3.14.56 \(\int \frac {-15-12 x^3}{25 x^2+10 x^5+x^8+(10 x+2 x^4) \log (5)+\log ^2(5)} \, dx\)

Optimal. Leaf size=19 \[ 1+\frac {1}{x+\frac {1}{3} \left (2 x+x^4+\log (5)\right )} \]

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Rubi [A]  time = 0.10, antiderivative size = 13, normalized size of antiderivative = 0.68, number of steps used = 3, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2074, 2099} \begin {gather*} \frac {3}{x^4+5 x+\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-15 - 12*x^3)/(25*x^2 + 10*x^5 + x^8 + (10*x + 2*x^4)*Log[5] + Log[5]^2),x]

[Out]

3/(5*x + x^4 + Log[5])

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]

Rule 2099

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Qn^(p + 1)
)/(n*(p + 1)*Coeff[Qn, x, n]), x] + Dist[Simplify[Pm - (Coeff[Pm, x, m]*D[Qn, x])/(n*Coeff[Qn, x, n])], Int[Qn
^p, x], x] /; EqQ[m, n - 1] && EqQ[D[Simplify[Pm - (Coeff[Pm, x, m]*D[Qn, x])/(n*Coeff[Qn, x, n])], x], 0]] /;
FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {15}{\left (5 x+x^4+\log (5)\right )^2}-\frac {12 x^3}{\left (5 x+x^4+\log (5)\right )^2}\right ) \, dx\\ &=-\left (12 \int \frac {x^3}{\left (5 x+x^4+\log (5)\right )^2} \, dx\right )-15 \int \frac {1}{\left (5 x+x^4+\log (5)\right )^2} \, dx\\ &=\frac {3}{5 x+x^4+\log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.68 \begin {gather*} \frac {3}{5 x+x^4+\log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-15 - 12*x^3)/(25*x^2 + 10*x^5 + x^8 + (10*x + 2*x^4)*Log[5] + Log[5]^2),x]

[Out]

3/(5*x + x^4 + Log[5])

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fricas [A]  time = 1.00, size = 13, normalized size = 0.68 \begin {gather*} \frac {3}{x^{4} + 5 \, x + \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^3-15)/(log(5)^2+(2*x^4+10*x)*log(5)+x^8+10*x^5+25*x^2),x, algorithm="fricas")

[Out]

3/(x^4 + 5*x + log(5))

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giac [A]  time = 0.36, size = 13, normalized size = 0.68 \begin {gather*} \frac {3}{x^{4} + 5 \, x + \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^3-15)/(log(5)^2+(2*x^4+10*x)*log(5)+x^8+10*x^5+25*x^2),x, algorithm="giac")

[Out]

3/(x^4 + 5*x + log(5))

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maple [A]  time = 0.06, size = 14, normalized size = 0.74

method	result	size
gosper	\(\frac {3}{x^{4}+\ln \relax (5)+5 x}\)	\(14\)
default	\(\frac {3}{x^{4}+\ln \relax (5)+5 x}\)	\(14\)
norman	\(\frac {3}{x^{4}+\ln \relax (5)+5 x}\)	\(14\)
risch	\(\frac {3}{x^{4}+\ln \relax (5)+5 x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12*x^3-15)/(ln(5)^2+(2*x^4+10*x)*ln(5)+x^8+10*x^5+25*x^2),x,method=_RETURNVERBOSE)

[Out]

3/(x^4+ln(5)+5*x)

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maxima [A]  time = 0.66, size = 13, normalized size = 0.68 \begin {gather*} \frac {3}{x^{4} + 5 \, x + \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x^3-15)/(log(5)^2+(2*x^4+10*x)*log(5)+x^8+10*x^5+25*x^2),x, algorithm="maxima")

[Out]

3/(x^4 + 5*x + log(5))

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mupad [B]  time = 0.11, size = 13, normalized size = 0.68 \begin {gather*} \frac {3}{x^4+5\,x+\ln \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12*x^3 + 15)/(log(5)*(10*x + 2*x^4) + log(5)^2 + 25*x^2 + 10*x^5 + x^8),x)

[Out]

3/(5*x + log(5) + x^4)

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sympy [A]  time = 0.35, size = 10, normalized size = 0.53 \begin {gather*} \frac {3}{x^{4} + 5 x + \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*x**3-15)/(ln(5)**2+(2*x**4+10*x)*ln(5)+x**8+10*x**5+25*x**2),x)

[Out]

3/(x**4 + 5*x + log(5))

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