3.77.93 \(\int \frac {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5+(3456+864 x-1010 x^2+194 x^3-12 x^4) \log (4+3 x)+(864+360 x-192 x^2+18 x^3) \log ^2(4+3 x)+(96+56 x-12 x^2) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{5184+720 x-1604 x^2+491 x^3-62 x^4+3 x^5+(3456+960 x-952 x^2+188 x^3-12 x^4) \log (4+3 x)+(864+368 x-186 x^2+18 x^3) \log ^2(4+3 x)+(96+56 x-12 x^2) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)} \, dx\)

Optimal. Leaf size=25 \[ \frac {x^2}{x+\frac {x^2}{(-6+x-\log (4+3 x))^2}} \]

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Rubi [F]  time = 3.14, 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 {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5+\left (3456+864 x-1010 x^2+194 x^3-12 x^4\right ) \log (4+3 x)+\left (864+360 x-192 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{5184+720 x-1604 x^2+491 x^3-62 x^4+3 x^5+\left (3456+960 x-952 x^2+188 x^3-12 x^4\right ) \log (4+3 x)+\left (864+368 x-186 x^2+18 x^3\right ) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5184 + 432*x - 1740*x^2 + 518*x^3 - 62*x^4 + 3*x^5 + (3456 + 864*x - 1010*x^2 + 194*x^3 - 12*x^4)*Log[4 +
3*x] + (864 + 360*x - 192*x^2 + 18*x^3)*Log[4 + 3*x]^2 + (96 + 56*x - 12*x^2)*Log[4 + 3*x]^3 + (4 + 3*x)*Log[
4 + 3*x]^4)/(5184 + 720*x - 1604*x^2 + 491*x^3 - 62*x^4 + 3*x^5 + (3456 + 960*x - 952*x^2 + 188*x^3 - 12*x^4)*
Log[4 + 3*x] + (864 + 368*x - 186*x^2 + 18*x^3)*Log[4 + 3*x]^2 + (96 + 56*x - 12*x^2)*Log[4 + 3*x]^3 + (4 + 3*
x)*Log[4 + 3*x]^4),x]

[Out]

