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 \[ \frac {9}{2} \log (a-x)-17 \log (2 a-x)+\frac {35}{2} \log (3 a-x) \]

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Rubi [A]  time = 0.05, antiderivative size = 33, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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

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IntegrateAlgebraic [A]  time = 0.02, size = 33, normalized size = 1.00 \[ \frac {9}{2} \log (a-x)-17 \log (2 a-x)+\frac {35}{2} \log (3 a-x) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(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

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fricas [A]  time = 0.73, size = 25, normalized size = 0.76 \[ \frac {9}{2} \, \log \left (-a + x\right ) - 17 \, \log \left (-2 \, a + x\right ) + \frac {35}{2} \, \log \left (-3 \, a + x\right ) \]

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)

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giac [A]  time = 0.97, size = 28, normalized size = 0.85 \[ \frac {9}{2} \, \log \left ({\left | -a + x \right |}\right ) - 17 \, \log \left ({\left | -2 \, a + x \right |}\right ) + \frac {35}{2} \, \log \left ({\left | -3 \, a + x \right |}\right ) \]

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))

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maple [A]  time = 0.04, size = 26, normalized size = 0.79

method	result	size
risch	\(-17 \ln \left (-2 a +x \right )+\frac {9 \ln \left (-a +x \right )}{2}+\frac {35 \ln \left (-3 a +x \right )}{2}\)	\(26\)
default	\(\frac {9 \ln \left (a -x \right )}{2}-17 \ln \left (2 a -x \right )+\frac {35 \ln \left (3 a -x \right )}{2}\)	\(30\)
norman	\(\frac {9 \ln \left (a -x \right )}{2}-17 \ln \left (2 a -x \right )+\frac {35 \ln \left (3 a -x \right )}{2}\)	\(30\)

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,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.44, size = 25, normalized size = 0.76 \[ \frac {9}{2} \, \log \left (-a + x\right ) - 17 \, \log \left (-2 \, a + x\right ) + \frac {35}{2} \, \log \left (-3 \, a + x\right ) \]

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)

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mupad [B]  time = 0.09, size = 25, normalized size = 0.76 \[ \frac {9\,\ln \left (x-a\right )}{2}-17\,\ln \left (x-2\,a\right )+\frac {35\,\ln \left (x-3\,a\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.26, size = 26, normalized size = 0.79 \[ \frac {35 \log {\left (- 3 a + x \right )}}{2} - 17 \log {\left (- 2 a + x \right )} + \frac {9 \log {\left (- a + x \right )}}{2} \]

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

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