3.102.23 \(\int \frac {7 x^2+(12+4 x) \log ^3(5 x) \log (3+x)+\log ^4(5 x) (-x+(-3-x) \log (3+x))}{147 x^2+49 x^3+(-42 x-14 x^2) \log ^4(5 x)+(3+x) \log ^8(5 x)} \, dx\)

Optimal. Leaf size=20 \[ \frac {x \log (3+x)}{7 x-\log ^4(5 x)} \]

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Rubi [F]  time = 1.19, 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 {7 x^2+(12+4 x) \log ^3(5 x) \log (3+x)+\log ^4(5 x) (-x+(-3-x) \log (3+x))}{147 x^2+49 x^3+\left (-42 x-14 x^2\right ) \log ^4(5 x)+(3+x) \log ^8(5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(7*x^2 + (12 + 4*x)*Log[5*x]^3*Log[3 + x] + Log[5*x]^4*(-x + (-3 - x)*Log[3 + x]))/(147*x^2 + 49*x^3 + (-4
2*x - 14*x^2)*Log[5*x]^4 + (3 + x)*Log[5*x]^8),x]

[Out]

Defer[Int][(7*x - Log[5*x]^4)^(-1), x] - 3*Defer[Int][1/((3 + x)*(7*x - Log[5*x]^4)), x] + 4*Defer[Int][(Log[5
*x]^3*Log[3 + x])/(-7*x + Log[5*x]^4)^2, x] - Defer[Int][(Log[5*x]^4*Log[3 + x])/(-7*x + Log[5*x]^4)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {7 x^2+4 (3+x) \log ^3(5 x) \log (3+x)-\log ^4(5 x) (x+(3+x) \log (3+x))}{(3+x) \left (7 x-\log ^4(5 x)\right )^2} \, dx\\ &=\int \left (\frac {x}{(3+x) \left (7 x-\log ^4(5 x)\right )}-\frac {(-4+\log (5 x)) \log ^3(5 x) \log (3+x)}{\left (-7 x+\log ^4(5 x)\right )^2}\right ) \, dx\\ &=\int \frac {x}{(3+x) \left (7 x-\log ^4(5 x)\right )} \, dx-\int \frac {(-4+\log (5 x)) \log ^3(5 x) \log (3+x)}{\left (-7 x+\log ^4(5 x)\right )^2} \, dx\\ &=\int \left (\frac {1}{7 x-\log ^4(5 x)}-\frac {3}{(3+x) \left (7 x-\log ^4(5 x)\right )}\right ) \, dx-\int \left (-\frac {4 \log ^3(5 x) \log (3+x)}{\left (-7 x+\log ^4(5 x)\right )^2}+\frac {\log ^4(5 x) \log (3+x)}{\left (-7 x+\log ^4(5 x)\right )^2}\right ) \, dx\\ &=-\left (3 \int \frac {1}{(3+x) \left (7 x-\log ^4(5 x)\right )} \, dx\right )+4 \int \frac {\log ^3(5 x) \log (3+x)}{\left (-7 x+\log ^4(5 x)\right )^2} \, dx+\int \frac {1}{7 x-\log ^4(5 x)} \, dx-\int \frac {\log ^4(5 x) \log (3+x)}{\left (-7 x+\log ^4(5 x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 20, normalized size = 1.00 \begin {gather*} \frac {x \log (3+x)}{7 x-\log ^4(5 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7*x^2 + (12 + 4*x)*Log[5*x]^3*Log[3 + x] + Log[5*x]^4*(-x + (-3 - x)*Log[3 + x]))/(147*x^2 + 49*x^3
+ (-42*x - 14*x^2)*Log[5*x]^4 + (3 + x)*Log[5*x]^8),x]

[Out]

(x*Log[3 + x])/(7*x - Log[5*x]^4)

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fricas [A]  time = 1.04, size = 19, normalized size = 0.95 \begin {gather*} -\frac {x \log \left (x + 3\right )}{\log \left (5 \, x\right )^{4} - 7 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3-x)*log(3+x)-x)*log(5*x)^4+(4*x+12)*log(3+x)*log(5*x)^3+7*x^2)/((3+x)*log(5*x)^8+(-14*x^2-42*x)
*log(5*x)^4+49*x^3+147*x^2),x, algorithm="fricas")

[Out]

-x*log(x + 3)/(log(5*x)^4 - 7*x)

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giac [A]  time = 0.40, size = 19, normalized size = 0.95 \begin {gather*} -\frac {x \log \left (x + 3\right )}{\log \left (5 \, x\right )^{4} - 7 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3-x)*log(3+x)-x)*log(5*x)^4+(4*x+12)*log(3+x)*log(5*x)^3+7*x^2)/((3+x)*log(5*x)^8+(-14*x^2-42*x)
*log(5*x)^4+49*x^3+147*x^2),x, algorithm="giac")

[Out]

-x*log(x + 3)/(log(5*x)^4 - 7*x)

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maple [A]  time = 0.35, size = 21, normalized size = 1.05

method	result	size
risch	\(\frac {\ln \left (3+x \right ) x}{7 x -\ln \left (5 x \right )^{4}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-3-x)*ln(3+x)-x)*ln(5*x)^4+(4*x+12)*ln(3+x)*ln(5*x)^3+7*x^2)/((3+x)*ln(5*x)^8+(-14*x^2-42*x)*ln(5*x)^4+
49*x^3+147*x^2),x,method=_RETURNVERBOSE)

[Out]

ln(3+x)/(7*x-ln(5*x)^4)*x

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maxima [B]  time = 0.47, size = 47, normalized size = 2.35 \begin {gather*} -\frac {x \log \left (x + 3\right )}{\log \relax (5)^{4} + 4 \, \log \relax (5)^{3} \log \relax (x) + 6 \, \log \relax (5)^{2} \log \relax (x)^{2} + 4 \, \log \relax (5) \log \relax (x)^{3} + \log \relax (x)^{4} - 7 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3-x)*log(3+x)-x)*log(5*x)^4+(4*x+12)*log(3+x)*log(5*x)^3+7*x^2)/((3+x)*log(5*x)^8+(-14*x^2-42*x)
*log(5*x)^4+49*x^3+147*x^2),x, algorithm="maxima")

[Out]

-x*log(x + 3)/(log(5)^4 + 4*log(5)^3*log(x) + 6*log(5)^2*log(x)^2 + 4*log(5)*log(x)^3 + log(x)^4 - 7*x)

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mupad [B]  time = 7.38, size = 20, normalized size = 1.00 \begin {gather*} \frac {x\,\ln \left (x+3\right )}{7\,x-{\ln \left (5\,x\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x^2 - log(5*x)^4*(x + log(x + 3)*(x + 3)) + log(5*x)^3*log(x + 3)*(4*x + 12))/(147*x^2 - log(5*x)^4*(42
*x + 14*x^2) + 49*x^3 + log(5*x)^8*(x + 3)),x)

[Out]

(x*log(x + 3))/(7*x - log(5*x)^4)

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sympy [A]  time = 0.52, size = 15, normalized size = 0.75 \begin {gather*} \frac {x \log {\left (x + 3 \right )}}{7 x - \log {\left (5 x \right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-3-x)*ln(3+x)-x)*ln(5*x)**4+(4*x+12)*ln(3+x)*ln(5*x)**3+7*x**2)/((3+x)*ln(5*x)**8+(-14*x**2-42*x)
*ln(5*x)**4+49*x**3+147*x**2),x)

[Out]

x*log(x + 3)/(7*x - log(5*x)**4)

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