3.87.10 \(\int \frac {36 x^3+12 x^4+(-36 x^3-9 x^4) \log (x)+(-18-9 x-x^2) \log ^2(5) \log ^3(x)+(-18 x^3-6 x^4+(18 x^3+6 x^4) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)) \log (3+x)}{(3 x^2+x^3) \log ^2(5) \log ^3(x)} \, dx\)

Optimal. Leaf size=28 \[ \frac {\left (-3-x+\frac {3 x^3}{\log ^2(5) \log ^2(x)}\right ) (-2+\log (3+x))}{x} \]

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Rubi [F]  time = 0.70, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^2(5) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(36*x^3 + 12*x^4 + (-36*x^3 - 9*x^4)*Log[x] + (-18 - 9*x - x^2)*Log[5]^2*Log[x]^3 + (-18*x^3 - 6*x^4 + (18
*x^3 + 6*x^4)*Log[x] + (9 + 3*x)*Log[5]^2*Log[x]^3)*Log[3 + x])/((3*x^2 + x^3)*Log[5]^2*Log[x]^3),x]

[Out]

6/x - Log[3 + x] - (3*Log[3 + x])/x - (9*Defer[Int][(x*(4 + x))/((3 + x)*Log[x]^2), x])/Log[5]^2 - (6*Defer[In
t][(x*(-2 + Log[3 + x]))/Log[x]^3, x])/Log[5]^2 + (6*Defer[Int][(x*Log[3 + x])/Log[x]^2, x])/Log[5]^2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{\left (3 x^2+x^3\right ) \log ^3(x)} \, dx}{\log ^2(5)}\\ &=\frac {\int \frac {36 x^3+12 x^4+\left (-36 x^3-9 x^4\right ) \log (x)+\left (-18-9 x-x^2\right ) \log ^2(5) \log ^3(x)+\left (-18 x^3-6 x^4+\left (18 x^3+6 x^4\right ) \log (x)+(9+3 x) \log ^2(5) \log ^3(x)\right ) \log (3+x)}{x^2 (3+x) \log ^3(x)} \, dx}{\log ^2(5)}\\ &=\frac {\int \left (-\frac {\log ^2(5) (6+x-3 \log (3+x))}{x^2}-\frac {6 x (-2+\log (3+x))}{\log ^3(x)}+\frac {-9 x (4+x)+6 x (3+x) \log (3+x)}{(3+x) \log ^2(x)}\right ) \, dx}{\log ^2(5)}\\ &=\frac {\int \frac {-9 x (4+x)+6 x (3+x) \log (3+x)}{(3+x) \log ^2(x)} \, dx}{\log ^2(5)}-\frac {6 \int \frac {x (-2+\log (3+x))}{\log ^3(x)} \, dx}{\log ^2(5)}-\int \frac {6+x-3 \log (3+x)}{x^2} \, dx\\ &=\frac {\int \left (-\frac {9 x (4+x)}{(3+x) \log ^2(x)}+\frac {6 x \log (3+x)}{\log ^2(x)}\right ) \, dx}{\log ^2(5)}-\frac {6 \int \frac {x (-2+\log (3+x))}{\log ^3(x)} \, dx}{\log ^2(5)}-\int \left (\frac {6+x}{x^2}-\frac {3 \log (3+x)}{x^2}\right ) \, dx\\ &=3 \int \frac {\log (3+x)}{x^2} \, dx-\frac {6 \int \frac {x (-2+\log (3+x))}{\log ^3(x)} \, dx}{\log ^2(5)}+\frac {6 \int \frac {x \log (3+x)}{\log ^2(x)} \, dx}{\log ^2(5)}-\frac {9 \int \frac {x (4+x)}{(3+x) \log ^2(x)} \, dx}{\log ^2(5)}-\int \frac {6+x}{x^2} \, dx\\ &=-\frac {3 \log (3+x)}{x}+3 \int \frac {1}{x (3+x)} \, dx-\frac {6 \int \frac {x (-2+\log (3+x))}{\log ^3(x)} \, dx}{\log ^2(5)}+\frac {6 \int \frac {x \log (3+x)}{\log ^2(x)} \, dx}{\log ^2(5)}-\frac {9 \int \frac {x (4+x)}{(3+x) \log ^2(x)} \, dx}{\log ^2(5)}-\int \left (\frac {6}{x^2}+\frac {1}{x}\right ) \, dx\\ &=\frac {6}{x}-\log (x)-\frac {3 \log (3+x)}{x}-\frac {6 \int \frac {x (-2+\log (3+x))}{\log ^3(x)} \, dx}{\log ^2(5)}+\frac {6 \int \frac {x \log (3+x)}{\log ^2(x)} \, dx}{\log ^2(5)}-\frac {9 \int \frac {x (4+x)}{(3+x) \log ^2(x)} \, dx}{\log ^2(5)}+\int \frac {1}{x} \, dx-\int \frac {1}{3+x} \, dx\\ &=\frac {6}{x}-\log (3+x)-\frac {3 \log (3+x)}{x}-\frac {6 \int \frac {x (-2+\log (3+x))}{\log ^3(x)} \, dx}{\log ^2(5)}+\frac {6 \int \frac {x \log (3+x)}{\log ^2(x)} \, dx}{\log ^2(5)}-\frac {9 \int \frac {x (4+x)}{(3+x) \log ^2(x)} \, dx}{\log ^2(5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 34, normalized size = 1.21 \begin {gather*} \frac {6+\frac {3 x^3 (-2+\log (3+x))}{\log ^2(5) \log ^2(x)}-(3+x) \log (3+x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36*x^3 + 12*x^4 + (-36*x^3 - 9*x^4)*Log[x] + (-18 - 9*x - x^2)*Log[5]^2*Log[x]^3 + (-18*x^3 - 6*x^4
+ (18*x^3 + 6*x^4)*Log[x] + (9 + 3*x)*Log[5]^2*Log[x]^3)*Log[3 + x])/((3*x^2 + x^3)*Log[5]^2*Log[x]^3),x]

