3.41.12 \(\int \frac {108-111 x-267 x^2+6 x^3+143 x^4+39 x^5+(72-30 x-99 x^2-27 x^3) \log (x)+(12+4 x) \log ^2(x)+(-72+30 x+99 x^2+27 x^3+(-24-8 x) \log (x)) \log (15+5 x)+(12+4 x) \log ^2(15+5 x)}{27-9 x-57 x^2+x^3+33 x^4+9 x^5+(18-20 x^2-6 x^3) \log (x)+(3+x) \log ^2(x)+(-18+20 x^2+6 x^3+(-6-2 x) \log (x)) \log (15+5 x)+(3+x) \log ^2(15+5 x)} \, dx\)

Optimal. Leaf size=29 \[ x \left (4+\frac {x (5+x)}{-3+x+3 x^2-\log (x)+\log (5 (3+x))}\right ) \]

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Rubi [F]  time = 1.88, 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 {108-111 x-267 x^2+6 x^3+143 x^4+39 x^5+\left (72-30 x-99 x^2-27 x^3\right ) \log (x)+(12+4 x) \log ^2(x)+\left (-72+30 x+99 x^2+27 x^3+(-24-8 x) \log (x)\right ) \log (15+5 x)+(12+4 x) \log ^2(15+5 x)}{27-9 x-57 x^2+x^3+33 x^4+9 x^5+\left (18-20 x^2-6 x^3\right ) \log (x)+(3+x) \log ^2(x)+\left (-18+20 x^2+6 x^3+(-6-2 x) \log (x)\right ) \log (15+5 x)+(3+x) \log ^2(15+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(108 - 111*x - 267*x^2 + 6*x^3 + 143*x^4 + 39*x^5 + (72 - 30*x - 99*x^2 - 27*x^3)*Log[x] + (12 + 4*x)*Log[
x]^2 + (-72 + 30*x + 99*x^2 + 27*x^3 + (-24 - 8*x)*Log[x])*Log[15 + 5*x] + (12 + 4*x)*Log[15 + 5*x]^2)/(27 - 9
*x - 57*x^2 + x^3 + 33*x^4 + 9*x^5 + (18 - 20*x^2 - 6*x^3)*Log[x] + (3 + x)*Log[x]^2 + (-18 + 20*x^2 + 6*x^3 +
(-6 - 2*x)*Log[x])*Log[15 + 5*x] + (3 + x)*Log[15 + 5*x]^2),x]

[Out]

4*x + 6*Defer[Int][(-3 + x + 3*x^2 - Log[x] + Log[5*(3 + x)])^(-2), x] + 3*Defer[Int][x/(-3 + x + 3*x^2 - Log[
x] + Log[5*(3 + x)])^2, x] - 5*Defer[Int][x^2/(-3 + x + 3*x^2 - Log[x] + Log[5*(3 + x)])^2, x] - 31*Defer[Int]
[x^3/(-3 + x + 3*x^2 - Log[x] + Log[5*(3 + x)])^2, x] - 6*Defer[Int][x^4/(-3 + x + 3*x^2 - Log[x] + Log[5*(3 +
x)])^2, x] - 18*Defer[Int][1/((3 + x)*(-3 + x + 3*x^2 - Log[x] + Log[5*(3 + x)])^2), x] + 10*Defer[Int][x/(-3
+ x + 3*x^2 - Log[x] + Log[5*(3 + x)]), x] + 3*Defer[Int][x^2/(-3 + x + 3*x^2 - Log[x] + Log[5*(3 + x)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {108-111 x-267 x^2+6 x^3+143 x^4+39 x^5+4 (3+x) \log ^2(x)+3 \left (-24+10 x+33 x^2+9 x^3\right ) \log (5 (3+x))+4 (3+x) \log ^2(5 (3+x))-(3+x) \log (x) \left (3 \left (-8+6 x+9 x^2\right )+8 \log (5 (3+x))\right )}{(3+x) \left (3-x-3 x^2+\log (x)-\log (5 (3+x))\right )^2} \, dx\\ &=\int \left (4-\frac {x \left (-15+12 x+98 x^2+49 x^3+6 x^4\right )}{(3+x) \left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2}+\frac {x (10+3 x)}{-3+x+3 x^2-\log (x)+\log (5 (3+x))}\right ) \, dx\\ &=4 x-\int \frac {x \left (-15+12 x+98 x^2+49 x^3+6 x^4\right )}{(3+x) \left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2} \, dx+\int \frac {x (10+3 x)}{-3+x+3 x^2-\log (x)+\log (5 (3+x))} \, dx\\ &=4 x-\int \left (-\frac {6}{\left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2}-\frac {3 x}{\left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2}+\frac {5 x^2}{\left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2}+\frac {31 x^3}{\left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2}+\frac {6 x^4}{\left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2}+\frac {18}{(3+x) \left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2}\right ) \, dx+\int \left (\frac {10 x}{-3+x+3 x^2-\log (x)+\log (5 (3+x))}+\frac {3 x^2}{-3+x+3 x^2-\log (x)+\log (5 (3+x))}\right ) \, dx\\ &=4 x+3 \int \frac {x}{\left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2} \, dx+3 \int \frac {x^2}{-3+x+3 x^2-\log (x)+\log (5 (3+x))} \, dx-5 \int \frac {x^2}{\left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2} \, dx+6 \int \frac {1}{\left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2} \, dx-6 \int \frac {x^4}{\left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2} \, dx+10 \int \frac {x}{-3+x+3 x^2-\log (x)+\log (5 (3+x))} \, dx-18 \int \frac {1}{(3+x) \left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2} \, dx-31 \int \frac {x^3}{\left (-3+x+3 x^2-\log (x)+\log (5 (3+x))\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 31, normalized size = 1.07 \begin {gather*} 4 x+\frac {x^2 (5+x)}{-3+x+3 x^2-\log (x)+\log (5 (3+x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(108 - 111*x - 267*x^2 + 6*x^3 + 143*x^4 + 39*x^5 + (72 - 30*x - 99*x^2 - 27*x^3)*Log[x] + (12 + 4*x
)*Log[x]^2 + (-72 + 30*x + 99*x^2 + 27*x^3 + (-24 - 8*x)*Log[x])*Log[15 + 5*x] + (12 + 4*x)*Log[15 + 5*x]^2)/(
27 - 9*x - 57*x^2 + x^3 + 33*x^4 + 9*x^5 + (18 - 20*x^2 - 6*x^3)*Log[x] + (3 + x)*Log[x]^2 + (-18 + 20*x^2 + 6
*x^3 + (-6 - 2*x)*Log[x])*Log[15 + 5*x] + (3 + x)*Log[15 + 5*x]^2),x]

