3.97 \(\int \frac{11 a^2-7 a x+5 x^2}{-6 a^3+11 a^2 x-6 a x^2+x^3} \, dx\)

Optimal. Leaf size=33 \[ \frac{9}{2} \log (a-x)-17 \log (2 a-x)+\frac{35}{2} \log (3 a-x) \]

[Out]

(9*Log[a - x])/2 - 17*Log[2*a - x] + (35*Log[3*a - x])/2

________________________________________________________________________________________

Rubi [A]  time = 0.0538914, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {2074} \[ \frac{9}{2} \log (a-x)-17 \log (2 a-x)+\frac{35}{2} \log (3 a-x) \]

Antiderivative was successfully verified.

[In]

Int[(11*a^2 - 7*a*x + 5*x^2)/(-6*a^3 + 11*a^2*x - 6*a*x^2 + x^3),x]

[Out]

(9*Log[a - x])/2 - 17*Log[2*a - x] + (35*Log[3*a - x])/2

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{align*} \int \frac{11 a^2-7 a x+5 x^2}{-6 a^3+11 a^2 x-6 a x^2+x^3} \, dx &=\int \left (-\frac{9}{2 (a-x)}+\frac{17}{2 a-x}-\frac{35}{2 (3 a-x)}\right ) \, dx\\ &=\frac{9}{2} \log (a-x)-17 \log (2 a-x)+\frac{35}{2} \log (3 a-x)\\ \end{align*}

Mathematica [A]  time = 0.014098, size = 29, normalized size = 0.88 \[ \frac{35}{2} \log (x-3 a)-17 \log (x-2 a)+\frac{9}{2} \log (x-a) \]

Antiderivative was successfully verified.

[In]

Integrate[(11*a^2 - 7*a*x + 5*x^2)/(-6*a^3 + 11*a^2*x - 6*a*x^2 + x^3),x]

[Out]

(35*Log[-3*a + x])/2 - 17*Log[-2*a + x] + (9*Log[-a + x])/2

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 26, normalized size = 0.8 \begin{align*}{\frac{35\,\ln \left ( x-3\,a \right ) }{2}}+{\frac{9\,\ln \left ( -a+x \right ) }{2}}-17\,\ln \left ( x-2\,a \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((11*a^2-7*a*x+5*x^2)/(-6*a^3+11*a^2*x-6*a*x^2+x^3),x)

[Out]

35/2*ln(x-3*a)+9/2*ln(-a+x)-17*ln(x-2*a)

________________________________________________________________________________________

Maxima [A]  time = 0.938507, size = 34, normalized size = 1.03 \begin{align*} \frac{9}{2} \, \log \left (-a + x\right ) - 17 \, \log \left (-2 \, a + x\right ) + \frac{35}{2} \, \log \left (-3 \, a + x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((11*a^2-7*a*x+5*x^2)/(-6*a^3+11*a^2*x-6*a*x^2+x^3),x, algorithm="maxima")

[Out]

9/2*log(-a + x) - 17*log(-2*a + x) + 35/2*log(-3*a + x)

________________________________________________________________________________________

Fricas [A]  time = 1.82003, size = 77, normalized size = 2.33 \begin{align*} \frac{9}{2} \, \log \left (-a + x\right ) - 17 \, \log \left (-2 \, a + x\right ) + \frac{35}{2} \, \log \left (-3 \, a + x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((11*a^2-7*a*x+5*x^2)/(-6*a^3+11*a^2*x-6*a*x^2+x^3),x, algorithm="fricas")

[Out]

9/2*log(-a + x) - 17*log(-2*a + x) + 35/2*log(-3*a + x)

________________________________________________________________________________________

Sympy [A]  time = 0.371312, size = 26, normalized size = 0.79 \begin{align*} \frac{35 \log{\left (- 3 a + x \right )}}{2} - 17 \log{\left (- 2 a + x \right )} + \frac{9 \log{\left (- a + x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((11*a**2-7*a*x+5*x**2)/(-6*a**3+11*a**2*x-6*a*x**2+x**3),x)

[Out]

35*log(-3*a + x)/2 - 17*log(-2*a + x) + 9*log(-a + x)/2

________________________________________________________________________________________

Giac [A]  time = 1.05728, size = 38, normalized size = 1.15 \begin{align*} \frac{9}{2} \, \log \left ({\left | -a + x \right |}\right ) - 17 \, \log \left ({\left | -2 \, a + x \right |}\right ) + \frac{35}{2} \, \log \left ({\left | -3 \, a + x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((11*a^2-7*a*x+5*x^2)/(-6*a^3+11*a^2*x-6*a*x^2+x^3),x, algorithm="giac")

[Out]

9/2*log(abs(-a + x)) - 17*log(abs(-2*a + x)) + 35/2*log(abs(-3*a + x))