3.59.45 \(\int \frac {-5-15 x+30 x^2+14 x^3+x^4}{10-15 x+10 x^3+x^4} \, dx\)

Optimal. Leaf size=20 \[ x+\log \left (-2+3 x-\left (2+\frac {x}{5}\right ) x^3\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6742, 1587} \begin {gather*} \log \left (x^4+10 x^3-15 x+10\right )+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 - 15*x + 30*x^2 + 14*x^3 + x^4)/(10 - 15*x + 10*x^3 + x^4),x]

[Out]

x + Log[10 - 15*x + 10*x^3 + x^4]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {15-30 x^2-4 x^3}{10-15 x+10 x^3+x^4}\right ) \, dx\\ &=x-\int \frac {15-30 x^2-4 x^3}{10-15 x+10 x^3+x^4} \, dx\\ &=x+\log \left (10-15 x+10 x^3+x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.80 \begin {gather*} x+\log \left (10-15 x+10 x^3+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 - 15*x + 30*x^2 + 14*x^3 + x^4)/(10 - 15*x + 10*x^3 + x^4),x]

[Out]

x + Log[10 - 15*x + 10*x^3 + x^4]

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fricas [A]  time = 0.79, size = 16, normalized size = 0.80 \begin {gather*} x + \log \left (x^{4} + 10 \, x^{3} - 15 \, x + 10\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+14*x^3+30*x^2-15*x-5)/(x^4+10*x^3-15*x+10),x, algorithm="fricas")

[Out]

x + log(x^4 + 10*x^3 - 15*x + 10)

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giac [A]  time = 0.13, size = 17, normalized size = 0.85 \begin {gather*} x + \log \left ({\left | x^{4} + 10 \, x^{3} - 15 \, x + 10 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+14*x^3+30*x^2-15*x-5)/(x^4+10*x^3-15*x+10),x, algorithm="giac")

[Out]

x + log(abs(x^4 + 10*x^3 - 15*x + 10))

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maple [A]  time = 0.04, size = 17, normalized size = 0.85

method	result	size
default	\(x +\ln \left (x^{4}+10 x^{3}-15 x +10\right )\)	\(17\)
norman	\(x +\ln \left (x^{4}+10 x^{3}-15 x +10\right )\)	\(17\)
risch	\(x +\ln \left (x^{4}+10 x^{3}-15 x +10\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+14*x^3+30*x^2-15*x-5)/(x^4+10*x^3-15*x+10),x,method=_RETURNVERBOSE)

[Out]

x+ln(x^4+10*x^3-15*x+10)

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maxima [A]  time = 0.44, size = 16, normalized size = 0.80 \begin {gather*} x + \log \left (x^{4} + 10 \, x^{3} - 15 \, x + 10\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+14*x^3+30*x^2-15*x-5)/(x^4+10*x^3-15*x+10),x, algorithm="maxima")

[Out]

x + log(x^4 + 10*x^3 - 15*x + 10)

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mupad [B]  time = 4.04, size = 16, normalized size = 0.80 \begin {gather*} x+\ln \left (x^4+10\,x^3-15\,x+10\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((30*x^2 - 15*x + 14*x^3 + x^4 - 5)/(10*x^3 - 15*x + x^4 + 10),x)

[Out]

x + log(10*x^3 - 15*x + x^4 + 10)

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sympy [A]  time = 0.09, size = 15, normalized size = 0.75 \begin {gather*} x + \log {\left (x^{4} + 10 x^{3} - 15 x + 10 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+14*x**3+30*x**2-15*x-5)/(x**4+10*x**3-15*x+10),x)

[Out]

x + log(x**4 + 10*x**3 - 15*x + 10)

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