3.59.51 \(\int \frac {1171875+777500 x+3750 x^3+2503 x^4+3 x^6+2 x^7}{11328125+1171875 x+388750 x^2+36250 x^3+3750 x^4+1247 x^5+29 x^6+3 x^7+x^8} \, dx\)

Optimal. Leaf size=22 \[ \log \left (-25+x-(2+x)^2+\frac {3}{\frac {625}{x^2}+x}\right ) \]

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Rubi [A]  time = 0.17, antiderivative size = 33, normalized size of antiderivative = 1.50, number of steps used = 4, number of rules used = 3, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2074, 260, 1587} \begin {gather*} \log \left (x^5+3 x^4+29 x^3+622 x^2+1875 x+18125\right )-\log \left (x^3+625\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1171875 + 777500*x + 3750*x^3 + 2503*x^4 + 3*x^6 + 2*x^7)/(11328125 + 1171875*x + 388750*x^2 + 36250*x^3
+ 3750*x^4 + 1247*x^5 + 29*x^6 + 3*x^7 + x^8),x]

[Out]

-Log[625 + x^3] + Log[18125 + 1875*x + 622*x^2 + 29*x^3 + 3*x^4 + x^5]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 {3 x^2}{625+x^3}+\frac {1875+1244 x+87 x^2+12 x^3+5 x^4}{18125+1875 x+622 x^2+29 x^3+3 x^4+x^5}\right ) \, dx\\ &=-\left (3 \int \frac {x^2}{625+x^3} \, dx\right )+\int \frac {1875+1244 x+87 x^2+12 x^3+5 x^4}{18125+1875 x+622 x^2+29 x^3+3 x^4+x^5} \, dx\\ &=-\log \left (625+x^3\right )+\log \left (18125+1875 x+622 x^2+29 x^3+3 x^4+x^5\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 1.50 \begin {gather*} -\log \left (625+x^3\right )+\log \left (18125+1875 x+622 x^2+29 x^3+3 x^4+x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1171875 + 777500*x + 3750*x^3 + 2503*x^4 + 3*x^6 + 2*x^7)/(11328125 + 1171875*x + 388750*x^2 + 3625
0*x^3 + 3750*x^4 + 1247*x^5 + 29*x^6 + 3*x^7 + x^8),x]

[Out]

-Log[625 + x^3] + Log[18125 + 1875*x + 622*x^2 + 29*x^3 + 3*x^4 + x^5]

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fricas [A]  time = 0.87, size = 33, normalized size = 1.50 \begin {gather*} \log \left (x^{5} + 3 \, x^{4} + 29 \, x^{3} + 622 \, x^{2} + 1875 \, x + 18125\right ) - \log \left (x^{3} + 625\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7+3*x^6+2503*x^4+3750*x^3+777500*x+1171875)/(x^8+3*x^7+29*x^6+1247*x^5+3750*x^4+36250*x^3+38875
0*x^2+1171875*x+11328125),x, algorithm="fricas")

[Out]

log(x^5 + 3*x^4 + 29*x^3 + 622*x^2 + 1875*x + 18125) - log(x^3 + 625)

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giac [A]  time = 0.19, size = 35, normalized size = 1.59 \begin {gather*} \log \left ({\left | x^{5} + 3 \, x^{4} + 29 \, x^{3} + 622 \, x^{2} + 1875 \, x + 18125 \right |}\right ) - \log \left ({\left | x^{3} + 625 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7+3*x^6+2503*x^4+3750*x^3+777500*x+1171875)/(x^8+3*x^7+29*x^6+1247*x^5+3750*x^4+36250*x^3+38875
0*x^2+1171875*x+11328125),x, algorithm="giac")

[Out]

log(abs(x^5 + 3*x^4 + 29*x^3 + 622*x^2 + 1875*x + 18125)) - log(abs(x^3 + 625))

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maple [A]  time = 0.04, size = 34, normalized size = 1.55

method	result	size
default	\(\ln \left (x^{5}+3 x^{4}+29 x^{3}+622 x^{2}+1875 x +18125\right )-\ln \left (x^{3}+625\right )\)	\(34\)
norman	\(\ln \left (x^{5}+3 x^{4}+29 x^{3}+622 x^{2}+1875 x +18125\right )-\ln \left (x^{3}+625\right )\)	\(34\)
risch	\(\ln \left (x^{5}+3 x^{4}+29 x^{3}+622 x^{2}+1875 x +18125\right )-\ln \left (x^{3}+625\right )\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^7+3*x^6+2503*x^4+3750*x^3+777500*x+1171875)/(x^8+3*x^7+29*x^6+1247*x^5+3750*x^4+36250*x^3+388750*x^2+
1171875*x+11328125),x,method=_RETURNVERBOSE)

[Out]

ln(x^5+3*x^4+29*x^3+622*x^2+1875*x+18125)-ln(x^3+625)

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maxima [A]  time = 0.35, size = 33, normalized size = 1.50 \begin {gather*} \log \left (x^{5} + 3 \, x^{4} + 29 \, x^{3} + 622 \, x^{2} + 1875 \, x + 18125\right ) - \log \left (x^{3} + 625\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^7+3*x^6+2503*x^4+3750*x^3+777500*x+1171875)/(x^8+3*x^7+29*x^6+1247*x^5+3750*x^4+36250*x^3+38875
0*x^2+1171875*x+11328125),x, algorithm="maxima")

[Out]

log(x^5 + 3*x^4 + 29*x^3 + 622*x^2 + 1875*x + 18125) - log(x^3 + 625)

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mupad [B]  time = 0.19, size = 33, normalized size = 1.50 \begin {gather*} \ln \left (x^5+3\,x^4+29\,x^3+622\,x^2+1875\,x+18125\right )-\ln \left (x^3+625\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((777500*x + 3750*x^3 + 2503*x^4 + 3*x^6 + 2*x^7 + 1171875)/(1171875*x + 388750*x^2 + 36250*x^3 + 3750*x^4
+ 1247*x^5 + 29*x^6 + 3*x^7 + x^8 + 11328125),x)

[Out]

log(1875*x + 622*x^2 + 29*x^3 + 3*x^4 + x^5 + 18125) - log(x^3 + 625)

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sympy [A]  time = 0.14, size = 31, normalized size = 1.41 \begin {gather*} - \log {\left (x^{3} + 625 \right )} + \log {\left (x^{5} + 3 x^{4} + 29 x^{3} + 622 x^{2} + 1875 x + 18125 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**7+3*x**6+2503*x**4+3750*x**3+777500*x+1171875)/(x**8+3*x**7+29*x**6+1247*x**5+3750*x**4+36250*
x**3+388750*x**2+1171875*x+11328125),x)

[Out]

-log(x**3 + 625) + log(x**5 + 3*x**4 + 29*x**3 + 622*x**2 + 1875*x + 18125)

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