4 x^{3}+5 x +15}{x \left (x +1\right )}\)	\(24\)
norman	\(\frac {-x^{4}+4 x^{3}-5 x -15}{\left (x +1\right ) x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^5+x^4+8*x^3+5*x^2+30*x+15)/(x^4+2*x^3+x^2),x,method=_RETURNVERBOSE)

[Out]

-x^2+5*x-15/x/(x+1)

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maxima [A]  time = 0.35, size = 18, normalized size = 0.62 \begin {gather*} -x^{2} + 5 \, x - \frac {15}{x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5+x^4+8*x^3+5*x^2+30*x+15)/(x^4+2*x^3+x^2),x, algorithm="maxima")

[Out]

-x^2 + 5*x - 15/(x^2 + x)

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mupad [B]  time = 0.05, size = 19, normalized size = 0.66 \begin {gather*} 5\,x-\frac {15}{x\,\left (x+1\right )}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((30*x + 5*x^2 + 8*x^3 + x^4 - 2*x^5 + 15)/(x^2 + 2*x^3 + x^4),x)

[Out]

5*x - 15/(x*(x + 1)) - x^2

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sympy [A]  time = 0.08, size = 12, normalized size = 0.41 \begin {gather*} - x^{2} + 5 x - \frac {15}{x^{2} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**5+x**4+8*x**3+5*x**2+30*x+15)/(x**4+2*x**3+x**2),x)

[Out]

-x**2 + 5*x - 15/(x**2 + x)

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