3.1.84 \(\int \frac {(2+x) (d+e x+f x^2+g x^3+h x^4+i x^5)}{4-5 x^2+x^4} \, dx\)

Optimal. Leaf size=96 \[ -\frac {1}{2} \log (1-x) (d+e+f+g+h+i)+\frac {1}{3} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac {1}{6} \log (x+1) (d-e+f-g+h-i)+x (g+2 h+5 i)+\frac {1}{2} x^2 (h+2 i)+\frac {i x^3}{3} \]

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Rubi [A]  time = 0.14, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {1586, 2074} \begin {gather*} -\frac {1}{2} \log (1-x) (d+e+f+g+h+i)+\frac {1}{3} \log (2-x) (d+2 e+4 f+8 g+16 h+32 i)+\frac {1}{6} \log (x+1) (d-e+f-g+h-i)+x (g+2 h+5 i)+\frac {1}{2} x^2 (h+2 i)+\frac {i x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + x)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4),x]

[Out]

(g + 2*h + 5*i)*x + ((h + 2*i)*x^2)/2 + (i*x^3)/3 - ((d + e + f + g + h + i)*Log[1 - x])/2 + ((d + 2*e + 4*f +
8*g + 16*h + 32*i)*Log[2 - x])/3 + ((d - e + f - g + h - i)*Log[1 + x])/6

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {align*} \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+84 x^5\right )}{4-5 x^2+x^4} \, dx &=\int \frac {d+e x+f x^2+g x^3+h x^4+84 x^5}{2-x-2 x^2+x^3} \, dx\\ &=\int \left (420 \left (1+\frac {1}{420} (g+2 h)\right )+\frac {2688+d+2 e+4 f+8 g+16 h}{3 (-2+x)}+\frac {-84-d-e-f-g-h}{2 (-1+x)}+(168+h) x+84 x^2+\frac {-84+d-e+f-g+h}{6 (1+x)}\right ) \, dx\\ &=(420+g+2 h) x+\frac {1}{2} (168+h) x^2+28 x^3-\frac {1}{2} (84+d+e+f+g+h) \log (1-x)+\frac {1}{3} (2688+d+2 e+4 f+8 g+16 h) \log (2-x)-\frac {1}{6} (84-d+e-f+g-h) \log (1+x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 91, normalized size = 0.95 \begin {gather*} \frac {1}{6} \left (-3 \log (1-x) (d+e+f+g+h+i)+2 \log (2-x) (d+2 e+4 (f+2 g+4 h+8 i))+\log (x+1) (d-e+f-g+h-i)+6 x (g+2 h+5 i)+3 x^2 (h+2 i)+2 i x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + x)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4),x]

[Out]

(6*(g + 2*h + 5*i)*x + 3*(h + 2*i)*x^2 + 2*i*x^3 - 3*(d + e + f + g + h + i)*Log[1 - x] + 2*(d + 2*e + 4*(f +
2*g + 4*h + 8*i))*Log[2 - x] + (d - e + f - g + h - i)*Log[1 + x])/6

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+x) \left (d+e x+f x^2+g x^3+h x^4+i x^5\right )}{4-5 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((2 + x)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4),x]

[Out]

IntegrateAlgebraic[((2 + x)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(4 - 5*x^2 + x^4), x]

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fricas [A]  time = 1.21, size = 82, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, i x^{3} + \frac {1}{2} \, {\left (h + 2 \, i\right )} x^{2} + {\left (g + 2 \, h + 5 \, i\right )} x + \frac {1}{6} \, {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac {1}{2} \, {\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)*(i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x, algorithm="fricas")

[Out]

1/3*i*x^3 + 1/2*(h + 2*i)*x^2 + (g + 2*h + 5*i)*x + 1/6*(d - e + f - g + h - i)*log(x + 1) - 1/2*(d + e + f +
g + h + i)*log(x - 1) + 1/3*(d + 2*e + 4*f + 8*g + 16*h + 32*i)*log(x - 2)

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giac [A]  time = 0.24, size = 90, normalized size = 0.94 \begin {gather*} \frac {1}{3} \, i x^{3} + \frac {1}{2} \, h x^{2} + i x^{2} + g x + 2 \, h x + 5 \, i x + \frac {1}{6} \, {\left (d + f - g + h - i - e\right )} \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, {\left (d + f + g + h + i + e\right )} \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{3} \, {\left (d + 4 \, f + 8 \, g + 16 \, h + 32 \, i + 2 \, e\right )} \log \left ({\left | x - 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)*(i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x, algorithm="giac")

[Out]

