3.87.23 \(\int \frac {96+96 x+4 x^2+8 x^3-2 x^4-2 x^5}{x^3+x^4} \, dx\)

Optimal. Leaf size=21 \[ 6-x^2+4 \left (-\frac {12}{x^2}+\log \left (x+x^2\right )\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1593, 1620} \begin {gather*} -x^2-\frac {48}{x^2}+4 \log (x)+4 \log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(96 + 96*x + 4*x^2 + 8*x^3 - 2*x^4 - 2*x^5)/(x^3 + x^4),x]

[Out]

-48/x^2 - x^2 + 4*Log[x] + 4*Log[1 + x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - 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 {96+96 x+4 x^2+8 x^3-2 x^4-2 x^5}{x^3 (1+x)} \, dx\\ &=\int \left (\frac {96}{x^3}+\frac {4}{x}-2 x+\frac {4}{1+x}\right ) \, dx\\ &=-\frac {48}{x^2}-x^2+4 \log (x)+4 \log (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {48}{x^2}-x^2+4 \log (x)+4 \log (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(96 + 96*x + 4*x^2 + 8*x^3 - 2*x^4 - 2*x^5)/(x^3 + x^4),x]

[Out]

-48/x^2 - x^2 + 4*Log[x] + 4*Log[1 + x]

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fricas [A]  time = 0.92, size = 21, normalized size = 1.00 \begin {gather*} -\frac {x^{4} - 4 \, x^{2} \log \left (x^{2} + x\right ) + 48}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5-2*x^4+8*x^3+4*x^2+96*x+96)/(x^4+x^3),x, algorithm="fricas")

[Out]

-(x^4 - 4*x^2*log(x^2 + x) + 48)/x^2

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giac [A]  time = 0.20, size = 23, normalized size = 1.10 \begin {gather*} -x^{2} - \frac {48}{x^{2}} + 4 \, \log \left ({\left | x + 1 \right |}\right ) + 4 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5-2*x^4+8*x^3+4*x^2+96*x+96)/(x^4+x^3),x, algorithm="giac")

[Out]

-x^2 - 48/x^2 + 4*log(abs(x + 1)) + 4*log(abs(x))

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maple [A]  time = 0.44, size = 20, normalized size = 0.95

method	result	size
risch	\(-x^{2}-\frac {48}{x^{2}}+4 \ln \left (x^{2}+x \right )\)	\(20\)
default	\(-x^{2}-\frac {48}{x^{2}}+4 \ln \relax (x )+4 \ln \left (x +1\right )\)	\(22\)
norman	\(\frac {-x^{4}-48}{x^{2}}+4 \ln \relax (x )+4 \ln \left (x +1\right )\)	\(23\)
meijerg	\(\frac {x \left (-3 x +6\right )}{3}-2 x +4 \ln \left (x +1\right )+4 \ln \relax (x )-\frac {48}{x^{2}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^5-2*x^4+8*x^3+4*x^2+96*x+96)/(x^4+x^3),x,method=_RETURNVERBOSE)

[Out]

-x^2-48/x^2+4*ln(x^2+x)

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maxima [A]  time = 0.36, size = 21, normalized size = 1.00 \begin {gather*} -x^{2} - \frac {48}{x^{2}} + 4 \, \log \left (x + 1\right ) + 4 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5-2*x^4+8*x^3+4*x^2+96*x+96)/(x^4+x^3),x, algorithm="maxima")

[Out]

-x^2 - 48/x^2 + 4*log(x + 1) + 4*log(x)

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mupad [B]  time = 0.06, size = 19, normalized size = 0.90 \begin {gather*} 4\,\ln \left (x\,\left (x+1\right )\right )-\frac {48}{x^2}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((96*x + 4*x^2 + 8*x^3 - 2*x^4 - 2*x^5 + 96)/(x^3 + x^4),x)

[Out]

4*log(x*(x + 1)) - 48/x^2 - x^2

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sympy [A]  time = 0.09, size = 15, normalized size = 0.71 \begin {gather*} - x^{2} + 4 \log {\left (x^{2} + x \right )} - \frac {48}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**5-2*x**4+8*x**3+4*x**2+96*x+96)/(x**4+x**3),x)

[Out]

-x**2 + 4*log(x**2 + x) - 48/x**2

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