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

Optimal. Leaf size=86 \[ \frac {6055-28981 x}{3174 \sqrt {2 x^2-x+3}}+\frac {5}{4} \sqrt {2 x^2-x+3}+\frac {373 x-53}{69 \left (2 x^2-x+3\right )^{3/2}}-\frac {71 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8 \sqrt {2}} \]

[Out]

1/69*(-53+373*x)/(2*x^2-x+3)^(3/2)-71/16*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+1/3174*(6055-28981*x)/(2*x^2-x
+3)^(1/2)+5/4*(2*x^2-x+3)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1660, 640, 619, 215} \[ \frac {6055-28981 x}{3174 \sqrt {2 x^2-x+3}}+\frac {5}{4} \sqrt {2 x^2-x+3}-\frac {53-373 x}{69 \left (2 x^2-x+3\right )^{3/2}}-\frac {71 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(53 - 373*x)/(69*(3 - x + 2*x^2)^(3/2)) + (6055 - 28981*x)/(3174*Sqrt[3 - x + 2*x^2]) + (5*Sqrt[3 - x + 2*x^2
])/4 - (71*ArcSinh[(1 - 4*x)/Sqrt[23]])/(8*Sqrt[2])

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 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

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 {(5+2 x) \left (2+x+3 x^2-x^3+5 x^4\right )}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=-\frac {53-373 x}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac {2}{69} \int \frac {-\frac {233}{4}+483 x^2+\frac {345 x^3}{2}}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac {53-373 x}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac {6055-28981 x}{3174 \sqrt {3-x+2 x^2}}+\frac {4 \int \frac {\frac {52371}{16}+\frac {7935 x}{8}}{\sqrt {3-x+2 x^2}} \, dx}{1587}\\ &=-\frac {53-373 x}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac {6055-28981 x}{3174 \sqrt {3-x+2 x^2}}+\frac {5}{4} \sqrt {3-x+2 x^2}+\frac {71}{8} \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {53-373 x}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac {6055-28981 x}{3174 \sqrt {3-x+2 x^2}}+\frac {5}{4} \sqrt {3-x+2 x^2}+\frac {71 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{8 \sqrt {46}}\\ &=-\frac {53-373 x}{69 \left (3-x+2 x^2\right )^{3/2}}+\frac {6055-28981 x}{3174 \sqrt {3-x+2 x^2}}+\frac {5}{4} \sqrt {3-x+2 x^2}-\frac {71 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{8 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 60, normalized size = 0.70 \[ \frac {31740 x^4-147664 x^3+185337 x^2-199290 x+102869}{6348 \left (2 x^2-x+3\right )^{3/2}}+\frac {71 \sinh ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{8 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(102869 - 199290*x + 185337*x^2 - 147664*x^3 + 31740*x^4)/(6348*(3 - x + 2*x^2)^(3/2)) + (71*ArcSinh[(-1 + 4*x
)/Sqrt[23]])/(8*Sqrt[2])

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fricas [A]  time = 0.72, size = 117, normalized size = 1.36 \[ \frac {112677 \, \sqrt {2} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8 \, {\left (31740 \, x^{4} - 147664 \, x^{3} + 185337 \, x^{2} - 199290 \, x + 102869\right )} \sqrt {2 \, x^{2} - x + 3}}{50784 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50784*(112677*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) + 8*(31740*x^4 - 147664*x^3 + 185337*x^2 - 199290*x + 102869)*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 = 0.21, size = 66, normalized size = 0.77 \[ -\frac {71}{16} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {{\left ({\left (4 \, {\left (7935 \, x - 36916\right )} x + 185337\right )} x - 199290\right )} x + 102869}{6348 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-71/16*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/6348*(((4*(7935*x - 36916)*x + 185337
)*x - 199290)*x + 102869)/(2*x^2 - x + 3)^(3/2)

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maple [B]  time = 0.01, size = 163, normalized size = 1.90 \[ \frac {5 x^{4}}{\left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {71 x^{3}}{12 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {401 x^{2}}{16 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {945 x}{128 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {71 x}{8 \sqrt {2 x^{2}-x +3}}+\frac {71 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{16}+\frac {\frac {643 x}{3174}-\frac {643}{12696}}{\sqrt {2 x^{2}-x +3}}-\frac {2327 \left (4 x -1\right )}{35328 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}+\frac {11749}{512 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}-\frac {71}{32 \sqrt {2 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

643/12696*(4*x-1)/(2*x^2-x+3)^(1/2)-2327/35328*(4*x-1)/(2*x^2-x+3)^(3/2)+5/(2*x^2-x+3)^(3/2)*x^4-71/12/(2*x^2-
x+3)^(3/2)*x^3+401/16/(2*x^2-x+3)^(3/2)*x^2-945/128/(2*x^2-x+3)^(3/2)*x+71/16*2^(1/2)*arcsinh(4/23*23^(1/2)*(x
-1/4))-71/8/(2*x^2-x+3)^(1/2)*x+11749/512/(2*x^2-x+3)^(3/2)-71/32/(2*x^2-x+3)^(1/2)

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maxima [B]  time = 0.97, size = 202, normalized size = 2.35 \[ \frac {5 \, x^{4}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {71}{12696} \, x {\left (\frac {284 \, x}{\sqrt {2 \, x^{2} - x + 3}} - \frac {3174 \, x^{2}}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {71}{\sqrt {2 \, x^{2} - x + 3}} + \frac {805 \, x}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {3243}{{\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} + \frac {71}{16} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {5041}{6348} \, \sqrt {2 \, x^{2} - x + 3} - \frac {10007 \, x}{3174 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {59 \, x^{2}}{2 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {2959}{2116 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {807 \, x}{92 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {7603}{276 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x^4/(2*x^2 - x + 3)^(3/2) + 71/12696*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)) + 71/16*sqrt(2)*arcsinh(1/23*sqrt(
23)*(4*x - 1)) - 5041/6348*sqrt(2*x^2 - x + 3) - 10007/3174*x/sqrt(2*x^2 - x + 3) + 59/2*x^2/(2*x^2 - x + 3)^(
3/2) - 2959/2116/sqrt(2*x^2 - x + 3) - 807/92*x/(2*x^2 - x + 3)^(3/2) + 7603/276/(2*x^2 - x + 3)^(3/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (2\,x+5\right )\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x^2-x+3\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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