3.20.42 \(\int \frac {-2600-1747 x+146 x^2-3 x^3+(-204-145 x+6 x^2) \log (4+3 x)+(-4-3 x) \log ^2(4+3 x)}{7500+5025 x-438 x^2+9 x^3+(600+426 x-18 x^2) \log (4+3 x)+(12+9 x) \log ^2(4+3 x)} \, dx\)

Optimal. Leaf size=27 \[ \log (2)+\frac {1}{3} \left (1-x+\frac {x}{-25+x-\log (4+3 x)}\right ) \]

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Rubi [F]  time = 0.80, 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 {-2600-1747 x+146 x^2-3 x^3+\left (-204-145 x+6 x^2\right ) \log (4+3 x)+(-4-3 x) \log ^2(4+3 x)}{7500+5025 x-438 x^2+9 x^3+\left (600+426 x-18 x^2\right ) \log (4+3 x)+(12+9 x) \log ^2(4+3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2600 - 1747*x + 146*x^2 - 3*x^3 + (-204 - 145*x + 6*x^2)*Log[4 + 3*x] + (-4 - 3*x)*Log[4 + 3*x]^2)/(7500
+ 5025*x - 438*x^2 + 9*x^3 + (600 + 426*x - 18*x^2)*Log[4 + 3*x] + (12 + 9*x)*Log[4 + 3*x]^2),x]

[Out]

-1/3*x + Defer[Int][(-25 + x - Log[4 + 3*x])^(-2), x]/3 - Defer[Int][x/(-25 + x - Log[4 + 3*x])^2, x]/3 - (4*D
efer[Int][1/((4 + 3*x)*(-25 + x - Log[4 + 3*x])^2), x])/3 + Defer[Int][(-25 + x - Log[4 + 3*x])^(-1), x]/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2600-1747 x+146 x^2-3 x^3+\left (-204-145 x+6 x^2\right ) \log (4+3 x)+(-4-3 x) \log ^2(4+3 x)}{3 (4+3 x) (25-x+\log (4+3 x))^2} \, dx\\ &=\frac {1}{3} \int \frac {-2600-1747 x+146 x^2-3 x^3+\left (-204-145 x+6 x^2\right ) \log (4+3 x)+(-4-3 x) \log ^2(4+3 x)}{(4+3 x) (25-x+\log (4+3 x))^2} \, dx\\ &=\frac {1}{3} \int \left (-1-\frac {x (1+3 x)}{(4+3 x) (-25+x-\log (4+3 x))^2}+\frac {1}{-25+x-\log (4+3 x)}\right ) \, dx\\ &=-\frac {x}{3}-\frac {1}{3} \int \frac {x (1+3 x)}{(4+3 x) (-25+x-\log (4+3 x))^2} \, dx+\frac {1}{3} \int \frac {1}{-25+x-\log (4+3 x)} \, dx\\ &=-\frac {x}{3}-\frac {1}{3} \int \left (-\frac {1}{(-25+x-\log (4+3 x))^2}+\frac {x}{(-25+x-\log (4+3 x))^2}+\frac {4}{(4+3 x) (-25+x-\log (4+3 x))^2}\right ) \, dx+\frac {1}{3} \int \frac {1}{-25+x-\log (4+3 x)} \, dx\\ &=-\frac {x}{3}+\frac {1}{3} \int \frac {1}{(-25+x-\log (4+3 x))^2} \, dx-\frac {1}{3} \int \frac {x}{(-25+x-\log (4+3 x))^2} \, dx+\frac {1}{3} \int \frac {1}{-25+x-\log (4+3 x)} \, dx-\frac {4}{3} \int \frac {1}{(4+3 x) (-25+x-\log (4+3 x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 20, normalized size = 0.74 \begin {gather*} -\frac {1}{3} x \left (1+\frac {1}{25-x+\log (4+3 x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2600 - 1747*x + 146*x^2 - 3*x^3 + (-204 - 145*x + 6*x^2)*Log[4 + 3*x] + (-4 - 3*x)*Log[4 + 3*x]^2)
/(7500 + 5025*x - 438*x^2 + 9*x^3 + (600 + 426*x - 18*x^2)*Log[4 + 3*x] + (12 + 9*x)*Log[4 + 3*x]^2),x]

[Out]

-1/3*(x*(1 + (25 - x + Log[4 + 3*x])^(-1)))

