3.76.48 \(\int \frac {-49-140 x-29 x^2+19 x^3+2 x^4}{49 x+14 x^2+x^3} \, dx\)

Optimal. Leaf size=27 \[ -13+x-x \left (3+\frac {4-x^2}{7+x}+\frac {\log (x)}{x}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 18, normalized size of antiderivative = 0.67, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1594, 27, 1620} \begin {gather*} x^2-9 x-\frac {315}{x+7}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-49 - 140*x - 29*x^2 + 19*x^3 + 2*x^4)/(49*x + 14*x^2 + x^3),x]

[Out]

-9*x + x^2 - 315/(7 + 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 {-49-140 x-29 x^2+19 x^3+2 x^4}{x \left (49+14 x+x^2\right )} \, dx\\ &=\int \frac {-49-140 x-29 x^2+19 x^3+2 x^4}{x (7+x)^2} \, dx\\ &=\int \left (-9-\frac {1}{x}+2 x+\frac {315}{(7+x)^2}\right ) \, dx\\ &=-9 x+x^2-\frac {315}{7+x}-\log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} -9 x+x^2-\frac {315}{7+x}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-49 - 140*x - 29*x^2 + 19*x^3 + 2*x^4)/(49*x + 14*x^2 + x^3),x]

[Out]

-9*x + x^2 - 315/(7 + x) - Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.54, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^{3} - 2 \, x^{2} - {\left (x + 7\right )} \log \relax (x) - 63 \, x - 315}{x + 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4+19*x^3-29*x^2-140*x-49)/(x^3+14*x^2+49*x),x, algorithm="fricas")

[Out]

(x^3 - 2*x^2 - (x + 7)*log(x) - 63*x - 315)/(x + 7)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 19, normalized size = 0.70 \begin {gather*} x^{2} - 9 \, x - \frac {315}{x + 7} - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4+19*x^3-29*x^2-140*x-49)/(x^3+14*x^2+49*x),x, algorithm="giac")

[Out]

x^2 - 9*x - 315/(x + 7) - log(abs(x))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 19, normalized size = 0.70

method	result	size
default	\(x^{2}-9 x -\frac {315}{x +7}-\ln \relax (x )\)	\(19\)
risch	\(x^{2}-9 x -\frac {315}{x +7}-\ln \relax (x )\)	\(19\)
norman	\(\frac {x^{3}-2 x^{2}+126}{x +7}-\ln \relax (x )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4+19*x^3-29*x^2-140*x-49)/(x^3+14*x^2+49*x),x,method=_RETURNVERBOSE)

[Out]

x^2-9*x-315/(x+7)-ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 18, normalized size = 0.67 \begin {gather*} x^{2} - 9 \, x - \frac {315}{x + 7} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4+19*x^3-29*x^2-140*x-49)/(x^3+14*x^2+49*x),x, algorithm="maxima")

[Out]

x^2 - 9*x - 315/(x + 7) - log(x)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 18, normalized size = 0.67 \begin {gather*} x^2-\ln \relax (x)-\frac {315}{x+7}-9\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(140*x + 29*x^2 - 19*x^3 - 2*x^4 + 49)/(49*x + 14*x^2 + x^3),x)

[Out]

x^2 - log(x) - 315/(x + 7) - 9*x

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 14, normalized size = 0.52 \begin {gather*} x^{2} - 9 x - \log {\relax (x )} - \frac {315}{x + 7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**4+19*x**3-29*x**2-140*x-49)/(x**3+14*x**2+49*x),x)

[Out]

x**2 - 9*x - log(x) - 315/(x + 7)

________________________________________________________________________________________