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3.75.35 \(\int \frac {980+294 x+49 x^2+4 x^3+(49+210 x+57 x^2+4 x^3) \log (5)}{245+70 x+12 x^2+x^3+(49 x+14 x^2+x^3) \log (5)} \, dx\)

Optimal. Leaf size=20 \[ -4+4 x+\log \left (5+x \left (\frac {x}{7+x}+\log (5)\right )\right ) \]

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Rubi [A]  time = 0.14, antiderivative size = 29, normalized size of antiderivative = 1.45, number of steps used = 3, number of rules used = 2, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2074, 628} \begin {gather*} \log \left (x^2 (1+\log (5))+x (5+7 \log (5))+35\right )+4 x-\log (x+7) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(980 + 294*x + 49*x^2 + 4*x^3 + (49 + 210*x + 57*x^2 + 4*x^3)*Log[5])/(245 + 70*x + 12*x^2 + x^3 + (49*x +
 14*x^2 + x^3)*Log[5]),x]

[Out]

4*x - Log[7 + x] + Log[35 + x^2*(1 + Log[5]) + x*(5 + 7*Log[5])]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

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 (4+\frac {1}{-7-x}+\frac {5+7 \log (5)+2 x (1+\log (5))}{35+x^2 (1+\log (5))+x (5+7 \log (5))}\right ) \, dx\\ &=4 x-\log (7+x)+\int \frac {5+7 \log (5)+2 x (1+\log (5))}{35+x^2 (1+\log (5))+x (5+7 \log (5))} \, dx\\ &=4 x-\log (7+x)+\log \left (35+x^2 (1+\log (5))+x (5+7 \log (5))\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 31, normalized size = 1.55 \begin {gather*} 4 (7+x)-\log (7+x)+\log \left (35+x^2 (1+\log (5))+x (5+7 \log (5))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(980 + 294*x + 49*x^2 + 4*x^3 + (49 + 210*x + 57*x^2 + 4*x^3)*Log[5])/(245 + 70*x + 12*x^2 + x^3 + (
49*x + 14*x^2 + x^3)*Log[5]),x]

[Out]

4*(7 + x) - Log[7 + x] + Log[35 + x^2*(1 + Log[5]) + x*(5 + 7*Log[5])]

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fricas [A]  time = 0.64, size = 29, normalized size = 1.45 \begin {gather*} 4 \, x + \log \left (x^{2} + {\left (x^{2} + 7 \, x\right )} \log \relax (5) + 5 \, x + 35\right ) - \log \left (x + 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+57*x^2+210*x+49)*log(5)+4*x^3+49*x^2+294*x+980)/((x^3+14*x^2+49*x)*log(5)+x^3+12*x^2+70*x+24
5),x, algorithm="fricas")

[Out]

4*x + log(x^2 + (x^2 + 7*x)*log(5) + 5*x + 35) - log(x + 7)

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giac [B]  time = 0.17, size = 42, normalized size = 2.10 \begin {gather*} \frac {4 \, {\left (x \log \relax (5) + x\right )}}{\log \relax (5) + 1} + \log \left (x^{2} \log \relax (5) + x^{2} + 7 \, x \log \relax (5) + 5 \, x + 35\right ) - \log \left ({\left | x + 7 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+57*x^2+210*x+49)*log(5)+4*x^3+49*x^2+294*x+980)/((x^3+14*x^2+49*x)*log(5)+x^3+12*x^2+70*x+24
5),x, algorithm="giac")

[Out]

4*(x*log(5) + x)/(log(5) + 1) + log(x^2*log(5) + x^2 + 7*x*log(5) + 5*x + 35) - log(abs(x + 7))

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maple [A]  time = 0.09, size = 31, normalized size = 1.55

method result size
default \(4 x -\ln \left (x +7\right )+\ln \left (x^{2} \ln \relax (5)+7 x \ln \relax (5)+x^{2}+5 x +35\right )\) \(31\)
norman \(4 x -\ln \left (x +7\right )+\ln \left (x^{2} \ln \relax (5)+7 x \ln \relax (5)+x^{2}+5 x +35\right )\) \(31\)
risch \(4 x -\ln \left (x +7\right )+\ln \left (\left (-\ln \relax (5)-1\right ) x^{2}+\left (-7 \ln \relax (5)-5\right ) x -35\right )\) \(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3+57*x^2+210*x+49)*ln(5)+4*x^3+49*x^2+294*x+980)/((x^3+14*x^2+49*x)*ln(5)+x^3+12*x^2+70*x+245),x,met
hod=_RETURNVERBOSE)

[Out]

4*x-ln(x+7)+ln(x^2*ln(5)+7*x*ln(5)+x^2+5*x+35)

