3.375 \(\int \frac {1+x^5}{-10 x-3 x^2+x^3} \, dx\)

Optimal. Leaf size=42 \[ \frac {x^3}{3}+\frac {3 x^2}{2}+19 x+\frac {3126}{35} \log (5-x)-\frac {\log (x)}{10}-\frac {31}{14} \log (x+2) \]

[Out]

19*x+3/2*x^2+1/3*x^3+3126/35*ln(5-x)-1/10*ln(x)-31/14*ln(2+x)

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Rubi [A]  time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1594, 1628} \[ \frac {x^3}{3}+\frac {3 x^2}{2}+19 x+\frac {3126}{35} \log (5-x)-\frac {\log (x)}{10}-\frac {31}{14} \log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x^5)/(-10*x - 3*x^2 + x^3),x]

[Out]

19*x + (3*x^2)/2 + x^3/3 + (3126*Log[5 - x])/35 - Log[x]/10 - (31*Log[2 + x])/14

Rule 1594

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

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin {align*} \int \frac {1+x^5}{-10 x-3 x^2+x^3} \, dx &=\int \frac {1+x^5}{x \left (-10-3 x+x^2\right )} \, dx\\ &=\int \left (19+\frac {3126}{35 (-5+x)}-\frac {1}{10 x}+3 x+x^2-\frac {31}{14 (2+x)}\right ) \, dx\\ &=19 x+\frac {3 x^2}{2}+\frac {x^3}{3}+\frac {3126}{35} \log (5-x)-\frac {\log (x)}{10}-\frac {31}{14} \log (2+x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 42, normalized size = 1.00 \[ \frac {x^3}{3}+\frac {3 x^2}{2}+19 x+\frac {3126}{35} \log (5-x)-\frac {\log (x)}{10}-\frac {31}{14} \log (x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^5)/(-10*x - 3*x^2 + x^3),x]

[Out]

19*x + (3*x^2)/2 + x^3/3 + (3126*Log[5 - x])/35 - Log[x]/10 - (31*Log[2 + x])/14

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fricas [A]  time = 0.59, size = 30, normalized size = 0.71 \[ \frac {1}{3} \, x^{3} + \frac {3}{2} \, x^{2} + 19 \, x - \frac {31}{14} \, \log \left (x + 2\right ) + \frac {3126}{35} \, \log \left (x - 5\right ) - \frac {1}{10} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)/(x^3-3*x^2-10*x),x, algorithm="fricas")

[Out]

1/3*x^3 + 3/2*x^2 + 19*x - 31/14*log(x + 2) + 3126/35*log(x - 5) - 1/10*log(x)

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giac [A]  time = 0.29, size = 33, normalized size = 0.79 \[ \frac {1}{3} \, x^{3} + \frac {3}{2} \, x^{2} + 19 \, x - \frac {31}{14} \, \log \left ({\left | x + 2 \right |}\right ) + \frac {3126}{35} \, \log \left ({\left | x - 5 \right |}\right ) - \frac {1}{10} \, \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)/(x^3-3*x^2-10*x),x, algorithm="giac")

[Out]

1/3*x^3 + 3/2*x^2 + 19*x - 31/14*log(abs(x + 2)) + 3126/35*log(abs(x - 5)) - 1/10*log(abs(x))

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maple [A]  time = 0.01, size = 31, normalized size = 0.74 \[ \frac {x^{3}}{3}+\frac {3 x^{2}}{2}+19 x -\frac {\ln \relax (x )}{10}+\frac {3126 \ln \left (x -5\right )}{35}-\frac {31 \ln \left (x +2\right )}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+1)/(x^3-3*x^2-10*x),x)

[Out]

1/3*x^3+3/2*x^2+19*x-31/14*ln(x+2)-1/10*ln(x)+3126/35*ln(x-5)

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maxima [A]  time = 0.88, size = 30, normalized size = 0.71 \[ \frac {1}{3} \, x^{3} + \frac {3}{2} \, x^{2} + 19 \, x - \frac {31}{14} \, \log \left (x + 2\right ) + \frac {3126}{35} \, \log \left (x - 5\right ) - \frac {1}{10} \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+1)/(x^3-3*x^2-10*x),x, algorithm="maxima")

[Out]

1/3*x^3 + 3/2*x^2 + 19*x - 31/14*log(x + 2) + 3126/35*log(x - 5) - 1/10*log(x)

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mupad [B]  time = 0.05, size = 30, normalized size = 0.71 \[ 19\,x-\frac {31\,\ln \left (x+2\right )}{14}+\frac {3126\,\ln \left (x-5\right )}{35}-\frac {\ln \relax (x)}{10}+\frac {3\,x^2}{2}+\frac {x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^5 + 1)/(10*x + 3*x^2 - x^3),x)

[Out]

19*x - (31*log(x + 2))/14 + (3126*log(x - 5))/35 - log(x)/10 + (3*x^2)/2 + x^3/3

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sympy [A]  time = 0.15, size = 36, normalized size = 0.86 \[ \frac {x^{3}}{3} + \frac {3 x^{2}}{2} + 19 x - \frac {\log {\relax (x )}}{10} + \frac {3126 \log {\left (x - 5 \right )}}{35} - \frac {31 \log {\left (x + 2 \right )}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5+1)/(x**3-3*x**2-10*x),x)

[Out]

x**3/3 + 3*x**2/2 + 19*x - log(x)/10 + 3126*log(x - 5)/35 - 31*log(x + 2)/14

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