3.103.64 \(\int \frac {9 x^2+x^2 \log (5)+(90 x+18 x^2+2 x^2 \log (5)) \log (\frac {-45-9 x-x \log (5)}{\log (5)})}{45+9 x+x \log (5)} \, dx\) [10264]

Optimal. Leaf size=20 \[ x^2 \log \left (-x+\frac {9 (-5-x)}{\log (5)}\right ) \]

[Out]

x^2*ln(9/ln(5)*(-x-5)-x)

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Rubi [A]
time = 0.17, antiderivative size = 23, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 5, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {6, 6820, 14, 45, 2442} \begin {gather*} x^2 \log \left (-\frac {x (9+\log (5))}{\log (5)}-\frac {45}{\log (5)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9*x^2 + x^2*Log[5] + (90*x + 18*x^2 + 2*x^2*Log[5])*Log[(-45 - 9*x - x*Log[5])/Log[5]])/(45 + 9*x + x*Log
[5]),x]

[Out]

x^2*Log[-45/Log[5] - (x*(9 + Log[5]))/Log[5]]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9 x^2+x^2 \log (5)+\left (90 x+18 x^2+2 x^2 \log (5)\right ) \log \left (\frac {-45-9 x-x \log (5)}{\log (5)}\right )}{45+x (9+\log (5))} \, dx\\ &=\int \frac {x^2 (9+\log (5))+\left (90 x+18 x^2+2 x^2 \log (5)\right ) \log \left (\frac {-45-9 x-x \log (5)}{\log (5)}\right )}{45+x (9+\log (5))} \, dx\\ &=\int x \left (\frac {x (9+\log (5))}{45+x (9+\log (5))}+2 \log \left (-\frac {45+x (9+\log (5))}{\log (5)}\right )\right ) \, dx\\ &=\int \left (\frac {x^2 (9+\log (5))}{45+x (9+\log (5))}+2 x \log \left (-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}\right )\right ) \, dx\\ &=2 \int x \log \left (-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}\right ) \, dx+(9+\log (5)) \int \frac {x^2}{45+x (9+\log (5))} \, dx\\ &=x^2 \log \left (-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}\right )+(9+\log (5)) \int \left (-\frac {45}{(9+\log (5))^2}+\frac {x}{9+\log (5)}+\frac {2025}{(9+\log (5))^2 (45+x (9+\log (5)))}\right ) \, dx+\frac {(9+\log (5)) \int \frac {x^2}{-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}} \, dx}{\log (5)}\\ &=\frac {x^2}{2}-\frac {45 x}{9+\log (5)}+\frac {2025 \log (45+x (9+\log (5)))}{(9+\log (5))^2}+x^2 \log \left (-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}\right )+\frac {(9+\log (5)) \int \left (\frac {45 \log (5)}{(9+\log (5))^2}-\frac {x \log (5)}{9+\log (5)}+\frac {2025 \log (5)}{(9+\log (5))^2 (-45-x (9+\log (5)))}\right ) \, dx}{\log (5)}\\ &=x^2 \log \left (-\frac {45}{\log (5)}-\frac {x (9+\log (5))}{\log (5)}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.05, size = 19, normalized size = 0.95 \begin {gather*} x^2 \log \left (-\frac {45+x (9+\log (5))}{\log (5)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9*x^2 + x^2*Log[5] + (90*x + 18*x^2 + 2*x^2*Log[5])*Log[(-45 - 9*x - x*Log[5])/Log[5]])/(45 + 9*x +
x*Log[5]),x]