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maxima [A]  time = 0.36, size = 29, normalized size = 1.45 \begin {gather*} 4 \, x + \log \left (x^{2} {\left (\log \relax (5) + 1\right )} + x {\left (7 \, \log \relax (5) + 5\right )} + 35\right ) - \log \left (x + 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+57*x^2+210*x+49)*log(5)+4*x^3+49*x^2+294*x+980)/((x^3+14*x^2+49*x)*log(5)+x^3+12*x^2+70*x+24
5),x, algorithm="maxima")

[Out]

4*x + log(x^2*(log(5) + 1) + x*(7*log(5) + 5) + 35) - log(x + 7)

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mupad [B]  time = 5.14, size = 2977, normalized size = 148.85 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((294*x + log(5)*(210*x + 57*x^2 + 4*x^3 + 49) + 49*x^2 + 4*x^3 + 980)/(70*x + log(5)*(49*x + 14*x^2 + x^3)
 + 12*x^2 + x^3 + 245),x)

[Out]

log((7*(3514*log(5) - 840*log(625) - 41923*log(5)*log(625) + 11760*log(5)*log(625)^2 - 93422*log(5)^2*log(625)
 - 52087*log(5)^3*log(625) + 1372*log(5)^4*log(625) + 87304*log(5)^2 + 185710*log(5)^3 + 98637*log(5)^4 - 5488
*log(5)^5 + 5040*log(625)^2 + 6860*log(5)^2*log(625)^2 + 35))/(2*log(5) + log(5)^2 + 1) - ((7*(4870*log(5) - 1
010*log(625) - 1983*log(5)*log(625) + 448*log(5)^2*log(625) + 2793*log(5)^3*log(625) + 1372*log(5)^4*log(625)
+ 8944*log(5)^2 - 1436*log(5)^3 - 11326*log(5)^4 - 5586*log(5)^5 + 230))/(2*log(5) + log(5)^2 + 1) - ((7*(676*
log(5) + 760*log(5)^2 + 174*log(5)^3 - 203*log(5)^4 - 98*log(5)^5 + 195))/(2*log(5) + log(5)^2 + 1) + (x*(438*
log(5) + 424*log(5)^2 - 36*log(5)^3 - 252*log(5)^4 - 98*log(5)^5 + 132))/(2*log(5) + log(5)^2 + 1))*(56*log(5)
 - 14*log(625) + (log(625) + 4)/(log(5) + 1) - 5) + (x*(3470*log(5) - 660*log(625) - 422*log(5)*log(625) + 317
8*log(5)^2*log(625) + 4998*log(5)^3*log(625) + 2058*log(5)^4*log(625) + 2700*log(5)^2 - 12356*log(5)^3 - 20146
*log(5)^4 - 8330*log(5)^5 + 230))/(2*log(5) + log(5)^2 + 1))*(56*log(5) - 14*log(625) + (log(625) + 4)/(log(5)
 + 1) - 5) + (x*(448*log(625) - 1400*log(5) - 85946*log(5)*log(625) + 32977*log(5)*log(625)^2 - 256802*log(5)^
2*log(625) - 246078*log(5)^3*log(625) - 74774*log(5)^4*log(625) + 166180*log(5)^2 + 499968*log(5)^3 + 479514*l
og(5)^4 + 145432*log(5)^5 + 11137*log(625)^2 + 31556*log(5)^2*log(625)^2 + 9604*log(5)^3*log(625)^2 + 98))/(2*
log(5) + log(5)^2 + 1))*(56*log(5) - 14*log(625) + (log(625) + 4)/(log(5) + 1) - 5) + (log((7*(3514*log(5) - 8
40*log(625) - 41923*log(5)*log(625) + 11760*log(5)*log(625)^2 - 93422*log(5)^2*log(625) - 52087*log(5)^3*log(6
25) + 1372*log(5)^4*log(625) + 87304*log(5)^2 + 185710*log(5)^3 + 98637*log(5)^4 - 5488*log(5)^5 + 5040*log(62
5)^2 + 6860*log(5)^2*log(625)^2 + 35))/(2*log(5) + log(5)^2 + 1) - (((7*(4870*log(5) - 1010*log(625) - 1983*lo
g(5)*log(625) + 448*log(5)^2*log(625) + 2793*log(5)^3*log(625) + 1372*log(5)^4*log(625) + 8944*log(5)^2 - 1436
*log(5)^3 - 11326*log(5)^4 - 5586*log(5)^5 + 230))/(2*log(5) + log(5)^2 + 1) - (((7*(676*log(5) + 760*log(5)^2
 + 174*log(5)^3 - 203*log(5)^4 - 98*log(5)^5 + 195))/(2*log(5) + log(5)^2 + 1) + (x*(438*log(5) + 424*log(5)^2
 - 36*log(5)^3 - 252*log(5)^4 - 98*log(5)^5 + 132))/(2*log(5) + log(5)^2 + 1))*(140*log(5) + 115*log(625) + 15
65*log(5)*log(625) - 500*log(5)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 125*log(625)*(49*log(5)^2 - 70*log(5)
- 115)^(1/2) + 2471*log(5)^2*log(625) + 343*log(5)^3*log(625) - 686*log(5)^4*log(625) - 5848*log(5)^2 - 9940*l