x - 16/(9*(36 - 11*x + x^2 + 12*Log[4 + 3*x] - 2*x*Log[4 + 3*x] + Log[4 + 3*x]^2)) + (32*Defer[Int][(36 - 11*x
+ x^2 + 12*Log[4 + 3*x] - 2*x*Log[4 + 3*x] + Log[4 + 3*x]^2)^(-2), x])/9 + (100*Defer[Int][x/(36 - 11*x + x^2
+ 12*Log[4 + 3*x] - 2*x*Log[4 + 3*x] + Log[4 + 3*x]^2)^2, x])/9 - 13*Defer[Int][x^2/(36 - 11*x + x^2 + 12*Log
[4 + 3*x] - 2*x*Log[4 + 3*x] + Log[4 + 3*x]^2)^2, x] + 2*Defer[Int][x^3/(36 - 11*x + x^2 + 12*Log[4 + 3*x] - 2
*x*Log[4 + 3*x] + Log[4 + 3*x]^2)^2, x] + (8*Defer[Int][Log[4 + 3*x]/(36 - 11*x + x^2 + 12*Log[4 + 3*x] - 2*x*
Log[4 + 3*x] + Log[4 + 3*x]^2)^2, x])/9 + 2*Defer[Int][(x*Log[4 + 3*x])/(36 - 11*x + x^2 + 12*Log[4 + 3*x] - 2
*x*Log[4 + 3*x] + Log[4 + 3*x]^2)^2, x] - 2*Defer[Int][(x^2*Log[4 + 3*x])/(36 - 11*x + x^2 + 12*Log[4 + 3*x] -
2*x*Log[4 + 3*x] + Log[4 + 3*x]^2)^2, x] - 2*Defer[Int][x/(36 - 11*x + x^2 + 12*Log[4 + 3*x] - 2*x*Log[4 + 3*
x] + Log[4 + 3*x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5184+432 x-1740 x^2+518 x^3-62 x^4+3 x^5-2 \left (-1728-432 x+505 x^2-97 x^3+6 x^4\right ) \log (4+3 x)+6 (-6+x)^2 (4+3 x) \log ^2(4+3 x)+\left (96+56 x-12 x^2\right ) \log ^3(4+3 x)+(4+3 x) \log ^4(4+3 x)}{(4+3 x) \left (36-11 x+x^2-2 (-6+x) \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx\\ &=\int \left (1+\frac {x^2 \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{(4+3 x) \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {2 x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)}\right ) \, dx\\ &=x-2 \int \frac {x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)} \, dx+\int \frac {x^2 \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{(4+3 x) \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx\\ &=x-2 \int \frac {x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)} \, dx+\int \left (-\frac {4 \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{9 \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}+\frac {x \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{3 \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}+\frac {16 \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{9 (4+3 x) \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}\right ) \, dx\\ &=x+\frac {1}{3} \int \frac {x \left (-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)\right )}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-\frac {4}{9} \int \frac {-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+\frac {16}{9} \int \frac {-8-31 x+6 x^2-2 \log (4+3 x)-6 x \log (4+3 x)}{(4+3 x) \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-2 \int \frac {x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)} \, dx\\ &=x-\frac {16}{9 \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )}+\frac {1}{3} \int \left (-\frac {8 x}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {31 x^2}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}+\frac {6 x^3}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {2 x \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {6 x^2 \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}\right ) \, dx-\frac {4}{9} \int \left (-\frac {8}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {31 x}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}+\frac {6 x^2}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {2 \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}-\frac {6 x \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2}\right ) \, dx-2 \int \frac {x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)} \, dx\\ &=x-\frac {16}{9 \left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )}-\frac {2}{3} \int \frac {x \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+\frac {8}{9} \int \frac {\log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+2 \int \frac {x^3}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-2 \int \frac {x^2 \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-2 \int \frac {x}{36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)} \, dx-\frac {8}{3} \int \frac {x}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-\frac {8}{3} \int \frac {x^2}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+\frac {8}{3} \int \frac {x \log (4+3 x)}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+\frac {32}{9} \int \frac {1}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx-\frac {31}{3} \int \frac {x^2}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx+\frac {124}{9} \int \frac {x}{\left (36-11 x+x^2+12 \log (4+3 x)-2 x \log (4+3 x)+\log ^2(4+3 x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 44, normalized size = 1.76 \begin {gather*} \frac {x (6-x+\log (4+3 x))^2}{36-11 x+x^2-2 (-6+x) \log (4+3 x)+\log ^2(4+3 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5184 + 432*x - 1740*x^2 + 518*x^3 - 62*x^4 + 3*x^5 + (3456 + 864*x - 1010*x^2 + 194*x^3 - 12*x^4)*L
og[4 + 3*x] + (864 + 360*x - 192*x^2 + 18*x^3)*Log[4 + 3*x]^2 + (96 + 56*x - 12*x^2)*Log[4 + 3*x]^3 + (4 + 3*x
)*Log[4 + 3*x]^4)/(5184 + 720*x - 1604*x^2 + 491*x^3 - 62*x^4 + 3*x^5 + (3456 + 960*x - 952*x^2 + 188*x^3 - 12
*x^4)*Log[4 + 3*x] + (864 + 368*x - 186*x^2 + 18*x^3)*Log[4 + 3*x]^2 + (96 + 56*x - 12*x^2)*Log[4 + 3*x]^3 + (
4 + 3*x)*Log[4 + 3*x]^4),x]

[Out]

(x*(6 - x + Log[4 + 3*x])^2)/(36 - 11*x + x^2 - 2*(-6 + x)*Log[4 + 3*x] + Log[4 + 3*x]^2)

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fricas [B]  time = 0.51, size = 67, normalized size = 2.68 \begin {gather*} \frac {x^{3} + x \log \left (3 \, x + 4\right )^{2} - 12 \, x^{2} - 2 \, {\left (x^{2} - 6 \, x\right )} \log \left (3 \, x + 4\right ) + 36 \, x}{x^{2} - 2 \, {\left (x - 6\right )} \log \left (3 \, x + 4\right ) + \log \left (3 \, x + 4\right )^{2} - 11 \, x + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4+3*x)*log(4+3*x)^4+(-12*x^2+56*x+96)*log(4+3*x)^3+(18*x^3-192*x^2+360*x+864)*log(4+3*x)^2+(-12*x^
4+194*x^3-1010*x^2+864*x+3456)*log(4+3*x)+3*x^5-62*x^4+518*x^3-1740*x^2+432*x+5184)/((4+3*x)*log(4+3*x)^4+(-12
*x^2+56*x+96)*log(4+3*x)^3+(18*x^3-186*x^2+368*x+864)*log(4+3*x)^2+(-12*x^4+188*x^3-952*x^2+960*x+3456)*log(4+
3*x)+3*x^5-62*x^4+491*x^3-1604*x^2+720*x+5184),x, algorithm="fricas")