[Out]

x^2*Log[-((45 + x*(9 + Log[5]))/Log[5])]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(235\) vs. \(2(20)=40\).
time = 1.12, size = 236, normalized size = 11.80

method	result	size
norman	\(x^{2} \ln \left (\frac {-x \ln \left (5\right )-9 x -45}{\ln \left (5\right )}\right )\)	\(21\)
risch	\(x^{2} \ln \left (\frac {-x \ln \left (5\right )-9 x -45}{\ln \left (5\right )}\right )\)	\(21\)
derivativedivides	\(-\frac {\ln \left (5\right ) \left (-\frac {2 \ln \left (5\right ) \left (\frac {\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2} \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{2}-\frac {\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2}}{4}\right )}{\ln \left (5\right )+9}-\frac {90 \left (\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right ) \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )+\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}+\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right )+9}-\frac {\ln \left (5\right ) \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2}}{2 \left (\ln \left (5\right )+9\right )}-\frac {90 \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right )+9}-\frac {2025 \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right ) \left (\ln \left (5\right )+9\right )}\right )}{\ln \left (5\right )+9}\)	\(236\)
default	\(-\frac {\ln \left (5\right ) \left (-\frac {2 \ln \left (5\right ) \left (\frac {\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2} \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{2}-\frac {\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2}}{4}\right )}{\ln \left (5\right )+9}-\frac {90 \left (\left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right ) \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )+\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}+\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right )+9}-\frac {\ln \left (5\right ) \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )^{2}}{2 \left (\ln \left (5\right )+9\right )}-\frac {90 \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right )+9}-\frac {2025 \ln \left (-\frac {\left (\ln \left (5\right )+9\right ) x}{\ln \left (5\right )}-\frac {45}{\ln \left (5\right )}\right )}{\ln \left (5\right ) \left (\ln \left (5\right )+9\right )}\right )}{\ln \left (5\right )+9}\)	\(236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2*ln(5)+18*x^2+90*x)*ln((-x*ln(5)-9*x-45)/ln(5))+x^2*ln(5)+9*x^2)/(x*ln(5)+9*x+45),x,method=_RETURNV
ERBOSE)

[Out]