og(5)^3 - 1470*log(5)^4 + 2744*log(5)^5 - 792*log(5)^2*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 84*log(5)^3*(49
*log(5)^2 - 70*log(5) - 115)^(1/2) + 392*log(5)^4*(49*log(5)^2 - 70*log(5) - 115)^(1/2) - 21*log(5)^2*log(625)
*(49*log(5)^2 - 70*log(5) - 115)^(1/2) - 98*log(5)^3*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 198*log(
5)*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 230))/(2*(300*log(5) + 206*log(5)^2 - 28*log(5)^3 - 49*log
(5)^4 + 115)) + (x*(3470*log(5) - 660*log(625) - 422*log(5)*log(625) + 3178*log(5)^2*log(625) + 4998*log(5)^3*
log(625) + 2058*log(5)^4*log(625) + 2700*log(5)^2 - 12356*log(5)^3 - 20146*log(5)^4 - 8330*log(5)^5 + 230))/(2
*log(5) + log(5)^2 + 1))*(140*log(5) + 115*log(625) + 1565*log(5)*log(625) - 500*log(5)*(49*log(5)^2 - 70*log(
5) - 115)^(1/2) + 125*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 2471*log(5)^2*log(625) + 343*log(5)^3*l
og(625) - 686*log(5)^4*log(625) - 5848*log(5)^2 - 9940*log(5)^3 - 1470*log(5)^4 + 2744*log(5)^5 - 792*log(5)^2
*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 84*log(5)^3*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 392*log(5)^4*(49*
log(5)^2 - 70*log(5) - 115)^(1/2) - 21*log(5)^2*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) - 98*log(5)^3*l
og(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 198*log(5)*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 23
0))/(2*(300*log(5) + 206*log(5)^2 - 28*log(5)^3 - 49*log(5)^4 + 115)) + (x*(448*log(625) - 1400*log(5) - 85946
*log(5)*log(625) + 32977*log(5)*log(625)^2 - 256802*log(5)^2*log(625) - 246078*log(5)^3*log(625) - 74774*log(5
)^4*log(625) + 166180*log(5)^2 + 499968*log(5)^3 + 479514*log(5)^4 + 145432*log(5)^5 + 11137*log(625)^2 + 3155
6*log(5)^2*log(625)^2 + 9604*log(5)^3*log(625)^2 + 98))/(2*log(5) + log(5)^2 + 1))*(140*log(5) + 115*log(625)
+ 1565*log(5)*log(625) - 500*log(5)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 125*log(625)*(49*log(5)^2 - 70*log
(5) - 115)^(1/2) + 2471*log(5)^2*log(625) + 343*log(5)^3*log(625) - 686*log(5)^4*log(625) - 5848*log(5)^2 - 99
40*log(5)^3 - 1470*log(5)^4 + 2744*log(5)^5 - 792*log(5)^2*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 84*log(5)^3
*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 392*log(5)^4*(49*log(5)^2 - 70*log(5) - 115)^(1/2) - 21*log(5)^2*log(
625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) - 98*log(5)^3*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 198*
log(5)*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 230))/(2*(300*log(5) + 206*log(5)^2 - 28*log(5)^3 - 49
*log(5)^4 + 115)) + (log((7*(3514*log(5) - 840*log(625) - 41923*log(5)*log(625) + 11760*log(5)*log(625)^2 - 93
422*log(5)^2*log(625) - 52087*log(5)^3*log(625) + 1372*log(5)^4*log(625) + 87304*log(5)^2 + 185710*log(5)^3 +
98637*log(5)^4 - 5488*log(5)^5 + 5040*log(625)^2 + 6860*log(5)^2*log(625)^2 + 35))/(2*log(5) + log(5)^2 + 1) -
 (((7*(4870*log(5) - 1010*log(625) - 1983*log(5)*log(625) + 448*log(5)^2*log(625) + 2793*log(5)^3*log(625) + 1
372*log(5)^4*log(625) + 8944*log(5)^2 - 1436*log(5)^3 - 11326*log(5)^4 - 5586*log(5)^5 + 230))/(2*log(5) + log
(5)^2 + 1) - (((7*(676*log(5) + 760*log(5)^2 + 174*log(5)^3 - 203*log(5)^4 - 98*log(5)^5 + 195))/(2*log(5) + l
og(5)^2 + 1) + (x*(438*log(5) + 424*log(5)^2 - 36*log(5)^3 - 252*log(5)^4 - 98*log(5)^5 + 132))/(2*log(5) + lo
