3.24.94 \(\int \frac {1953125+4 x-781250 x^2-2 x^3}{390625 x+x^2} \, dx\) [2394]

Optimal. Leaf size=19 \[ 2-x^2+5 (5+\log (x))-\log (390625+x) \]

[Out]

27+5*ln(x)-x^2-ln(x+390625)

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Rubi [A]
time = 0.02, antiderivative size = 16, normalized size of antiderivative = 0.84, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1607, 1634} \begin {gather*} -x^2+5 \log (x)-\log (x+390625) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1953125 + 4*x - 781250*x^2 - 2*x^3)/(390625*x + x^2),x]

[Out]

-x^2 + 5*Log[x] - Log[390625 + x]

Rule 1607

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 1634

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 {1953125+4 x-781250 x^2-2 x^3}{x (390625+x)} \, dx\\ &=\int \left (\frac {1}{-390625-x}+\frac {5}{x}-2 x\right ) \, dx\\ &=-x^2+5 \log (x)-\log (390625+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.84 \begin {gather*} -x^2+5 \log (x)-\log (390625+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1953125 + 4*x - 781250*x^2 - 2*x^3)/(390625*x + x^2),x]

[Out]

-x^2 + 5*Log[x] - Log[390625 + x]

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Maple [A]
time = 0.24, size = 17, normalized size = 0.89

method	result	size
default	\(-x^{2}-\ln \left (x +390625\right )+5 \ln \left (x \right )\)	\(17\)
norman	\(-x^{2}-\ln \left (x +390625\right )+5 \ln \left (x \right )\)	\(17\)
risch	\(-x^{2}-\ln \left (x +390625\right )+5 \ln \left (x \right )\)	\(17\)
meijerg	\(-\ln \left (1+\frac {x}{390625}\right )+5 \ln \left (x \right )-40 \ln \left (5\right )+\frac {390625 x \left (-\frac {3 x}{390625}+6\right )}{3}-781250 x\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^3-781250*x^2+4*x+1953125)/(x^2+390625*x),x,method=_RETURNVERBOSE)

[Out]

-x^2-ln(x+390625)+5*ln(x)

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Maxima [A]
time = 0.26, size = 16, normalized size = 0.84 \begin {gather*} -x^{2} - \log \left (x + 390625\right ) + 5 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3-781250*x^2+4*x+1953125)/(x^2+390625*x),x, algorithm="maxima")

[Out]

-x^2 - log(x + 390625) + 5*log(x)

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Fricas [A]
time = 0.35, size = 16, normalized size = 0.84 \begin {gather*} -x^{2} - \log \left (x + 390625\right ) + 5 \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3-781250*x^2+4*x+1953125)/(x^2+390625*x),x, algorithm="fricas")

[Out]

-x^2 - log(x + 390625) + 5*log(x)

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Sympy [A]
time = 0.03, size = 12, normalized size = 0.63 \begin {gather*} - x^{2} + 5 \log {\left (x \right )} - \log {\left (x + 390625 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**3-781250*x**2+4*x+1953125)/(x**2+390625*x),x)

[Out]

-x**2 + 5*log(x) - log(x + 390625)

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Giac [A]
time = 0.40, size = 18, normalized size = 0.95 \begin {gather*} -x^{2} - \log \left ({\left | x + 390625 \right |}\right ) + 5 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3-781250*x^2+4*x+1953125)/(x^2+390625*x),x, algorithm="giac")

[Out]

-x^2 - log(abs(x + 390625)) + 5*log(abs(x))

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Mupad [B]
time = 0.05, size = 16, normalized size = 0.84 \begin {gather*} 5\,\ln \left (x\right )-\ln \left (x+390625\right )-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - 781250*x^2 - 2*x^3 + 1953125)/(390625*x + x^2),x)

[Out]

5*log(x) - log(x + 390625) - x^2

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