-1/(ln(5)+9)*ln(5)*(-2/(ln(5)+9)*ln(5)*(1/2*(-(ln(5)+9)/ln(5)*x-45/ln(5))^2*ln(-(ln(5)+9)/ln(5)*x-45/ln(5))-1/
4*(-(ln(5)+9)/ln(5)*x-45/ln(5))^2)-90/(ln(5)+9)*((-(ln(5)+9)/ln(5)*x-45/ln(5))*ln(-(ln(5)+9)/ln(5)*x-45/ln(5))
+(ln(5)+9)/ln(5)*x+45/ln(5))-1/2/(ln(5)+9)*ln(5)*(-(ln(5)+9)/ln(5)*x-45/ln(5))^2-90/(ln(5)+9)*(-(ln(5)+9)/ln(5
)*x-45/ln(5))-2025/ln(5)/(ln(5)+9)*ln(-(ln(5)+9)/ln(5)*x-45/ln(5)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (18) = 36\).
time = 0.47, size = 577, normalized size = 28.85 \begin {gather*} {\left (\frac {x^{2} {\left (\log \left (5\right ) + 9\right )} - 90 \, x}{\log \left (5\right )^{2} + 18 \, \log \left (5\right ) + 81} + \frac {4050 \, \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )}{\log \left (5\right )^{3} + 27 \, \log \left (5\right )^{2} + 243 \, \log \left (5\right ) + 729}\right )} \log \left (5\right ) \log \left (-x - \frac {9 \, x}{\log \left (5\right )} - \frac {45}{\log \left (5\right )}\right ) + \frac {1}{2} \, {\left (\frac {x^{2} {\left (\log \left (5\right ) + 9\right )} - 90 \, x}{\log \left (5\right )^{2} + 18 \, \log \left (5\right ) + 81} + \frac {4050 \, \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )}{\log \left (5\right )^{3} + 27 \, \log \left (5\right )^{2} + 243 \, \log \left (5\right ) + 729}\right )} \log \left (5\right ) - \frac {{\left ({\left (\log \left (5\right )^{3} + 18 \, \log \left (5\right )^{2} + 81 \, \log \left (5\right )\right )} x^{2} + 4050 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )^{2} - 270 \, {\left (\log \left (5\right )^{2} + 9 \, \log \left (5\right )\right )} x + 12150 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )\right )} {\left (\frac {9}{\log \left (5\right )} + 1\right )} \log \left (5\right )}{2 \, {\left (\log \left (5\right )^{4} + 36 \, \log \left (5\right )^{3} + 486 \, \log \left (5\right )^{2} + 2916 \, \log \left (5\right ) + 6561\right )}} + 9 \, {\left (\frac {x^{2} {\left (\log \left (5\right ) + 9\right )} - 90 \, x}{\log \left (5\right )^{2} + 18 \, \log \left (5\right ) + 81} + \frac {4050 \, \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )}{\log \left (5\right )^{3} + 27 \, \log \left (5\right )^{2} + 243 \, \log \left (5\right ) + 729}\right )} \log \left (-x - \frac {9 \, x}{\log \left (5\right )} - \frac {45}{\log \left (5\right )}\right ) + 90 \, {\left (\frac {x}{\log \left (5\right ) + 9} - \frac {45 \, \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )}{\log \left (5\right )^{2} + 18 \, \log \left (5\right ) + 81}\right )} \log \left (-x - \frac {9 \, x}{\log \left (5\right )} - \frac {45}{\log \left (5\right )}\right ) - \frac {9 \, {\left ({\left (\log \left (5\right )^{3} + 18 \, \log \left (5\right )^{2} + 81 \, \log \left (5\right )\right )} x^{2} + 4050 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )^{2} - 270 \, {\left (\log \left (5\right )^{2} + 9 \, \log \left (5\right )\right )} x + 12150 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )\right )} {\left (\frac {9}{\log \left (5\right )} + 1\right )}}{2 \, {\left (\log \left (5\right )^{4} + 36 \, \log \left (5\right )^{3} + 486 \, \log \left (5\right )^{2} + 2916 \, \log \left (5\right ) + 6561\right )}} + \frac {45 \, {\left (45 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )^{2} - 2 \, {\left (\log \left (5\right )^{2} + 9 \, \log \left (5\right )\right )} x + 90 \, \log \left (5\right ) \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )\right )} {\left (\frac {9}{\log \left (5\right )} + 1\right )}}{\log \left (5\right )^{3} + 27 \, \log \left (5\right )^{2} + 243 \, \log \left (5\right ) + 729} + \frac {9 \, {\left (x^{2} {\left (\log \left (5\right ) + 9\right )} - 90 \, x\right )}}{2 \, {\left (\log \left (5\right )^{2} + 18 \, \log \left (5\right ) + 81\right )}} + \frac {18225 \, \log \left (x {\left (\log \left (5\right ) + 9\right )} + 45\right )}{\log \left (5\right )^{3} + 27 \, \log \left (5\right )^{2} + 243 \, \log \left (5\right ) + 729} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(5)+18*x^2+90*x)*log((-x*log(5)-9*x-45)/log(5))+x^2*log(5)+9*x^2)/(x*log(5)+9*x+45),x, al
gorithm="maxima")

[Out]