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fricas [A]  time = 1.01, size = 31, normalized size = 1.15 \begin {gather*} -\frac {x^{2} - x \log \left (3 \, x + 4\right ) - 26 \, x}{3 \, {\left (x - \log \left (3 \, x + 4\right ) - 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-4)*log(4+3*x)^2+(6*x^2-145*x-204)*log(4+3*x)-3*x^3+146*x^2-1747*x-2600)/((9*x+12)*log(4+3*x)^
2+(-18*x^2+426*x+600)*log(4+3*x)+9*x^3-438*x^2+5025*x+7500),x, algorithm="fricas")

[Out]

-1/3*(x^2 - x*log(3*x + 4) - 26*x)/(x - log(3*x + 4) - 25)

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giac [A]  time = 0.25, size = 20, normalized size = 0.74 \begin {gather*} -\frac {1}{3} \, x + \frac {x}{3 \, {\left (x - \log \left (3 \, x + 4\right ) - 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-4)*log(4+3*x)^2+(6*x^2-145*x-204)*log(4+3*x)-3*x^3+146*x^2-1747*x-2600)/((9*x+12)*log(4+3*x)^
2+(-18*x^2+426*x+600)*log(4+3*x)+9*x^3-438*x^2+5025*x+7500),x, algorithm="giac")

[Out]

-1/3*x + 1/3*x/(x - log(3*x + 4) - 25)

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maple [A]  time = 0.07, size = 21, normalized size = 0.78

method	result	size
risch	\(-\frac {x}{3}+\frac {x}{3 x -3 \ln \left (4+3 x \right )-75}\)	\(21\)
norman	\(\frac {\frac {\ln \left (4+3 x \right )^{2}}{3}+17 x -\frac {x^{2}}{3}-\frac {625}{3}}{x -\ln \left (4+3 x \right )-25}+\frac {\ln \left (4+3 x \right )}{3}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3*x-4)*ln(4+3*x)^2+(6*x^2-145*x-204)*ln(4+3*x)-3*x^3+146*x^2-1747*x-2600)/((9*x+12)*ln(4+3*x)^2+(-18*x^
2+426*x+600)*ln(4+3*x)+9*x^3-438*x^2+5025*x+7500),x,method=_RETURNVERBOSE)

[Out]

-1/3*x+1/3*x/(x-ln(4+3*x)-25)

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maxima [A]  time = 0.52, size = 31, normalized size = 1.15 \begin {gather*} -\frac {x^{2} - x \log \left (3 \, x + 4\right ) - 26 \, x}{3 \, {\left (x - \log \left (3 \, x + 4\right ) - 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-4)*log(4+3*x)^2+(6*x^2-145*x-204)*log(4+3*x)-3*x^3+146*x^2-1747*x-2600)/((9*x+12)*log(4+3*x)^
2+(-18*x^2+426*x+600)*log(4+3*x)+9*x^3-438*x^2+5025*x+7500),x, algorithm="maxima")

[Out]

-1/3*(x^2 - x*log(3*x + 4) - 26*x)/(x - log(3*x + 4) - 25)

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mupad [B]  time = 0.25, size = 41, normalized size = 1.52 \begin {gather*} -\frac {25\,x+\ln \left (3\,x+4\right )+x\,\ln \left (3\,x+4\right )-x^2+25}{3\,\left (\ln \left (3\,x+4\right )-x+25\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1747*x + log(3*x + 4)*(145*x - 6*x^2 + 204) + log(3*x + 4)^2*(3*x + 4) - 146*x^2 + 3*x^3 + 2600)/(5025*x
+ log(3*x + 4)*(426*x - 18*x^2 + 600) + log(3*x + 4)^2*(9*x + 12) - 438*x^2 + 9*x^3 + 7500),x)

[Out]

-(25*x + log(3*x + 4) + x*log(3*x + 4) - x^2 + 25)/(3*(log(3*x + 4) - x + 25))

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sympy [A]  time = 0.15, size = 19, normalized size = 0.70 \begin {gather*} - \frac {x}{3} - \frac {x}{- 3 x + 3 \log {\left (3 x + 4 \right )} + 75} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3*x-4)*ln(4+3*x)**2+(6*x**2-145*x-204)*ln(4+3*x)-3*x**3+146*x**2-1747*x-2600)/((9*x+12)*ln(4+3*x)
**2+(-18*x**2+426*x+600)*ln(4+3*x)+9*x**3-438*x**2+5025*x+7500),x)

[Out]

-x/3 - x/(-3*x + 3*log(3*x + 4) + 75)

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