[Out]

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

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fricas [A]  time = 0.55, size = 49, normalized size = 1.69 \begin {gather*} \frac {13 \, x^{3} + 9 \, x^{2} + 4 \, x \log \left (5 \, x + 15\right ) - 4 \, x \log \relax (x) - 12 \, x}{3 \, x^{2} + x + \log \left (5 \, x + 15\right ) - \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x+12)*log(5*x+15)^2+((-8*x-24)*log(x)+27*x^3+99*x^2+30*x-72)*log(5*x+15)+(4*x+12)*log(x)^2+(-27*
x^3-99*x^2-30*x+72)*log(x)+39*x^5+143*x^4+6*x^3-267*x^2-111*x+108)/((3+x)*log(5*x+15)^2+((-2*x-6)*log(x)+6*x^3
+20*x^2-18)*log(5*x+15)+(3+x)*log(x)^2+(-6*x^3-20*x^2+18)*log(x)+9*x^5+33*x^4+x^3-57*x^2-9*x+27),x, algorithm=
"fricas")

[Out]

(13*x^3 + 9*x^2 + 4*x*log(5*x + 15) - 4*x*log(x) - 12*x)/(3*x^2 + x + log(5*x + 15) - log(x) - 3)

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giac [A]  time = 0.31, size = 34, normalized size = 1.17 \begin {gather*} 4 \, x + \frac {x^{3} + 5 \, x^{2}}{3 \, x^{2} + x + \log \left (5 \, x + 15\right ) - \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x+12)*log(5*x+15)^2+((-8*x-24)*log(x)+27*x^3+99*x^2+30*x-72)*log(5*x+15)+(4*x+12)*log(x)^2+(-27*
x^3-99*x^2-30*x+72)*log(x)+39*x^5+143*x^4+6*x^3-267*x^2-111*x+108)/((3+x)*log(5*x+15)^2+((-2*x-6)*log(x)+6*x^3
+20*x^2-18)*log(5*x+15)+(3+x)*log(x)^2+(-6*x^3-20*x^2+18)*log(x)+9*x^5+33*x^4+x^3-57*x^2-9*x+27),x, algorithm=
"giac")

[Out]

4*x + (x^3 + 5*x^2)/(3*x^2 + x + log(5*x + 15) - log(x) - 3)