g(5)^2 + 1))*(140*log(5) + 115*log(625) + 1565*log(5)*log(625) + 500*log(5)*(49*log(5)^2 - 70*log(5) - 115)^(1
/2) - 125*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 2471*log(5)^2*log(625) + 343*log(5)^3*log(625) - 68
6*log(5)^4*log(625) - 5848*log(5)^2 - 9940*log(5)^3 - 1470*log(5)^4 + 2744*log(5)^5 + 792*log(5)^2*(49*log(5)^
2 - 70*log(5) - 115)^(1/2) - 84*log(5)^3*(49*log(5)^2 - 70*log(5) - 115)^(1/2) - 392*log(5)^4*(49*log(5)^2 - 7
0*log(5) - 115)^(1/2) + 21*log(5)^2*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 98*log(5)^3*log(625)*(49*
log(5)^2 - 70*log(5) - 115)^(1/2) - 198*log(5)*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 230))/(2*(300*
log(5) + 206*log(5)^2 - 28*log(5)^3 - 49*log(5)^4 + 115)) + (x*(3470*log(5) - 660*log(625) - 422*log(5)*log(62
5) + 3178*log(5)^2*log(625) + 4998*log(5)^3*log(625) + 2058*log(5)^4*log(625) + 2700*log(5)^2 - 12356*log(5)^3
 - 20146*log(5)^4 - 8330*log(5)^5 + 230))/(2*log(5) + log(5)^2 + 1))*(140*log(5) + 115*log(625) + 1565*log(5)*
log(625) + 500*log(5)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) - 125*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/
2) + 2471*log(5)^2*log(625) + 343*log(5)^3*log(625) - 686*log(5)^4*log(625) - 5848*log(5)^2 - 9940*log(5)^3 -
1470*log(5)^4 + 2744*log(5)^5 + 792*log(5)^2*(49*log(5)^2 - 70*log(5) - 115)^(1/2) - 84*log(5)^3*(49*log(5)^2
- 70*log(5) - 115)^(1/2) - 392*log(5)^4*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 21*log(5)^2*log(625)*(49*log(5
)^2 - 70*log(5) - 115)^(1/2) + 98*log(5)^3*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) - 198*log(5)*log(625
)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 230))/(2*(300*log(5) + 206*log(5)^2 - 28*log(5)^3 - 49*log(5)^4 + 11
5)) + (x*(448*log(625) - 1400*log(5) - 85946*log(5)*log(625) + 32977*log(5)*log(625)^2 - 256802*log(5)^2*log(6
25) - 246078*log(5)^3*log(625) - 74774*log(5)^4*log(625) + 166180*log(5)^2 + 499968*log(5)^3 + 479514*log(5)^4
 + 145432*log(5)^5 + 11137*log(625)^2 + 31556*log(5)^2*log(625)^2 + 9604*log(5)^3*log(625)^2 + 98))/(2*log(5)
+ log(5)^2 + 1))*(140*log(5) + 115*log(625) + 1565*log(5)*log(625) + 500*log(5)*(49*log(5)^2 - 70*log(5) - 115
)^(1/2) - 125*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 2471*log(5)^2*log(625) + 343*log(5)^3*log(625)
- 686*log(5)^4*log(625) - 5848*log(5)^2 - 9940*log(5)^3 - 1470*log(5)^4 + 2744*log(5)^5 + 792*log(5)^2*(49*log
(5)^2 - 70*log(5) - 115)^(1/2) - 84*log(5)^3*(49*log(5)^2 - 70*log(5) - 115)^(1/2) - 392*log(5)^4*(49*log(5)^2
 - 70*log(5) - 115)^(1/2) + 21*log(5)^2*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 98*log(5)^3*log(625)*
(49*log(5)^2 - 70*log(5) - 115)^(1/2) - 198*log(5)*log(625)*(49*log(5)^2 - 70*log(5) - 115)^(1/2) + 230))/(2*(
300*log(5) + 206*log(5)^2 - 28*log(5)^3 - 49*log(5)^4 + 115)) + (x*(log(625) + 4))/(log(5) + 1)

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sympy [A]  time = 1.19, size = 32, normalized size = 1.60 \begin {gather*} 4 x - \log {\left (x + 7 \right )} + \log {\left (x^{2} + \frac {x \left (5 + 7 \log {\relax (5 )}\right )}{1 + \log {\relax (5 )}} + \frac {35}{1 + \log {\relax (5 )}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3+57*x**2+210*x+49)*ln(5)+4*x**3+49*x**2+294*x+980)/((x**3+14*x**2+49*x)*ln(5)+x**3+12*x**2+7
0*x+245),x)

[Out]

4*x - log(x + 7) + log(x**2 + x*(5 + 7*log(5))/(1 + log(5)) + 35/(1 + log(5)))

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