((x^2*(log(5) + 9) - 90*x)/(log(5)^2 + 18*log(5) + 81) + 4050*log(x*(log(5) + 9) + 45)/(log(5)^3 + 27*log(5)^2
+ 243*log(5) + 729))*log(5)*log(-x - 9*x/log(5) - 45/log(5)) + 1/2*((x^2*(log(5) + 9) - 90*x)/(log(5)^2 + 18*
log(5) + 81) + 4050*log(x*(log(5) + 9) + 45)/(log(5)^3 + 27*log(5)^2 + 243*log(5) + 729))*log(5) - 1/2*((log(5
)^3 + 18*log(5)^2 + 81*log(5))*x^2 + 4050*log(5)*log(x*(log(5) + 9) + 45)^2 - 270*(log(5)^2 + 9*log(5))*x + 12
150*log(5)*log(x*(log(5) + 9) + 45))*(9/log(5) + 1)*log(5)/(log(5)^4 + 36*log(5)^3 + 486*log(5)^2 + 2916*log(5
) + 6561) + 9*((x^2*(log(5) + 9) - 90*x)/(log(5)^2 + 18*log(5) + 81) + 4050*log(x*(log(5) + 9) + 45)/(log(5)^3
+ 27*log(5)^2 + 243*log(5) + 729))*log(-x - 9*x/log(5) - 45/log(5)) + 90*(x/(log(5) + 9) - 45*log(x*(log(5) +
9) + 45)/(log(5)^2 + 18*log(5) + 81))*log(-x - 9*x/log(5) - 45/log(5)) - 9/2*((log(5)^3 + 18*log(5)^2 + 81*lo
g(5))*x^2 + 4050*log(5)*log(x*(log(5) + 9) + 45)^2 - 270*(log(5)^2 + 9*log(5))*x + 12150*log(5)*log(x*(log(5)
+ 9) + 45))*(9/log(5) + 1)/(log(5)^4 + 36*log(5)^3 + 486*log(5)^2 + 2916*log(5) + 6561) + 45*(45*log(5)*log(x*
(log(5) + 9) + 45)^2 - 2*(log(5)^2 + 9*log(5))*x + 90*log(5)*log(x*(log(5) + 9) + 45))*(9/log(5) + 1)/(log(5)^
3 + 27*log(5)^2 + 243*log(5) + 729) + 9/2*(x^2*(log(5) + 9) - 90*x)/(log(5)^2 + 18*log(5) + 81) + 18225*log(x*
(log(5) + 9) + 45)/(log(5)^3 + 27*log(5)^2 + 243*log(5) + 729)

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Fricas [A]
time = 0.39, size = 20, normalized size = 1.00 \begin {gather*} x^{2} \log \left (-\frac {x \log \left (5\right ) + 9 \, x + 45}{\log \left (5\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(5)+18*x^2+90*x)*log((-x*log(5)-9*x-45)/log(5))+x^2*log(5)+9*x^2)/(x*log(5)+9*x+45),x, al
gorithm="fricas")

[Out]

x^2*log(-(x*log(5) + 9*x + 45)/log(5))

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Sympy [A]
time = 0.07, size = 19, normalized size = 0.95 \begin {gather*} x^{2} \log {\left (\frac {- 9 x - x \log {\left (5 \right )} - 45}{\log {\left (5 \right )}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2*ln(5)+18*x**2+90*x)*ln((-x*ln(5)-9*x-45)/ln(5))+x**2*ln(5)+9*x**2)/(x*ln(5)+9*x+45),x)

[Out]

x**2*log((-9*x - x*log(5) - 45)/log(5))

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Giac [A]
time = 0.42, size = 24, normalized size = 1.20 \begin {gather*} x^{2} \log \left (-x \log \left (5\right ) - 9 \, x - 45\right ) - x^{2} \log \left (\log \left (5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2*log(5)+18*x^2+90*x)*log((-x*log(5)-9*x-45)/log(5))+x^2*log(5)+9*x^2)/(x*log(5)+9*x+45),x, al
gorithm="giac")

[Out]

x^2*log(-x*log(5) - 9*x - 45) - x^2*log(log(5))

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Mupad [B]
time = 0.23, size = 22, normalized size = 1.10 \begin {gather*} -x^2\,\left (\ln \left (\ln \left (5\right )\right )-\ln \left (-9\,x-x\,\ln \left (5\right )-45\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-(9*x + x*log(5) + 45)/log(5))*(90*x + 2*x^2*log(5) + 18*x^2) + x^2*log(5) + 9*x^2)/(9*x + x*log(5) +
45),x)

[Out]

-x^2*(log(log(5)) - log(- 9*x - x*log(5) - 45))

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