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maple [A]  time = 0.05, size = 32, normalized size = 1.10

method	result	size
risch	\(4 x +\frac {x^{2} \left (5+x \right )}{\ln \left (5 x +15\right )-3+3 x^{2}-\ln \relax (x )+x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x+12)*ln(5*x+15)^2+((-8*x-24)*ln(x)+27*x^3+99*x^2+30*x-72)*ln(5*x+15)+(4*x+12)*ln(x)^2+(-27*x^3-99*x^2
-30*x+72)*ln(x)+39*x^5+143*x^4+6*x^3-267*x^2-111*x+108)/((3+x)*ln(5*x+15)^2+((-2*x-6)*ln(x)+6*x^3+20*x^2-18)*l
n(5*x+15)+(3+x)*ln(x)^2+(-6*x^3-20*x^2+18)*ln(x)+9*x^5+33*x^4+x^3-57*x^2-9*x+27),x,method=_RETURNVERBOSE)

[Out]

4*x+x^2*(5+x)/(ln(5*x+15)-3+3*x^2-ln(x)+x)

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maxima [A]  time = 1.25, size = 51, normalized size = 1.76 \begin {gather*} \frac {13 \, x^{3} + 9 \, x^{2} + 4 \, x {\left (\log \relax (5) - 3\right )} + 4 \, x \log \left (x + 3\right ) - 4 \, x \log \relax (x)}{3 \, x^{2} + x + \log \relax (5) + \log \left (x + 3\right ) - \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x+12)*log(5*x+15)^2+((-8*x-24)*log(x)+27*x^3+99*x^2+30*x-72)*log(5*x+15)+(4*x+12)*log(x)^2+(-27*
x^3-99*x^2-30*x+72)*log(x)+39*x^5+143*x^4+6*x^3-267*x^2-111*x+108)/((3+x)*log(5*x+15)^2+((-2*x-6)*log(x)+6*x^3
+20*x^2-18)*log(5*x+15)+(3+x)*log(x)^2+(-6*x^3-20*x^2+18)*log(x)+9*x^5+33*x^4+x^3-57*x^2-9*x+27),x, algorithm=
"maxima")

[Out]

(13*x^3 + 9*x^2 + 4*x*(log(5) - 3) + 4*x*log(x + 3) - 4*x*log(x))/(3*x^2 + x + log(5) + log(x + 3) - log(x) -
3)

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mupad [B]  time = 3.85, size = 66, normalized size = 2.28 \begin {gather*} \frac {19\,\ln \relax (x)-19\,\ln \left (5\,x+15\right )-163\,x+48\,x\,\ln \left (5\,x+15\right )-48\,x\,\ln \relax (x)+51\,x^2+156\,x^3+57}{12\,\left (x+\ln \left (5\,x+15\right )-\ln \relax (x)+3\,x^2-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(5*x + 15)*(30*x - log(x)*(8*x + 24) + 99*x^2 + 27*x^3 - 72) - 111*x + log(5*x + 15)^2*(4*x + 12) - 26
7*x^2 + 6*x^3 + 143*x^4 + 39*x^5 + log(x)^2*(4*x + 12) - log(x)*(30*x + 99*x^2 + 27*x^3 - 72) + 108)/(log(5*x
+ 15)^2*(x + 3) - log(x)*(20*x^2 + 6*x^3 - 18) - 9*x - log(5*x + 15)*(log(x)*(2*x + 6) - 20*x^2 - 6*x^3 + 18)
+ log(x)^2*(x + 3) - 57*x^2 + x^3 + 33*x^4 + 9*x^5 + 27),x)

[Out]

(19*log(x) - 19*log(5*x + 15) - 163*x + 48*x*log(5*x + 15) - 48*x*log(x) + 51*x^2 + 156*x^3 + 57)/(12*(x + log
(5*x + 15) - log(x) + 3*x^2 - 3))

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sympy [A]  time = 0.37, size = 29, normalized size = 1.00 \begin {gather*} 4 x + \frac {x^{3} + 5 x^{2}}{3 x^{2} + x - \log {\relax (x )} + \log {\left (5 x + 15 \right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x+12)*ln(5*x+15)**2+((-8*x-24)*ln(x)+27*x**3+99*x**2+30*x-72)*ln(5*x+15)+(4*x+12)*ln(x)**2+(-27*
x**3-99*x**2-30*x+72)*ln(x)+39*x**5+143*x**4+6*x**3-267*x**2-111*x+108)/((3+x)*ln(5*x+15)**2+((-2*x-6)*ln(x)+6
*x**3+20*x**2-18)*ln(5*x+15)+(3+x)*ln(x)**2+(-6*x**3-20*x**2+18)*ln(x)+9*x**5+33*x**4+x**3-57*x**2-9*x+27),x)

[Out]

4*x + (x**3 + 5*x**2)/(3*x**2 + x - log(x) + log(5*x + 15) - 3)

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