3.36.70 \(\int \frac {-30+20 x+85 x^2-140 x^3-75 x^4+90 x^5}{-3+6 x+11 x^2-17 x^3-12 x^4+9 x^5} \, dx\)

Optimal. Leaf size=36 \[ 5 \left (2 x-\log (x)-\log \left (\frac {1}{-((-2+x) x)+\frac {x}{1-x-3 x^2}}\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 35, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2074, 628, 1587} \begin {gather*} -5 \log \left (-3 x^2-x+1\right )+5 \log \left (3 x^3-5 x^2-3 x+3\right )+10 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-30 + 20*x + 85*x^2 - 140*x^3 - 75*x^4 + 90*x^5)/(-3 + 6*x + 11*x^2 - 17*x^3 - 12*x^4 + 9*x^5),x]

[Out]

10*x - 5*Log[1 - x - 3*x^2] + 5*Log[3 - 3*x - 5*x^2 + 3*x^3]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

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 (10-\frac {5 (1+6 x)}{-1+x+3 x^2}+\frac {5 \left (-3-10 x+9 x^2\right )}{3-3 x-5 x^2+3 x^3}\right ) \, dx\\ &=10 x-5 \int \frac {1+6 x}{-1+x+3 x^2} \, dx+5 \int \frac {-3-10 x+9 x^2}{3-3 x-5 x^2+3 x^3} \, dx\\ &=10 x-5 \log \left (1-x-3 x^2\right )+5 \log \left (3-3 x-5 x^2+3 x^3\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 35, normalized size = 0.97 \begin {gather*} 5 \left (2 x-\log \left (1-x-3 x^2\right )+\log \left (3-3 x-5 x^2+3 x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-30 + 20*x + 85*x^2 - 140*x^3 - 75*x^4 + 90*x^5)/(-3 + 6*x + 11*x^2 - 17*x^3 - 12*x^4 + 9*x^5),x]

[Out]

5*(2*x - Log[1 - x - 3*x^2] + Log[3 - 3*x - 5*x^2 + 3*x^3])

________________________________________________________________________________________

fricas [A]  time = 0.66, size = 33, normalized size = 0.92 \begin {gather*} 10 \, x + 5 \, \log \left (3 \, x^{3} - 5 \, x^{2} - 3 \, x + 3\right ) - 5 \, \log \left (3 \, x^{2} + x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((90*x^5-75*x^4-140*x^3+85*x^2+20*x-30)/(9*x^5-12*x^4-17*x^3+11*x^2+6*x-3),x, algorithm="fricas")

[Out]

10*x + 5*log(3*x^3 - 5*x^2 - 3*x + 3) - 5*log(3*x^2 + x - 1)

________________________________________________________________________________________

giac [A]  time = 0.23, size = 35, normalized size = 0.97 \begin {gather*} 10 \, x + 5 \, \log \left ({\left | 3 \, x^{3} - 5 \, x^{2} - 3 \, x + 3 \right |}\right ) - 5 \, \log \left ({\left | 3 \, x^{2} + x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((90*x^5-75*x^4-140*x^3+85*x^2+20*x-30)/(9*x^5-12*x^4-17*x^3+11*x^2+6*x-3),x, algorithm="giac")

[Out]

10*x + 5*log(abs(3*x^3 - 5*x^2 - 3*x + 3)) - 5*log(abs(3*x^2 + x - 1))

________________________________________________________________________________________

maple [A]  time = 0.03, size = 34, normalized size = 0.94

method	result	size
default	\(10 x -5 \ln \left (3 x^{2}+x -1\right )+5 \ln \left (3 x^{3}-5 x^{2}-3 x +3\right )\)	\(34\)
norman	\(10 x -5 \ln \left (3 x^{2}+x -1\right )+5 \ln \left (3 x^{3}-5 x^{2}-3 x +3\right )\)	\(34\)
risch	\(10 x -5 \ln \left (3 x^{2}+x -1\right )+5 \ln \left (3 x^{3}-5 x^{2}-3 x +3\right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((90*x^5-75*x^4-140*x^3+85*x^2+20*x-30)/(9*x^5-12*x^4-17*x^3+11*x^2+6*x-3),x,method=_RETURNVERBOSE)

[Out]

10*x-5*ln(3*x^2+x-1)+5*ln(3*x^3-5*x^2-3*x+3)

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 33, normalized size = 0.92 \begin {gather*} 10 \, x + 5 \, \log \left (3 \, x^{3} - 5 \, x^{2} - 3 \, x + 3\right ) - 5 \, \log \left (3 \, x^{2} + x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((90*x^5-75*x^4-140*x^3+85*x^2+20*x-30)/(9*x^5-12*x^4-17*x^3+11*x^2+6*x-3),x, algorithm="maxima")

[Out]

10*x + 5*log(3*x^3 - 5*x^2 - 3*x + 3) - 5*log(3*x^2 + x - 1)

________________________________________________________________________________________

mupad [B]  time = 2.16, size = 31, normalized size = 0.86 \begin {gather*} 10\,x-5\,\ln \left (3\,x^2+x-1\right )+5\,\ln \left (x^3-\frac {5\,x^2}{3}-x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x + 85*x^2 - 140*x^3 - 75*x^4 + 90*x^5 - 30)/(6*x + 11*x^2 - 17*x^3 - 12*x^4 + 9*x^5 - 3),x)

[Out]

10*x - 5*log(x + 3*x^2 - 1) + 5*log(x^3 - (5*x^2)/3 - x + 1)

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 32, normalized size = 0.89 \begin {gather*} 10 x - 5 \log {\left (3 x^{2} + x - 1 \right )} + 5 \log {\left (3 x^{3} - 5 x^{2} - 3 x + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((90*x**5-75*x**4-140*x**3+85*x**2+20*x-30)/(9*x**5-12*x**4-17*x**3+11*x**2+6*x-3),x)

[Out]

10*x - 5*log(3*x**2 + x - 1) + 5*log(3*x**3 - 5*x**2 - 3*x + 3)

________________________________________________________________________________________