3.254 \(\int \frac {(1+2 x) (1+3 x+4 x^2)}{(2-x+3 x^2)^{5/2}} \, dx\)

Optimal. Leaf size=47 \[ -\frac {4 (3889-4290 x)}{14283 \sqrt {3 x^2-x+2}}-\frac {2 (367 x+73)}{621 \left (3 x^2-x+2\right )^{3/2}} \]

[Out]

-2/621*(73+367*x)/(3*x^2-x+2)^(3/2)-4/14283*(3889-4290*x)/(3*x^2-x+2)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1660, 636} \[ -\frac {4 (3889-4290 x)}{14283 \sqrt {3 x^2-x+2}}-\frac {2 (367 x+73)}{621 \left (3 x^2-x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + 2*x)*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(5/2),x]

[Out]

(-2*(73 + 367*x))/(621*(2 - x + 3*x^2)^(3/2)) - (4*(3889 - 4290*x))/(14283*Sqrt[2 - x + 3*x^2])

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rubi steps

\begin {align*} \int \frac {(1+2 x) \left (1+3 x+4 x^2\right )}{\left (2-x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (73+367 x)}{621 \left (2-x+3 x^2\right )^{3/2}}+\frac {2}{69} \int \frac {\frac {577}{9}+92 x}{\left (2-x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (73+367 x)}{621 \left (2-x+3 x^2\right )^{3/2}}-\frac {4 (3889-4290 x)}{14283 \sqrt {2-x+3 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 33, normalized size = 0.70 \[ \frac {2 \left (2860 x^3-3546 x^2+1833 x-1915\right )}{1587 \left (3 x^2-x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + 2*x)*(1 + 3*x + 4*x^2))/(2 - x + 3*x^2)^(5/2),x]

[Out]

(2*(-1915 + 1833*x - 3546*x^2 + 2860*x^3))/(1587*(2 - x + 3*x^2)^(3/2))

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fricas [A]  time = 0.80, size = 51, normalized size = 1.09 \[ \frac {2 \, {\left (2860 \, x^{3} - 3546 \, x^{2} + 1833 \, x - 1915\right )} \sqrt {3 \, x^{2} - x + 2}}{1587 \, {\left (9 \, x^{4} - 6 \, x^{3} + 13 \, x^{2} - 4 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2-x+2)^(5/2),x, algorithm="fricas")

[Out]

2/1587*(2860*x^3 - 3546*x^2 + 1833*x - 1915)*sqrt(3*x^2 - x + 2)/(9*x^4 - 6*x^3 + 13*x^2 - 4*x + 4)

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giac [A]  time = 0.20, size = 28, normalized size = 0.60 \[ \frac {2 \, {\left ({\left (2 \, {\left (1430 \, x - 1773\right )} x + 1833\right )} x - 1915\right )}}{1587 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2-x+2)^(5/2),x, algorithm="giac")

[Out]

2/1587*((2*(1430*x - 1773)*x + 1833)*x - 1915)/(3*x^2 - x + 2)^(3/2)

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maple [A]  time = 0.00, size = 30, normalized size = 0.64 \[ \frac {\frac {5720}{1587} x^{3}-\frac {2364}{529} x^{2}+\frac {1222}{529} x -\frac {3830}{1587}}{\left (3 x^{2}-x +2\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+1)*(4*x^2+3*x+1)/(3*x^2-x+2)^(5/2),x)

[Out]

2/1587/(3*x^2-x+2)^(3/2)*(2860*x^3-3546*x^2+1833*x-1915)

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maxima [A]  time = 0.44, size = 76, normalized size = 1.62 \[ \frac {5720 \, x}{4761 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {8 \, x^{2}}{3 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {2860}{14283 \, \sqrt {3 \, x^{2} - x + 2}} - \frac {182 \, x}{621 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} - \frac {1250}{621 \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x^2+3*x+1)/(3*x^2-x+2)^(5/2),x, algorithm="maxima")

[Out]

5720/4761*x/sqrt(3*x^2 - x + 2) - 8/3*x^2/(3*x^2 - x + 2)^(3/2) - 2860/14283/sqrt(3*x^2 - x + 2) - 182/621*x/(
3*x^2 - x + 2)^(3/2) - 1250/621/(3*x^2 - x + 2)^(3/2)

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mupad [B]  time = 4.20, size = 49, normalized size = 1.04 \[ -\frac {442\,x-5720\,x\,\left (3\,x^2-x+2\right )+15556\,x^2+11490}{\sqrt {3\,x^2-x+2}\,\left (14283\,x^2-4761\,x+9522\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x + 1)*(3*x + 4*x^2 + 1))/(3*x^2 - x + 2)^(5/2),x)

[Out]

-(442*x - 5720*x*(3*x^2 - x + 2) + 15556*x^2 + 11490)/((3*x^2 - x + 2)^(1/2)*(14283*x^2 - 4761*x + 9522))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (2 x + 1\right ) \left (4 x^{2} + 3 x + 1\right )}{\left (3 x^{2} - x + 2\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)*(4*x**2+3*x+1)/(3*x**2-x+2)**(5/2),x)

[Out]

Integral((2*x + 1)*(4*x**2 + 3*x + 1)/(3*x**2 - x + 2)**(5/2), x)

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