3.41.69 $\int \frac{-104+9x+1803x^2+(-4+147x^2)\log (x)+3x^2{\log }^2(x)}{1875x^2+150x^2\log (x)+3x^2{\log }^2(x)}dx$

Optimal. Leaf size=20 $$x-\frac{3-\frac{4}{3x}+x}{25+\log (x)}$$

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Rubi [F]  time = 0.40, 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 = {} $$\begin{array}{c}\int \frac{-104+9x+1803x^2+(-4+147x^2)\log (x)+3x^2{\log }^2(x)}{1875x^2+150x^2\log (x)+3x^2{\log }^2(x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(-104 + 9*x + 1803*x^2 + (-4 + 147*x^2)*Log[x] + 3*x^2*Log[x]^2)/(1875*x^2 + 150*x^2*Log[x] + 3*x^2*Log[x]
^2),x]

[Out]

x - (4*E^25*ExpIntegralEi[-25 - Log[x]])/3 - ExpIntegralEi[25 + Log[x]]/E^25 + Defer[Int][(-4 + 9*x + 3*x^2)/(
x^2*(25 + Log[x])^2), x]/3

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-104+9x+1803x^2+(-4+147x^2)\log (x)+3x^2{\log }^2(x)}{3x^2(25+\log (x){)}^2}dx\\ & =\frac{1}{3}\int \frac{-104+9x+1803x^2+(-4+147x^2)\log (x)+3x^2{\log }^2(x)}{x^2(25+\log (x){)}^2}dx\\ & =\frac{1}{3}\int (3+\frac{-4+9x+3x^2}{x^2(25+\log (x){)}^2}+\frac{-4-3x^2}{x^2(25+\log (x))})dx\\ & =x+\frac{1}{3}\int \frac{-4+9x+3x^2}{x^2(25+\log (x){)}^2}dx+\frac{1}{3}\int \frac{-4-3x^2}{x^2(25+\log (x))}dx\\ & =x+\frac{1}{3}\int \frac{-4+9x+3x^2}{x^2(25+\log (x){)}^2}dx+\frac{1}{3}\int (-\frac{3}{25+\log (x)}-\frac{4}{x^2(25+\log (x))})dx\\ & =x+\frac{1}{3}\int \frac{-4+9x+3x^2}{x^2(25+\log (x){)}^2}dx-\frac{4}{3}\int \frac{1}{x^2(25+\log (x))}dx-\int \frac{1}{25+\log (x)}dx\\ & =x+\frac{1}{3}\int \frac{-4+9x+3x^2}{x^2(25+\log (x){)}^2}dx-\frac{4}{3}\mathrm{Subst}(\int \frac{e^{-x}}{25+x}dx,x,\log (x))-\mathrm{Subst}(\int \frac{e^x}{25+x}dx,x,\log (x))\\ & =x-\frac{4}{3}{e}^{25}\text{Ei}(-25-\log (x))-\frac{\text{Ei}(25+\log (x))}{e^{25}}+\frac{1}{3}\int \frac{-4+9x+3x^2}{x^2(25+\log (x){)}^2}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.11, size = 28, normalized size = 1.40 $$\begin{array}{c}\frac{1}{3}(3x+\frac{4-9x-3x^2}{x(25+\log (x))})\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-104 + 9*x + 1803*x^2 + (-4 + 147*x^2)*Log[x] + 3*x^2*Log[x]^2)/(1875*x^2 + 150*x^2*Log[x] + 3*x^2*
Log[x]^2),x]

[Out]

(3*x + (4 - 9*x - 3*x^2)/(x*(25 + Log[x])))/3

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fricas [A]  time = 1.00, size = 29, normalized size = 1.45 $$\begin{array}{c}\frac{3x^2\log (x)+72x^2-9x+4}{3(x\log (x)+25x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(x)^2+(147*x^2-4)*log(x)+1803*x^2+9*x-104)/(3*x^2*log(x)^2+150*x^2*log(x)+1875*x^2),x, alg
orithm="fricas")

[Out]

1/3*(3*x^2*log(x) + 72*x^2 - 9*x + 4)/(x*log(x) + 25*x)

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giac [A]  time = 0.23, size = 24, normalized size = 1.20 $$\begin{array}{c}x-\frac{3x^2+9x-4}{3(x\log (x)+25x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(x)^2+(147*x^2-4)*log(x)+1803*x^2+9*x-104)/(3*x^2*log(x)^2+150*x^2*log(x)+1875*x^2),x, alg
orithm="giac")

[Out]

x - 1/3*(3*x^2 + 9*x - 4)/(x*log(x) + 25*x)

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maple [A]  time = 0.02, size = 24, normalized size = 1.20

method	result	size
risch	$x-\frac{3x^2+9x-4}{3x(\ln (x)+25)}$	$24$
norman	$\frac{\frac{4}{3}+x^2\ln (x)-3x+24x^2}{(\ln (x)+25)x}$	$27$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2*ln(x)^2+(147*x^2-4)*ln(x)+1803*x^2+9*x-104)/(3*x^2*ln(x)^2+150*x^2*ln(x)+1875*x^2),x,method=_RETURN
VERBOSE)

[Out]

x-1/3/x*(3*x^2+9*x-4)/(ln(x)+25)

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maxima [A]  time = 0.38, size = 29, normalized size = 1.45 $$\begin{array}{c}\frac{3x^2\log (x)+72x^2-9x+4}{3(x\log (x)+25x)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2*log(x)^2+(147*x^2-4)*log(x)+1803*x^2+9*x-104)/(3*x^2*log(x)^2+150*x^2*log(x)+1875*x^2),x, alg
orithm="maxima")

[Out]

1/3*(3*x^2*log(x) + 72*x^2 - 9*x + 4)/(x*log(x) + 25*x)

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mupad [B]  time = 3.20, size = 31, normalized size = 1.55 $$\begin{array}{c}\frac{x^2+\frac{3x}{25}}{x}-\frac{x^2+3x-\frac{4}{3}}{x(\ln (x)+25)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x + 3*x^2*log(x)^2 + 1803*x^2 + log(x)*(147*x^2 - 4) - 104)/(150*x^2*log(x) + 3*x^2*log(x)^2 + 1875*x^2
),x)

[Out]

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

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sympy [A]  time = 0.11, size = 20, normalized size = 1.00 $$\begin{array}{c}x+\frac{-3x^2-9x+4}{3x\log (x)+75x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2*ln(x)**2+(147*x**2-4)*ln(x)+1803*x**2+9*x-104)/(3*x**2*ln(x)**2+150*x**2*ln(x)+1875*x**2),x)

[Out]

x + (-3*x**2 - 9*x + 4)/(3*x*log(x) + 75*x)

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