3.18.100 \(\int \frac {950-60 x-180 x^2+24 x^3}{725+475 x-15 x^2-30 x^3+3 x^4} \, dx\)

Optimal. Leaf size=22 \[ 2 \log \left (2+x+3 \left (-3-x+\frac {x^2}{5}\right )^2\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {1587} \begin {gather*} 2 \log \left (3 x^4-30 x^3-15 x^2+475 x+725\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(950 - 60*x - 180*x^2 + 24*x^3)/(725 + 475*x - 15*x^2 - 30*x^3 + 3*x^4),x]

[Out]

2*Log[725 + 475*x - 15*x^2 - 30*x^3 + 3*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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 \log \left (725+475 x-15 x^2-30 x^3+3 x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.05 \begin {gather*} 2 \log \left (725+475 x-15 x^2-30 x^3+3 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(950 - 60*x - 180*x^2 + 24*x^3)/(725 + 475*x - 15*x^2 - 30*x^3 + 3*x^4),x]

[Out]

2*Log[725 + 475*x - 15*x^2 - 30*x^3 + 3*x^4]

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fricas [A]  time = 0.93, size = 23, normalized size = 1.05 \begin {gather*} 2 \, \log \left (3 \, x^{4} - 30 \, x^{3} - 15 \, x^{2} + 475 \, x + 725\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^3-180*x^2-60*x+950)/(3*x^4-30*x^3-15*x^2+475*x+725),x, algorithm="fricas")

[Out]

2*log(3*x^4 - 30*x^3 - 15*x^2 + 475*x + 725)

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giac [A]  time = 0.34, size = 24, normalized size = 1.09 \begin {gather*} 2 \, \log \left ({\left | 3 \, x^{4} - 30 \, x^{3} - 15 \, x^{2} + 475 \, x + 725 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^3-180*x^2-60*x+950)/(3*x^4-30*x^3-15*x^2+475*x+725),x, algorithm="giac")

[Out]

2*log(abs(3*x^4 - 30*x^3 - 15*x^2 + 475*x + 725))

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maple [A]  time = 0.02, size = 24, normalized size = 1.09

method	result	size
default	\(2 \ln \left (3 x^{4}-30 x^{3}-15 x^{2}+475 x +725\right )\)	\(24\)
norman	\(2 \ln \left (3 x^{4}-30 x^{3}-15 x^{2}+475 x +725\right )\)	\(24\)
risch	\(2 \ln \left (3 x^{4}-30 x^{3}-15 x^{2}+475 x +725\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*x^3-180*x^2-60*x+950)/(3*x^4-30*x^3-15*x^2+475*x+725),x,method=_RETURNVERBOSE)

[Out]

2*ln(3*x^4-30*x^3-15*x^2+475*x+725)

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maxima [A]  time = 0.61, size = 23, normalized size = 1.05 \begin {gather*} 2 \, \log \left (3 \, x^{4} - 30 \, x^{3} - 15 \, x^{2} + 475 \, x + 725\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x^3-180*x^2-60*x+950)/(3*x^4-30*x^3-15*x^2+475*x+725),x, algorithm="maxima")

[Out]

2*log(3*x^4 - 30*x^3 - 15*x^2 + 475*x + 725)

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mupad [B]  time = 0.10, size = 21, normalized size = 0.95 \begin {gather*} 2\,\ln \left (x^4-10\,x^3-5\,x^2+\frac {475\,x}{3}+\frac {725}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(60*x + 180*x^2 - 24*x^3 - 950)/(475*x - 15*x^2 - 30*x^3 + 3*x^4 + 725),x)

[Out]

2*log((475*x)/3 - 5*x^2 - 10*x^3 + x^4 + 725/3)

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sympy [A]  time = 0.09, size = 22, normalized size = 1.00 \begin {gather*} 2 \log {\left (3 x^{4} - 30 x^{3} - 15 x^{2} + 475 x + 725 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*x**3-180*x**2-60*x+950)/(3*x**4-30*x**3-15*x**2+475*x+725),x)

[Out]

2*log(3*x**4 - 30*x**3 - 15*x**2 + 475*x + 725)

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