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

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

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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 {gather*} \int \frac {-104+9 x+1803 x^2+\left (-4+147 x^2\right ) \log (x)+3 x^2 \log ^2(x)}{1875 x^2+150 x^2 \log (x)+3 x^2 \log ^2(x)} \, dx \end {gather*}

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 {gather*} \begin {aligned} \text {integral} &=\int \frac {-104+9 x+1803 x^2+\left (-4+147 x^2\right ) \log (x)+3 x^2 \log ^2(x)}{3 x^2 (25+\log (x))^2} \, dx\\ &=\frac {1}{3} \int \frac {-104+9 x+1803 x^2+\left (-4+147 x^2\right ) \log (x)+3 x^2 \log ^2(x)}{x^2 (25+\log (x))^2} \, dx\\ &=\frac {1}{3} \int \left (3+\frac {-4+9 x+3 x^2}{x^2 (25+\log (x))^2}+\frac {-4-3 x^2}{x^2 (25+\log (x))}\right ) \, dx\\ &=x+\frac {1}{3} \int \frac {-4+9 x+3 x^2}{x^2 (25+\log (x))^2} \, dx+\frac {1}{3} \int \frac {-4-3 x^2}{x^2 (25+\log (x))} \, dx\\ &=x+\frac {1}{3} \int \frac {-4+9 x+3 x^2}{x^2 (25+\log (x))^2} \, dx+\frac {1}{3} \int \left (-\frac {3}{25+\log (x)}-\frac {4}{x^2 (25+\log (x))}\right ) \, dx\\ &=x+\frac {1}{3} \int \frac {-4+9 x+3 x^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+9 x+3 x^2}{x^2 (25+\log (x))^2} \, dx-\frac {4}{3} \operatorname {Subst}\left (\int \frac {e^{-x}}{25+x} \, dx,x,\log (x)\right )-\operatorname {Subst}\left (\int \frac {e^x}{25+x} \, dx,x,\log (x)\right )\\ &=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+9 x+3 x^2}{x^2 (25+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}

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

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 {gather*} \frac {3 \, x^{2} \log \relax (x) + 72 \, x^{2} - 9 \, x + 4}{3 \, {\left (x \log \relax (x) + 25 \, x\right )}} \end {gather*}

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 {gather*} x - \frac {3 \, x^{2} + 9 \, x - 4}{3 \, {\left (x \log \relax (x) + 25 \, x\right )}} \end {gather*}

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	\(x -\frac {3 x^{2}+9 x -4}{3 x \left (\ln \relax (x )+25\right )}\)	\(24\)
norman	\(\frac {\frac {4}{3}+x^{2} \ln \relax (x )-3 x +24 x^{2}}{\left (\ln \relax (x )+25\right ) x}\)	\(27\)

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 {gather*} \frac {3 \, x^{2} \log \relax (x) + 72 \, x^{2} - 9 \, x + 4}{3 \, {\left (x \log \relax (x) + 25 \, x\right )}} \end {gather*}

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 {gather*} \frac {x^2+\frac {3\,x}{25}}{x}-\frac {x^2+3\,x-\frac {4}{3}}{x\,\left (\ln \relax (x)+25\right )} \end {gather*}

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 {gather*} x + \frac {- 3 x^{2} - 9 x + 4}{3 x \log {\relax (x )} + 75 x} \end {gather*}

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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