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

Optimal. Leaf size=68 \[ \frac {121 (19-7 x)}{276 \left (2 x^2-x+3\right )^{3/2}}-\frac {11 (2336 x+7351)}{6348 \sqrt {2 x^2-x+3}}-\frac {25 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4 \sqrt {2}} \]

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Rubi [A]  time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1660, 12, 619, 215} \begin {gather*} \frac {121 (19-7 x)}{276 \left (2 x^2-x+3\right )^{3/2}}-\frac {11 (2336 x+7351)}{6348 \sqrt {2 x^2-x+3}}-\frac {25 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(121*(19 - 7*x))/(276*(3 - x + 2*x^2)^(3/2)) - (11*(7351 + 2336*x))/(6348*Sqrt[3 - x + 2*x^2]) - (25*ArcSinh[(
1 - 4*x)/Sqrt[23]])/(4*Sqrt[2])

Rule 12

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/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 {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=\frac {121 (19-7 x)}{276 \left (3-x+2 x^2\right )^{3/2}}+\frac {2}{69} \int \frac {\frac {131}{16}+\frac {5865 x}{8}+\frac {1725 x^2}{4}}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=\frac {121 (19-7 x)}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac {11 (7351+2336 x)}{6348 \sqrt {3-x+2 x^2}}+\frac {4 \int \frac {39675}{16 \sqrt {3-x+2 x^2}} \, dx}{1587}\\ &=\frac {121 (19-7 x)}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac {11 (7351+2336 x)}{6348 \sqrt {3-x+2 x^2}}+\frac {25}{4} \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {121 (19-7 x)}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac {11 (7351+2336 x)}{6348 \sqrt {3-x+2 x^2}}+\frac {25 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{4 \sqrt {46}}\\ &=\frac {121 (19-7 x)}{276 \left (3-x+2 x^2\right )^{3/2}}-\frac {11 (7351+2336 x)}{6348 \sqrt {3-x+2 x^2}}-\frac {25 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 55, normalized size = 0.81 \begin {gather*} \frac {25 \sinh ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{4 \sqrt {2}}-\frac {11 \left (2336 x^3+6183 x^2+714 x+8623\right )}{3174 \left (2 x^2-x+3\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*(8623 + 714*x + 6183*x^2 + 2336*x^3))/(3174*(3 - x + 2*x^2)^(3/2)) + (25*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(4
*Sqrt[2])

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IntegrateAlgebraic [A]  time = 0.74, size = 70, normalized size = 1.03 \begin {gather*} -\frac {25 \log \left (2 \sqrt {2} \sqrt {2 x^2-x+3}-4 x+1\right )}{4 \sqrt {2}}-\frac {11 \left (2336 x^3+6183 x^2+714 x+8623\right )}{3174 \left (2 x^2-x+3\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*(8623 + 714*x + 6183*x^2 + 2336*x^3))/(3174*(3 - x + 2*x^2)^(3/2)) - (25*Log[1 - 4*x + 2*Sqrt[2]*Sqrt[3 -
x + 2*x^2]])/(4*Sqrt[2])

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fricas [B]  time = 0.41, size = 112, normalized size = 1.65 \begin {gather*} \frac {39675 \, \sqrt {2} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) - 88 \, {\left (2336 \, x^{3} + 6183 \, x^{2} + 714 \, x + 8623\right )} \sqrt {2 \, x^{2} - x + 3}}{25392 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25392*(39675*sqrt(2)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^
2 + 16*x - 25) - 88*(2336*x^3 + 6183*x^2 + 714*x + 8623)*sqrt(2*x^2 - x + 3))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x +
9)

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giac [A]  time = 0.27, size = 61, normalized size = 0.90 \begin {gather*} -\frac {25}{8} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac {11 \, {\left ({\left ({\left (2336 \, x + 6183\right )} x + 714\right )} x + 8623\right )}}{3174 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/8*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) - 11/3174*(((2336*x + 6183)*x + 714)*x + 8
623)/(2*x^2 - x + 3)^(3/2)

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maple [B]  time = 0.01, size = 146, normalized size = 2.15 \begin {gather*} -\frac {25 x^{3}}{6 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {145 x^{2}}{8 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {319 x}{64 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {25 x}{4 \sqrt {2 x^{2}-x +3}}+\frac {25 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8}-\frac {15775}{768 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {\frac {8493 x}{1472}-\frac {8493}{5888}}{\left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {\frac {2267 x}{529}-\frac {2267}{2116}}{\sqrt {2 x^{2}-x +3}}-\frac {25}{16 \sqrt {2 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/6/(2*x^2-x+3)^(3/2)*x^3-145/8/(2*x^2-x+3)^(3/2)*x^2-319/64/(2*x^2-x+3)^(3/2)*x-15775/768/(2*x^2-x+3)^(3/2)
+8493/5888*(4*x-1)/(2*x^2-x+3)^(3/2)+2267/2116*(4*x-1)/(2*x^2-x+3)^(1/2)-25/4/(2*x^2-x+3)^(1/2)*x-25/16/(2*x^2
-x+3)^(1/2)+25/8*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))

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maxima [B]  time = 0.98, size = 185, normalized size = 2.72 \begin {gather*} \frac {25}{6348} \, x {\left (\frac {284 \, x}{\sqrt {2 \, x^{2} - x + 3}} - \frac {3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {71}{\sqrt {2 \, x^{2} - x + 3}} + \frac {805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} + \frac {25}{8} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {1775}{3174} \, \sqrt {2 \, x^{2} - x + 3} + \frac {1017 \, x}{529 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {15 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {6413}{3174 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {x}{138 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {2593}{138 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/6348*x*(284*x/sqrt(2*x^2 - x + 3) - 3174*x^2/(2*x^2 - x + 3)^(3/2) - 71/sqrt(2*x^2 - x + 3) + 805*x/(2*x^2
- x + 3)^(3/2) - 3243/(2*x^2 - x + 3)^(3/2)) + 25/8*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 1775/3174*sqrt(
2*x^2 - x + 3) + 1017/529*x/sqrt(2*x^2 - x + 3) - 15*x^2/(2*x^2 - x + 3)^(3/2) - 6413/3174/sqrt(2*x^2 - x + 3)
- 1/138*x/(2*x^2 - x + 3)^(3/2) - 2593/138/(2*x^2 - x + 3)^(3/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (5\,x^2+3\,x+2\right )}^2}{{\left (2\,x^2-x+3\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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