### 3.258 $$\int \frac{1+x^5}{-10 x-3 x^2+x^3} \, dx$$

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

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Rubi [A]  time = 0.0426682, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, 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*}

Mathematica [A]  time = 0.0067155, size = 42, normalized size = 1. $\frac{x^3}{3}+\frac{3 x^2}{2}+19 x+\frac{3126}{35} \log (5-x)-\frac{\log (x)}{10}-\frac{31}{14} \log (x+2)$

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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Maple [A]  time = 0.007, size = 31, normalized size = 0.7 \begin{align*}{\frac{{x}^{3}}{3}}+{\frac{3\,{x}^{2}}{2}}+19\,x-{\frac{31\,\ln \left ( 2+x \right ) }{14}}-{\frac{\ln \left ( x \right ) }{10}}+{\frac{3126\,\ln \left ( -5+x \right ) }{35}} \end{align*}

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(2+x)-1/10*ln(x)+3126/35*ln(-5+x)

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Maxima [A]  time = 0.970599, size = 41, normalized size = 0.98 \begin{align*} \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 \left (x\right ) \end{align*}

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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Fricas [A]  time = 1.70858, size = 108, normalized size = 2.57 \begin{align*} \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 \left (x\right ) \end{align*}

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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Sympy [A]  time = 0.129621, size = 36, normalized size = 0.86 \begin{align*} \frac{x^{3}}{3} + \frac{3 x^{2}}{2} + 19 x - \frac{\log{\left (x \right )}}{10} + \frac{3126 \log{\left (x - 5 \right )}}{35} - \frac{31 \log{\left (x + 2 \right )}}{14} \end{align*}

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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Giac [A]  time = 1.06358, size = 45, normalized size = 1.07 \begin{align*} \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 ) \end{align*}

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))