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

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

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

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

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

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

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 $$\begin{array}{c}\frac{13x^3+9x^2+4x\log (5x+15)-4x\log (x)-12x}{3x^2+x+\log (5x+15)-\log (x)-3}\end{array}$$

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 $$\begin{array}{c}4x+\frac{x^3+5x^2}{3x^2+x+\log (5x+15)-\log (x)-3}\end{array}$$

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	$4x+\frac{x^2(5+x)}{\ln (5x+15)-3+3x^2-\ln (x)+x}$	$32$

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 $$\begin{array}{c}\frac{13x^3+9x^2+4x(\log (5)-3)+4x\log (x+3)-4x\log (x)}{3x^2+x+\log (5)+\log (x+3)-\log (x)-3}\end{array}$$

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 $$\begin{array}{c}\frac{19\ln (x)-19\ln (5x+15)-163x+48x\ln (5x+15)-48x\ln (x)+51x^2+156x^3+57}{12(x+\ln (5x+15)-\ln (x)+3x^2-3)}\end{array}$$

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 $$\begin{array}{c}4x+\frac{x^3+5x^2}{3x^2+x-\log (x)+\log (5x+15)-3}\end{array}$$

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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