3.97.72 \(\int \frac {75-53 x+148 x^2+6 x^3}{75 x+3 x^2} \, dx\)

Optimal. Leaf size=36 \[ \frac {-4 x+x \left (\frac {2 (5-x)}{3}+x^2\right )}{x}-\log \left (\frac {(25+x)^2}{x}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 17, normalized size of antiderivative = 0.47, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1593, 1620} \begin {gather*} x^2-\frac {2 x}{3}+\log (x)-2 \log (x+25) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(75 - 53*x + 148*x^2 + 6*x^3)/(75*x + 3*x^2),x]

[Out]

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

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {75-53 x+148 x^2+6 x^3}{x (75+3 x)} \, dx\\ &=\int \left (-\frac {2}{3}+\frac {1}{x}+2 x-\frac {2}{25+x}\right ) \, dx\\ &=-\frac {2 x}{3}+x^2+\log (x)-2 \log (25+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 0.47 \begin {gather*} -\frac {2 x}{3}+x^2+\log (x)-2 \log (25+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(75 - 53*x + 148*x^2 + 6*x^3)/(75*x + 3*x^2),x]

[Out]

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

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fricas [A]  time = 0.68, size = 15, normalized size = 0.42 \begin {gather*} x^{2} - \frac {2}{3} \, x - 2 \, \log \left (x + 25\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^3+148*x^2-53*x+75)/(3*x^2+75*x),x, algorithm="fricas")

[Out]

x^2 - 2/3*x - 2*log(x + 25) + log(x)

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giac [A]  time = 0.14, size = 17, normalized size = 0.47 \begin {gather*} x^{2} - \frac {2}{3} \, x - 2 \, \log \left ({\left | x + 25 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^3+148*x^2-53*x+75)/(3*x^2+75*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.09, size = 16, normalized size = 0.44

method	result	size
default	\(x^{2}-\frac {2 x}{3}+\ln \relax (x )-2 \ln \left (x +25\right )\)	\(16\)
norman	\(x^{2}-\frac {2 x}{3}+\ln \relax (x )-2 \ln \left (x +25\right )\)	\(16\)
risch	\(x^{2}-\frac {2 x}{3}+\ln \relax (x )-2 \ln \left (x +25\right )\)	\(16\)
meijerg	\(\ln \relax (x )-2 \ln \relax (5)-2 \ln \left (1+\frac {x}{25}\right )-\frac {25 x \left (-\frac {3 x}{25}+6\right )}{3}+\frac {148 x}{3}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x^3+148*x^2-53*x+75)/(3*x^2+75*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.48, size = 15, normalized size = 0.42 \begin {gather*} x^{2} - \frac {2}{3} \, x - 2 \, \log \left (x + 25\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x^3+148*x^2-53*x+75)/(3*x^2+75*x),x, algorithm="maxima")

[Out]

x^2 - 2/3*x - 2*log(x + 25) + log(x)

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mupad [B]  time = 7.48, size = 15, normalized size = 0.42 \begin {gather*} \ln \relax (x)-2\,\ln \left (x+25\right )-\frac {2\,x}{3}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((148*x^2 - 53*x + 6*x^3 + 75)/(75*x + 3*x^2),x)

[Out]

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

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sympy [A]  time = 0.10, size = 17, normalized size = 0.47 \begin {gather*} x^{2} - \frac {2 x}{3} + \log {\relax (x )} - 2 \log {\left (x + 25 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x**3+148*x**2-53*x+75)/(3*x**2+75*x),x)

[Out]

x**2 - 2*x/3 + log(x) - 2*log(x + 25)

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