[Out]

(x^3 + x*log(3*x + 4)^2 - 12*x^2 - 2*(x^2 - 6*x)*log(3*x + 4) + 36*x)/(x^2 - 2*(x - 6)*log(3*x + 4) + log(3*x
+ 4)^2 - 11*x + 36)

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giac [A]  time = 0.34, size = 42, normalized size = 1.68 \begin {gather*} x - \frac {x^{2}}{x^{2} - 2 \, x \log \left (3 \, x + 4\right ) + \log \left (3 \, x + 4\right )^{2} - 11 \, x + 12 \, \log \left (3 \, x + 4\right ) + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4+3*x)*log(4+3*x)^4+(-12*x^2+56*x+96)*log(4+3*x)^3+(18*x^3-192*x^2+360*x+864)*log(4+3*x)^2+(-12*x^
4+194*x^3-1010*x^2+864*x+3456)*log(4+3*x)+3*x^5-62*x^4+518*x^3-1740*x^2+432*x+5184)/((4+3*x)*log(4+3*x)^4+(-12
*x^2+56*x+96)*log(4+3*x)^3+(18*x^3-186*x^2+368*x+864)*log(4+3*x)^2+(-12*x^4+188*x^3-952*x^2+960*x+3456)*log(4+
3*x)+3*x^5-62*x^4+491*x^3-1604*x^2+720*x+5184),x, algorithm="giac")

[Out]

x - x^2/(x^2 - 2*x*log(3*x + 4) + log(3*x + 4)^2 - 11*x + 12*log(3*x + 4) + 36)

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maple [A]  time = 0.04, size = 43, normalized size = 1.72

method	result	size
risch	\(x -\frac {x^{2}}{\ln \left (4+3 x \right )^{2}-2 \ln \left (4+3 x \right ) x +x^{2}+12 \ln \left (4+3 x \right )-11 x +36}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4+3*x)*ln(4+3*x)^4+(-12*x^2+56*x+96)*ln(4+3*x)^3+(18*x^3-192*x^2+360*x+864)*ln(4+3*x)^2+(-12*x^4+194*x^3
-1010*x^2+864*x+3456)*ln(4+3*x)+3*x^5-62*x^4+518*x^3-1740*x^2+432*x+5184)/((4+3*x)*ln(4+3*x)^4+(-12*x^2+56*x+9
6)*ln(4+3*x)^3+(18*x^3-186*x^2+368*x+864)*ln(4+3*x)^2+(-12*x^4+188*x^3-952*x^2+960*x+3456)*ln(4+3*x)+3*x^5-62*
x^4+491*x^3-1604*x^2+720*x+5184),x,method=_RETURNVERBOSE)

[Out]

x-x^2/(ln(4+3*x)^2-2*ln(4+3*x)*x+x^2+12*ln(4+3*x)-11*x+36)

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maxima [B]  time = 0.41, size = 67, normalized size = 2.68 \begin {gather*} \frac {x^{3} + x \log \left (3 \, x + 4\right )^{2} - 12 \, x^{2} - 2 \, {\left (x^{2} - 6 \, x\right )} \log \left (3 \, x + 4\right ) + 36 \, x}{x^{2} - 2 \, {\left (x - 6\right )} \log \left (3 \, x + 4\right ) + \log \left (3 \, x + 4\right )^{2} - 11 \, x + 36} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4+3*x)*log(4+3*x)^4+(-12*x^2+56*x+96)*log(4+3*x)^3+(18*x^3-192*x^2+360*x+864)*log(4+3*x)^2+(-12*x^
4+194*x^3-1010*x^2+864*x+3456)*log(4+3*x)+3*x^5-62*x^4+518*x^3-1740*x^2+432*x+5184)/((4+3*x)*log(4+3*x)^4+(-12
*x^2+56*x+96)*log(4+3*x)^3+(18*x^3-186*x^2+368*x+864)*log(4+3*x)^2+(-12*x^4+188*x^3-952*x^2+960*x+3456)*log(4+
3*x)+3*x^5-62*x^4+491*x^3-1604*x^2+720*x+5184),x, algorithm="maxima")