1/3*i*x^3 + 1/2*h*x^2 + i*x^2 + g*x + 2*h*x + 5*i*x + 1/6*(d + f - g + h - i - e)*log(abs(x + 1)) - 1/2*(d + f
+ g + h + i + e)*log(abs(x - 1)) + 1/3*(d + 4*f + 8*g + 16*h + 32*i + 2*e)*log(abs(x - 2))

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maple [A]  time = 0.01, size = 156, normalized size = 1.62 \begin {gather*} \frac {i \,x^{3}}{3}+\frac {h \,x^{2}}{2}+i \,x^{2}+\frac {d \ln \left (x -2\right )}{3}-\frac {d \ln \left (x -1\right )}{2}+\frac {d \ln \left (x +1\right )}{6}+\frac {2 e \ln \left (x -2\right )}{3}-\frac {e \ln \left (x -1\right )}{2}-\frac {e \ln \left (x +1\right )}{6}+\frac {4 f \ln \left (x -2\right )}{3}-\frac {f \ln \left (x -1\right )}{2}+\frac {f \ln \left (x +1\right )}{6}+g x +\frac {8 g \ln \left (x -2\right )}{3}-\frac {g \ln \left (x -1\right )}{2}-\frac {g \ln \left (x +1\right )}{6}+2 h x +\frac {16 h \ln \left (x -2\right )}{3}-\frac {h \ln \left (x -1\right )}{2}+\frac {h \ln \left (x +1\right )}{6}+5 i x +\frac {32 i \ln \left (x -2\right )}{3}-\frac {i \ln \left (x -1\right )}{2}-\frac {i \ln \left (x +1\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+2)*(i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x)

[Out]

1/3*d*ln(x-2)+2/3*e*ln(x-2)+4/3*f*ln(x-2)+8/3*g*ln(x-2)+16/3*h*ln(x-2)+32/3*i*ln(x-2)+1/3*i*x^3+1/2*h*x^2+i*x^
2+g*x+2*h*x+5*i*x+1/6*d*ln(x+1)-1/6*e*ln(x+1)+1/6*f*ln(x+1)-1/6*g*ln(x+1)+1/6*h*ln(x+1)-1/6*i*ln(x+1)-1/2*d*ln
(x-1)-1/2*e*ln(x-1)-1/2*f*ln(x-1)-1/2*g*ln(x-1)-1/2*h*ln(x-1)-1/2*i*ln(x-1)

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maxima [A]  time = 0.45, size = 82, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, i x^{3} + \frac {1}{2} \, {\left (h + 2 \, i\right )} x^{2} + {\left (g + 2 \, h + 5 \, i\right )} x + \frac {1}{6} \, {\left (d - e + f - g + h - i\right )} \log \left (x + 1\right ) - \frac {1}{2} \, {\left (d + e + f + g + h + i\right )} \log \left (x - 1\right ) + \frac {1}{3} \, {\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h + 32 \, i\right )} \log \left (x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)*(i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x, algorithm="maxima")

[Out]

1/3*i*x^3 + 1/2*(h + 2*i)*x^2 + (g + 2*h + 5*i)*x + 1/6*(d - e + f - g + h - i)*log(x + 1) - 1/2*(d + e + f +
g + h + i)*log(x - 1) + 1/3*(d + 2*e + 4*f + 8*g + 16*h + 32*i)*log(x - 2)

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mupad [B]  time = 0.88, size = 99, normalized size = 1.03 \begin {gather*} x\,\left (g+2\,h+5\,i\right )+\frac {i\,x^3}{3}-\ln \left (x-1\right )\,\left (\frac {d}{2}+\frac {e}{2}+\frac {f}{2}+\frac {g}{2}+\frac {h}{2}+\frac {i}{2}\right )+\ln \left (x+1\right )\,\left (\frac {d}{6}-\frac {e}{6}+\frac {f}{6}-\frac {g}{6}+\frac {h}{6}-\frac {i}{6}\right )+\ln \left (x-2\right )\,\left (\frac {d}{3}+\frac {2\,e}{3}+\frac {4\,f}{3}+\frac {8\,g}{3}+\frac {16\,h}{3}+\frac {32\,i}{3}\right )+x^2\,\left (\frac {h}{2}+i\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 2)*(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5))/(x^4 - 5*x^2 + 4),x)

[Out]

x*(g + 2*h + 5*i) + (i*x^3)/3 - log(x - 1)*(d/2 + e/2 + f/2 + g/2 + h/2 + i/2) + log(x + 1)*(d/6 - e/6 + f/6 -
g/6 + h/6 - i/6) + log(x - 2)*(d/3 + (2*e)/3 + (4*f)/3 + (8*g)/3 + (16*h)/3 + (32*i)/3) + x^2*(h/2 + i)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)*(i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4),x)

[Out]

Timed out

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