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

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Rubi [A]  time = 0.0476253, antiderivative size = 47, normalized size of antiderivative = 1., 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 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
]

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]

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*}

Mathematica [A]  time = 0.0602662, size = 33, normalized size = 0.7 $\frac{2 \left (2860 x^3-3546 x^2+1833 x-1915\right )}{1587 \left (3 x^2-x+2\right )^{3/2}}$

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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Maple [A]  time = 0.046, size = 30, normalized size = 0.6 \begin{align*}{\frac{5720\,{x}^{3}-7092\,{x}^{2}+3666\,x-3830}{1587} \left ( 3\,{x}^{2}-x+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((1+2*x)*(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 = 1.00424, size = 103, normalized size = 2.19 \begin{align*} \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}}} \end{align*}

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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Fricas [A]  time = 1.00179, size = 136, normalized size = 2.89 \begin{align*} \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 )}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

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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Giac [A]  time = 1.18293, size = 38, normalized size = 0.81 \begin{align*} \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}}} \end{align*}

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)