[Out]

(x^3 + x*log(3*x + 4)^2 - 12*x^2 - 2*(x^2 - 6*x)*log(3*x + 4) + 36*x)/(x^2 - 2*(x - 6)*log(3*x + 4) + log(3*x
+ 4)^2 - 11*x + 36)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {432\,x+{\ln \left (3\,x+4\right )}^2\,\left (18\,x^3-192\,x^2+360\,x+864\right )+{\ln \left (3\,x+4\right )}^4\,\left (3\,x+4\right )+{\ln \left (3\,x+4\right )}^3\,\left (-12\,x^2+56\,x+96\right )-1740\,x^2+518\,x^3-62\,x^4+3\,x^5+\ln \left (3\,x+4\right )\,\left (-12\,x^4+194\,x^3-1010\,x^2+864\,x+3456\right )+5184}{720\,x+{\ln \left (3\,x+4\right )}^2\,\left (18\,x^3-186\,x^2+368\,x+864\right )+{\ln \left (3\,x+4\right )}^4\,\left (3\,x+4\right )+{\ln \left (3\,x+4\right )}^3\,\left (-12\,x^2+56\,x+96\right )-1604\,x^2+491\,x^3-62\,x^4+3\,x^5+\ln \left (3\,x+4\right )\,\left (-12\,x^4+188\,x^3-952\,x^2+960\,x+3456\right )+5184} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((432*x + log(3*x + 4)^2*(360*x - 192*x^2 + 18*x^3 + 864) + log(3*x + 4)^4*(3*x + 4) + log(3*x + 4)^3*(56*x
- 12*x^2 + 96) - 1740*x^2 + 518*x^3 - 62*x^4 + 3*x^5 + log(3*x + 4)*(864*x - 1010*x^2 + 194*x^3 - 12*x^4 + 34
56) + 5184)/(720*x + log(3*x + 4)^2*(368*x - 186*x^2 + 18*x^3 + 864) + log(3*x + 4)^4*(3*x + 4) + log(3*x + 4)
^3*(56*x - 12*x^2 + 96) - 1604*x^2 + 491*x^3 - 62*x^4 + 3*x^5 + log(3*x + 4)*(960*x - 952*x^2 + 188*x^3 - 12*x
^4 + 3456) + 5184),x)

[Out]

int((432*x + log(3*x + 4)^2*(360*x - 192*x^2 + 18*x^3 + 864) + log(3*x + 4)^4*(3*x + 4) + log(3*x + 4)^3*(56*x
- 12*x^2 + 96) - 1740*x^2 + 518*x^3 - 62*x^4 + 3*x^5 + log(3*x + 4)*(864*x - 1010*x^2 + 194*x^3 - 12*x^4 + 34
56) + 5184)/(720*x + log(3*x + 4)^2*(368*x - 186*x^2 + 18*x^3 + 864) + log(3*x + 4)^4*(3*x + 4) + log(3*x + 4)
^3*(56*x - 12*x^2 + 96) - 1604*x^2 + 491*x^3 - 62*x^4 + 3*x^5 + log(3*x + 4)*(960*x - 952*x^2 + 188*x^3 - 12*x
^4 + 3456) + 5184), x)

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sympy [A]  time = 0.26, size = 32, normalized size = 1.28 \begin {gather*} - \frac {x^{2}}{x^{2} - 11 x + \left (12 - 2 x\right ) \log {\left (3 x + 4 \right )} + \log {\left (3 x + 4 \right )}^{2} + 36} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4+3*x)*ln(4+3*x)**4+(-12*x**2+56*x+96)*ln(4+3*x)**3+(18*x**3-192*x**2+360*x+864)*ln(4+3*x)**2+(-12
*x**4+194*x**3-1010*x**2+864*x+3456)*ln(4+3*x)+3*x**5-62*x**4+518*x**3-1740*x**2+432*x+5184)/((4+3*x)*ln(4+3*x
)**4+(-12*x**2+56*x+96)*ln(4+3*x)**3+(18*x**3-186*x**2+368*x+864)*ln(4+3*x)**2+(-12*x**4+188*x**3-952*x**2+960
*x+3456)*ln(4+3*x)+3*x**5-62*x**4+491*x**3-1604*x**2+720*x+5184),x)

[Out]

-x**2/(x**2 - 11*x + (12 - 2*x)*log(3*x + 4) + log(3*x + 4)**2 + 36) + x

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