3.62.86 \(\int \frac {3970-2520 x-9360 x^2-6400 x^3-1440 x^4}{-33-463 x-315 x^2+298 x^3+316 x^4+72 x^5} \, dx\)

Optimal. Leaf size=27 \[ 5 \log \left (\frac {2}{\left (-4-x+\left (3 x-\frac {5}{3+2 x}\right )^2\right )^2}\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2074, 1587} \begin {gather*} 20 \log (2 x+3)-10 \log \left (-36 x^4-104 x^3+7 x^2+147 x+11\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3970 - 2520*x - 9360*x^2 - 6400*x^3 - 1440*x^4)/(-33 - 463*x - 315*x^2 + 298*x^3 + 316*x^4 + 72*x^5),x]

[Out]

20*Log[3 + 2*x] - 10*Log[11 + 147*x + 7*x^2 - 104*x^3 - 36*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 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 {40}{3+2 x}-\frac {10 \left (-147-14 x+312 x^2+144 x^3\right )}{-11-147 x-7 x^2+104 x^3+36 x^4}\right ) \, dx\\ &=20 \log (3+2 x)-10 \int \frac {-147-14 x+312 x^2+144 x^3}{-11-147 x-7 x^2+104 x^3+36 x^4} \, dx\\ &=20 \log (3+2 x)-10 \log \left (11+147 x+7 x^2-104 x^3-36 x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 32, normalized size = 1.19 \begin {gather*} -10 \left (-2 \log (3+2 x)+\log \left (11+147 x+7 x^2-104 x^3-36 x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3970 - 2520*x - 9360*x^2 - 6400*x^3 - 1440*x^4)/(-33 - 463*x - 315*x^2 + 298*x^3 + 316*x^4 + 72*x^5
),x]

[Out]

-10*(-2*Log[3 + 2*x] + Log[11 + 147*x + 7*x^2 - 104*x^3 - 36*x^4])

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fricas [A]  time = 0.55, size = 32, normalized size = 1.19 \begin {gather*} -10 \, \log \left (36 \, x^{4} + 104 \, x^{3} - 7 \, x^{2} - 147 \, x - 11\right ) + 20 \, \log \left (2 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1440*x^4-6400*x^3-9360*x^2-2520*x+3970)/(72*x^5+316*x^4+298*x^3-315*x^2-463*x-33),x, algorithm="fr
icas")

[Out]

-10*log(36*x^4 + 104*x^3 - 7*x^2 - 147*x - 11) + 20*log(2*x + 3)

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giac [A]  time = 0.18, size = 34, normalized size = 1.26 \begin {gather*} -10 \, \log \left ({\left | 36 \, x^{4} + 104 \, x^{3} - 7 \, x^{2} - 147 \, x - 11 \right |}\right ) + 20 \, \log \left ({\left | 2 \, x + 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1440*x^4-6400*x^3-9360*x^2-2520*x+3970)/(72*x^5+316*x^4+298*x^3-315*x^2-463*x-33),x, algorithm="gi
ac")

[Out]

-10*log(abs(36*x^4 + 104*x^3 - 7*x^2 - 147*x - 11)) + 20*log(abs(2*x + 3))

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maple [A]  time = 0.04, size = 33, normalized size = 1.22

method	result	size
default	\(-10 \ln \left (36 x^{4}+104 x^{3}-7 x^{2}-147 x -11\right )+20 \ln \left (2 x +3\right )\)	\(33\)
norman	\(-10 \ln \left (36 x^{4}+104 x^{3}-7 x^{2}-147 x -11\right )+20 \ln \left (2 x +3\right )\)	\(33\)
risch	\(-10 \ln \left (36 x^{4}+104 x^{3}-7 x^{2}-147 x -11\right )+20 \ln \left (2 x +3\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1440*x^4-6400*x^3-9360*x^2-2520*x+3970)/(72*x^5+316*x^4+298*x^3-315*x^2-463*x-33),x,method=_RETURNVERBOS
E)

[Out]

-10*ln(36*x^4+104*x^3-7*x^2-147*x-11)+20*ln(2*x+3)

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maxima [A]  time = 0.35, size = 32, normalized size = 1.19 \begin {gather*} -10 \, \log \left (36 \, x^{4} + 104 \, x^{3} - 7 \, x^{2} - 147 \, x - 11\right ) + 20 \, \log \left (2 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1440*x^4-6400*x^3-9360*x^2-2520*x+3970)/(72*x^5+316*x^4+298*x^3-315*x^2-463*x-33),x, algorithm="ma
xima")

[Out]

-10*log(36*x^4 + 104*x^3 - 7*x^2 - 147*x - 11) + 20*log(2*x + 3)

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mupad [B]  time = 4.12, size = 30, normalized size = 1.11 \begin {gather*} 20\,\ln \left (x+\frac {3}{2}\right )-10\,\ln \left (36\,x^4+104\,x^3-7\,x^2-147\,x-11\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2520*x + 9360*x^2 + 6400*x^3 + 1440*x^4 - 3970)/(463*x + 315*x^2 - 298*x^3 - 316*x^4 - 72*x^5 + 33),x)

[Out]

20*log(x + 3/2) - 10*log(104*x^3 - 7*x^2 - 147*x + 36*x^4 - 11)

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sympy [A]  time = 0.12, size = 31, normalized size = 1.15 \begin {gather*} 20 \log {\left (2 x + 3 \right )} - 10 \log {\left (36 x^{4} + 104 x^{3} - 7 x^{2} - 147 x - 11 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1440*x**4-6400*x**3-9360*x**2-2520*x+3970)/(72*x**5+316*x**4+298*x**3-315*x**2-463*x-33),x)

[Out]

20*log(2*x + 3) - 10*log(36*x**4 + 104*x**3 - 7*x**2 - 147*x - 11)

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