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

Optimal. Leaf size=47 \[ -\frac{71 (1-4 x)}{529 \sqrt{2 x^2-x+3}}-\frac{11 (3 x+5)}{69 \left (2 x^2-x+3\right )^{3/2}} \]

[Out]

(-11*(5 + 3*x))/(69*(3 - x + 2*x^2)^(3/2)) - (71*(1 - 4*x))/(529*Sqrt[3 - x + 2*x^2])

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Rubi [A]  time = 0.0223468, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1660, 12, 613} \[ -\frac{71 (1-4 x)}{529 \sqrt{2 x^2-x+3}}-\frac{11 (3 x+5)}{69 \left (2 x^2-x+3\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*(5 + 3*x))/(69*(3 - x + 2*x^2)^(3/2)) - (71*(1 - 4*x))/(529*Sqrt[3 - x + 2*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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 613

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

Rubi steps

\begin{align*} \int \frac{2+3 x+5 x^2}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=-\frac{11 (5+3 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{213}{4 \left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac{11 (5+3 x)}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac{71}{46} \int \frac{1}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac{11 (5+3 x)}{69 \left (3-x+2 x^2\right )^{3/2}}-\frac{71 (1-4 x)}{529 \sqrt{3-x+2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.104221, size = 33, normalized size = 0.7 \[ \frac{2 \left (852 x^3-639 x^2+1005 x-952\right )}{1587 \left (2 x^2-x+3\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-952 + 1005*x - 639*x^2 + 852*x^3))/(1587*(3 - x + 2*x^2)^(3/2))

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Maple [A]  time = 0.045, size = 30, normalized size = 0.6 \begin{align*}{\frac{1704\,{x}^{3}-1278\,{x}^{2}+2010\,x-1904}{1587} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1587/(2*x^2-x+3)^(3/2)*(852*x^3-639*x^2+1005*x-952)

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Maxima [A]  time = 1.07712, size = 80, normalized size = 1.7 \begin{align*} \frac{284 \, x}{529 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{71}{529 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{11 \, x}{23 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{55}{69 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

284/529*x/sqrt(2*x^2 - x + 3) - 71/529/sqrt(2*x^2 - x + 3) - 11/23*x/(2*x^2 - x + 3)^(3/2) - 55/69/(2*x^2 - x
+ 3)^(3/2)

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Fricas [A]  time = 1.34092, size = 132, normalized size = 2.81 \begin{align*} \frac{2 \,{\left (852 \, x^{3} - 639 \, x^{2} + 1005 \, x - 952\right )} \sqrt{2 \, x^{2} - x + 3}}{1587 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1587*(852*x^3 - 639*x^2 + 1005*x - 952)*sqrt(2*x^2 - x + 3)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 x^{2} + 3 x + 2}{\left (2 x^{2} - x + 3\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15732, size = 39, normalized size = 0.83 \begin{align*} \frac{2 \,{\left (3 \,{\left (71 \,{\left (4 \, x - 3\right )} x + 335\right )} x - 952\right )}}{1587 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1587*(3*(71*(4*x - 3)*x + 335)*x - 952)/(2*x^2 - x + 3)^(3/2)