3.58.89 \(\int \frac {-4-188 x-2275 x^2+1250 x^3}{4+196 x+2306 x^2-2325 x^3+625 x^4} \, dx\)

Optimal. Leaf size=29 \[ \frac {1}{4}+\log \left (-(2-x)^2-\frac {x}{5+\frac {1}{5 x}}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 25, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2074, 1587} \begin {gather*} \log \left (25 x^3-94 x^2+96 x+4\right )-\log (25 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 - 188*x - 2275*x^2 + 1250*x^3)/(4 + 196*x + 2306*x^2 - 2325*x^3 + 625*x^4),x]

[Out]

-Log[1 + 25*x] + Log[4 + 96*x - 94*x^2 + 25*x^3]

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 {25}{1+25 x}+\frac {96-188 x+75 x^2}{4+96 x-94 x^2+25 x^3}\right ) \, dx\\ &=-\log (1+25 x)+\int \frac {96-188 x+75 x^2}{4+96 x-94 x^2+25 x^3} \, dx\\ &=-\log (1+25 x)+\log \left (4+96 x-94 x^2+25 x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.86 \begin {gather*} -\log (1+25 x)+\log \left (4+96 x-94 x^2+25 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 188*x - 2275*x^2 + 1250*x^3)/(4 + 196*x + 2306*x^2 - 2325*x^3 + 625*x^4),x]

[Out]

-Log[1 + 25*x] + Log[4 + 96*x - 94*x^2 + 25*x^3]

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fricas [A]  time = 0.74, size = 25, normalized size = 0.86 \begin {gather*} \log \left (25 \, x^{3} - 94 \, x^{2} + 96 \, x + 4\right ) - \log \left (25 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1250*x^3-2275*x^2-188*x-4)/(625*x^4-2325*x^3+2306*x^2+196*x+4),x, algorithm="fricas")

[Out]

log(25*x^3 - 94*x^2 + 96*x + 4) - log(25*x + 1)

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giac [A]  time = 0.22, size = 27, normalized size = 0.93 \begin {gather*} \log \left ({\left | 25 \, x^{3} - 94 \, x^{2} + 96 \, x + 4 \right |}\right ) - \log \left ({\left | 25 \, x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1250*x^3-2275*x^2-188*x-4)/(625*x^4-2325*x^3+2306*x^2+196*x+4),x, algorithm="giac")

[Out]

log(abs(25*x^3 - 94*x^2 + 96*x + 4)) - log(abs(25*x + 1))

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maple [A]  time = 0.03, size = 26, normalized size = 0.90

method	result	size
default	\(\ln \left (25 x^{3}-94 x^{2}+96 x +4\right )-\ln \left (25 x +1\right )\)	\(26\)
norman	\(\ln \left (25 x^{3}-94 x^{2}+96 x +4\right )-\ln \left (25 x +1\right )\)	\(26\)
risch	\(\ln \left (25 x^{3}-94 x^{2}+96 x +4\right )-\ln \left (25 x +1\right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1250*x^3-2275*x^2-188*x-4)/(625*x^4-2325*x^3+2306*x^2+196*x+4),x,method=_RETURNVERBOSE)

[Out]

ln(25*x^3-94*x^2+96*x+4)-ln(25*x+1)

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maxima [A]  time = 0.36, size = 25, normalized size = 0.86 \begin {gather*} \log \left (25 \, x^{3} - 94 \, x^{2} + 96 \, x + 4\right ) - \log \left (25 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1250*x^3-2275*x^2-188*x-4)/(625*x^4-2325*x^3+2306*x^2+196*x+4),x, algorithm="maxima")

[Out]

log(25*x^3 - 94*x^2 + 96*x + 4) - log(25*x + 1)

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mupad [B]  time = 4.19, size = 23, normalized size = 0.79 \begin {gather*} \ln \left (25\,x^3-94\,x^2+96\,x+4\right )-\ln \left (x+\frac {1}{25}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(188*x + 2275*x^2 - 1250*x^3 + 4)/(196*x + 2306*x^2 - 2325*x^3 + 625*x^4 + 4),x)

[Out]

log(96*x - 94*x^2 + 25*x^3 + 4) - log(x + 1/25)

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sympy [A]  time = 0.10, size = 22, normalized size = 0.76 \begin {gather*} - \log {\left (25 x + 1 \right )} + \log {\left (25 x^{3} - 94 x^{2} + 96 x + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1250*x**3-2275*x**2-188*x-4)/(625*x**4-2325*x**3+2306*x**2+196*x+4),x)

[Out]

-log(25*x + 1) + log(25*x**3 - 94*x**2 + 96*x + 4)

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