[Out]

(6 + (3*x^3*(-2 + Log[3 + x]))/(Log[5]^2*Log[x]^2) - (3 + x)*Log[3 + x])/x

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fricas [A]  time = 0.91, size = 52, normalized size = 1.86 \begin {gather*} \frac {6 \, \log \relax (5)^{2} \log \relax (x)^{2} - 6 \, x^{3} - {\left ({\left (x + 3\right )} \log \relax (5)^{2} \log \relax (x)^{2} - 3 \, x^{3}\right )} \log \left (x + 3\right )}{x \log \relax (5)^{2} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x+9)*log(5)^2*log(x)^3+(6*x^4+18*x^3)*log(x)-6*x^4-18*x^3)*log(3+x)+(-x^2-9*x-18)*log(5)^2*log(
x)^3+(-9*x^4-36*x^3)*log(x)+12*x^4+36*x^3)/(x^3+3*x^2)/log(5)^2/log(x)^3,x, algorithm="fricas")

[Out]

(6*log(5)^2*log(x)^2 - 6*x^3 - ((x + 3)*log(5)^2*log(x)^2 - 3*x^3)*log(x + 3))/(x*log(5)^2*log(x)^2)

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giac [B]  time = 0.16, size = 58, normalized size = 2.07 \begin {gather*} -\frac {\log \relax (5)^{2} \log \left (x + 3\right ) + 3 \, {\left (\frac {\log \relax (5)^{2}}{x} - \frac {x^{2}}{\log \relax (x)^{2}}\right )} \log \left (x + 3\right ) - \frac {6 \, \log \relax (5)^{2}}{x} + \frac {6 \, x^{2}}{\log \relax (x)^{2}}}{\log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x+9)*log(5)^2*log(x)^3+(6*x^4+18*x^3)*log(x)-6*x^4-18*x^3)*log(3+x)+(-x^2-9*x-18)*log(5)^2*log(
x)^3+(-9*x^4-36*x^3)*log(x)+12*x^4+36*x^3)/(x^3+3*x^2)/log(5)^2/log(x)^3,x, algorithm="giac")

[Out]

-(log(5)^2*log(x + 3) + 3*(log(5)^2/x - x^2/log(x)^2)*log(x + 3) - 6*log(5)^2/x + 6*x^2/log(x)^2)/log(5)^2

