3.64.18 \(\int \frac {-15 e^3+3 x^2+2 x^3}{15 e^3 x+x^2+3 x^3+x^4} \, dx\)

Optimal. Leaf size=22 \[ \log \left (\frac {15 e^3}{x}+3 x+\frac {x+x^3}{x}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2074, 1587} \begin {gather*} \log \left (x^3+3 x^2+x+15 e^3\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[x] + Log[15*E^3 + x + 3*x^2 + x^3]

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 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {1}{x}+\frac {1+6 x+3 x^2}{15 e^3+x+3 x^2+x^3}\right ) \, dx\\ &=-\log (x)+\int \frac {1+6 x+3 x^2}{15 e^3+x+3 x^2+x^3} \, dx\\ &=-\log (x)+\log \left (15 e^3+x+3 x^2+x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 21, normalized size = 0.95 \begin {gather*} -\log (x)+\log \left (15 e^3+x+3 x^2+x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[x] + Log[15*E^3 + x + 3*x^2 + x^3]

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fricas [A]  time = 0.66, size = 20, normalized size = 0.91 \begin {gather*} \log \left (x^{3} + 3 \, x^{2} + x + 15 \, e^{3}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^3 + 3*x^2 + x + 15*e^3) - log(x)

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giac [A]  time = 0.12, size = 22, normalized size = 1.00 \begin {gather*} \log \left ({\left | x^{3} + 3 \, x^{2} + x + 15 \, e^{3} \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(x^3 + 3*x^2 + x + 15*e^3)) - log(abs(x))

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maple [A]  time = 0.07, size = 21, normalized size = 0.95

method	result	size
default	\(\ln \left (x^{3}+3 x^{2}+15 \,{\mathrm e}^{3}+x \right )-\ln \relax (x )\)	\(21\)
norman	\(\ln \left (x^{3}+3 x^{2}+15 \,{\mathrm e}^{3}+x \right )-\ln \relax (x )\)	\(21\)
risch	\(\ln \left (x^{3}+3 x^{2}+15 \,{\mathrm e}^{3}+x \right )-\ln \relax (x )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x^3+3*x^2+15*exp(3)+x)-ln(x)

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maxima [A]  time = 0.37, size = 20, normalized size = 0.91 \begin {gather*} \log \left (x^{3} + 3 \, x^{2} + x + 15 \, e^{3}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^3 + 3*x^2 + x + 15*e^3) - log(x)

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mupad [B]  time = 4.37, size = 20, normalized size = 0.91 \begin {gather*} \ln \left (x^3+3\,x^2+x+15\,{\mathrm {e}}^3\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 - 15*exp(3) + 2*x^3)/(15*x*exp(3) + x^2 + 3*x^3 + x^4),x)

[Out]

log(x + 15*exp(3) + 3*x^2 + x^3) - log(x)

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sympy [A]  time = 0.73, size = 19, normalized size = 0.86 \begin {gather*} - \log {\relax (x )} + \log {\left (x^{3} + 3 x^{2} + x + 15 e^{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-15*exp(3)+2*x**3+3*x**2)/(15*x*exp(3)+x**4+3*x**3+x**2),x)

[Out]

-log(x) + log(x**3 + 3*x**2 + x + 15*exp(3))

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