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maple [B]  time = 0.41, size = 77, normalized size = 2.75

method	result	size
risch	\(-\frac {3 \left (\ln \relax (x )^{2} \ln \relax (5)^{2}-x^{3}\right ) \ln \left (3+x \right )}{\ln \relax (5)^{2} x \ln \relax (x )^{2}}-\frac {\ln \relax (5)^{2} \ln \left (3+x \right ) x \ln \relax (x )^{2}-6 \ln \relax (x )^{2} \ln \relax (5)^{2}+6 x^{3}}{\ln \relax (5)^{2} x \ln \relax (x )^{2}}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*x+9)*ln(5)^2*ln(x)^3+(6*x^4+18*x^3)*ln(x)-6*x^4-18*x^3)*ln(3+x)+(-x^2-9*x-18)*ln(5)^2*ln(x)^3+(-9*x^4
-36*x^3)*ln(x)+12*x^4+36*x^3)/(x^3+3*x^2)/ln(5)^2/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

-3/ln(5)^2*(ln(x)^2*ln(5)^2-x^3)/x/ln(x)^2*ln(3+x)-1/ln(5)^2*(ln(5)^2*ln(3+x)*x*ln(x)^2-6*ln(x)^2*ln(5)^2+6*x^
3)/x/ln(x)^2

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maxima [B]  time = 0.48, size = 58, normalized size = 2.07 \begin {gather*} \frac {6 \, \log \relax (5)^{2} \log \relax (x)^{2} - 6 \, x^{3} + {\left (3 \, x^{3} - {\left (x \log \relax (5)^{2} + 3 \, \log \relax (5)^{2}\right )} \log \relax (x)^{2}\right )} \log \left (x + 3\right )}{x \log \relax (5)^{2} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x+9)*log(5)^2*log(x)^3+(6*x^4+18*x^3)*log(x)-6*x^4-18*x^3)*log(3+x)+(-x^2-9*x-18)*log(5)^2*log(
x)^3+(-9*x^4-36*x^3)*log(x)+12*x^4+36*x^3)/(x^3+3*x^2)/log(5)^2/log(x)^3,x, algorithm="maxima")

[Out]

(6*log(5)^2*log(x)^2 - 6*x^3 + (3*x^3 - (x*log(5)^2 + 3*log(5)^2)*log(x)^2)*log(x + 3))/(x*log(5)^2*log(x)^2)

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mupad [B]  time = 5.79, size = 51, normalized size = 1.82 \begin {gather*} \frac {6}{x}-\frac {3\,\ln \left (x+3\right )}{x}-\ln \left (x+3\right )-\frac {6\,x^2}{{\ln \relax (5)}^2\,{\ln \relax (x)}^2}+\frac {3\,x^2\,\ln \left (x+3\right )}{{\ln \relax (5)}^2\,{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + 3)*(log(x)*(18*x^3 + 6*x^4) - 18*x^3 - 6*x^4 + log(5)^2*log(x)^3*(3*x + 9)) - log(x)*(36*x^3 + 9*
x^4) + 36*x^3 + 12*x^4 - log(5)^2*log(x)^3*(9*x + x^2 + 18))/(log(5)^2*log(x)^3*(3*x^2 + x^3)),x)

[Out]

6/x - (3*log(x + 3))/x - log(x + 3) - (6*x^2)/(log(5)^2*log(x)^2) + (3*x^2*log(x + 3))/(log(5)^2*log(x)^2)

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sympy [B]  time = 0.50, size = 56, normalized size = 2.00 \begin {gather*} - \frac {6 x^{2}}{\log {\relax (5 )}^{2} \log {\relax (x )}^{2}} - \log {\left (x + 3 \right )} + \frac {\left (3 x^{3} - 3 \log {\relax (5 )}^{2} \log {\relax (x )}^{2}\right ) \log {\left (x + 3 \right )}}{x \log {\relax (5 )}^{2} \log {\relax (x )}^{2}} + \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x+9)*ln(5)**2*ln(x)**3+(6*x**4+18*x**3)*ln(x)-6*x**4-18*x**3)*ln(3+x)+(-x**2-9*x-18)*ln(5)**2*l
n(x)**3+(-9*x**4-36*x**3)*ln(x)+12*x**4+36*x**3)/(x**3+3*x**2)/ln(5)**2/ln(x)**3,x)

[Out]

-6*x**2/(log(5)**2*log(x)**2) - log(x + 3) + (3*x**3 - 3*log(5)**2*log(x)**2)*log(x + 3)/(x*log(5)**2*log(x)**
2) + 6/x

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