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

Optimal. Leaf size=105 \[ \frac{121 (10679-6744 x)}{8464 \sqrt{2 x^2-x+3}}+\frac{125}{16} x \sqrt{2 x^2-x+3}+\frac{3175}{64} \sqrt{2 x^2-x+3}-\frac{1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}-\frac{7495 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}} \]

[Out]

(-1331*(17 - 45*x))/(1104*(3 - x + 2*x^2)^(3/2)) + (121*(10679 - 6744*x))/(8464*Sqrt[3 - x + 2*x^2]) + (3175*S
qrt[3 - x + 2*x^2])/64 + (125*x*Sqrt[3 - x + 2*x^2])/16 - (7495*ArcSinh[(1 - 4*x)/Sqrt[23]])/(128*Sqrt[2])

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Rubi [A]  time = 0.105366, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1660, 1661, 640, 619, 215} \[ \frac{121 (10679-6744 x)}{8464 \sqrt{2 x^2-x+3}}+\frac{125}{16} x \sqrt{2 x^2-x+3}+\frac{3175}{64} \sqrt{2 x^2-x+3}-\frac{1331 (17-45 x)}{1104 \left (2 x^2-x+3\right )^{3/2}}-\frac{7495 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*(17 - 45*x))/(1104*(3 - x + 2*x^2)^(3/2)) + (121*(10679 - 6744*x))/(8464*Sqrt[3 - x + 2*x^2]) + (3175*S
qrt[3 - x + 2*x^2])/64 + (125*x*Sqrt[3 - x + 2*x^2])/16 - (7495*ArcSinh[(1 - 4*x)/Sqrt[23]])/(128*Sqrt[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 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

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 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 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]

Rubi steps

\begin{align*} \int \frac{\left (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=-\frac{1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{-\frac{91275}{64}-\frac{57201 x}{32}+\frac{66585 x^2}{16}+\frac{39675 x^3}{8}+\frac{8625 x^4}{4}}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac{1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac{121 (10679-6744 x)}{8464 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{\frac{1452105}{64}+\frac{277725 x}{8}+\frac{198375 x^2}{16}}{\sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=-\frac{1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac{121 (10679-6744 x)}{8464 \sqrt{3-x+2 x^2}}+\frac{125}{16} x \sqrt{3-x+2 x^2}+\frac{\int \frac{\frac{214245}{4}+\frac{5038725 x}{32}}{\sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=-\frac{1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac{121 (10679-6744 x)}{8464 \sqrt{3-x+2 x^2}}+\frac{3175}{64} \sqrt{3-x+2 x^2}+\frac{125}{16} x \sqrt{3-x+2 x^2}+\frac{7495}{128} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac{121 (10679-6744 x)}{8464 \sqrt{3-x+2 x^2}}+\frac{3175}{64} \sqrt{3-x+2 x^2}+\frac{125}{16} x \sqrt{3-x+2 x^2}+\frac{7495 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{128 \sqrt{46}}\\ &=-\frac{1331 (17-45 x)}{1104 \left (3-x+2 x^2\right )^{3/2}}+\frac{121 (10679-6744 x)}{8464 \sqrt{3-x+2 x^2}}+\frac{3175}{64} \sqrt{3-x+2 x^2}+\frac{125}{16} x \sqrt{3-x+2 x^2}-\frac{7495 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{128 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.335113, size = 65, normalized size = 0.62 \[ \frac{3174000 x^5+16980900 x^4-29423976 x^3+101546529 x^2-62463282 x+89784565}{101568 \left (2 x^2-x+3\right )^{3/2}}+\frac{7495 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{128 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(89784565 - 62463282*x + 101546529*x^2 - 29423976*x^3 + 16980900*x^4 + 3174000*x^5)/(101568*(3 - x + 2*x^2)^(3
/2)) + (7495*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(128*Sqrt[2])

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Maple [B]  time = 0.057, size = 180, normalized size = 1.7 \begin{align*}{\frac{125\,{x}^{5}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{7495\,{x}^{3}}{192} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{222809\,{x}^{2}}{256} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{7495\,\sqrt{2}}{256}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{-3391139+13564556\,x}{203136}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{-14081711+56326844\,x}{565248} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{281177\,x}{2048} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{2675\,{x}^{4}}{16} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{7495\,x}{128}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{7495}{512}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{20961031}{24576} \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)^3/(2*x^2-x+3)^(5/2),x)

[Out]

125/4*x^5/(2*x^2-x+3)^(3/2)-7495/192*x^3/(2*x^2-x+3)^(3/2)+222809/256*x^2/(2*x^2-x+3)^(3/2)+7495/256*2^(1/2)*a
rcsinh(4/23*23^(1/2)*(x-1/4))-3391139/203136*(-1+4*x)/(2*x^2-x+3)^(1/2)-14081711/565248*(-1+4*x)/(2*x^2-x+3)^(
3/2)-281177/2048*x/(2*x^2-x+3)^(3/2)+2675/16*x^4/(2*x^2-x+3)^(3/2)-7495/128*x/(2*x^2-x+3)^(1/2)-7495/512/(2*x^
2-x+3)^(1/2)[phone]/24576/(2*x^2-x+3)^(3/2)

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Maxima [B]  time = 1.50078, size = 296, normalized size = 2.82 \begin{align*} \frac{125 \, x^{5}}{4 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2675 \, x^{4}}{16 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{7495}{203136} \, 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{7495}{256} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{532145}{101568} \, \sqrt{2 \, x^{2} - x + 3} - \frac{4515389 \, x}{50784 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{7197 \, x^{2}}{8 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{396211}{50784 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{269783 \, x}{1104 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1002137}{1104 \,{\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)^3/(2*x^2-x+3)^(5/2),x, algorithm="maxima")

[Out]

125/4*x^5/(2*x^2 - x + 3)^(3/2) + 2675/16*x^4/(2*x^2 - x + 3)^(3/2) + 7495/203136*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)) + 7495/256*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 532145/101568*sqrt(2*x^2 - x + 3) - 4515389/507
84*x/sqrt(2*x^2 - x + 3) + 7197/8*x^2/(2*x^2 - x + 3)^(3/2) + 396211/50784/sqrt(2*x^2 - x + 3) - 269783/1104*x
/(2*x^2 - x + 3)^(3/2) + 1002137/1104/(2*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.37072, size = 370, normalized size = 3.52 \begin{align*} \frac{11894565 \, \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 (3174000 \, x^{5} + 16980900 \, x^{4} - 29423976 \, x^{3} + 101546529 \, x^{2} - 62463282 \, x + 89784565\right )} \sqrt{2 \, x^{2} - x + 3}}{812544 \,{\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)^3/(2*x^2-x+3)^(5/2),x, algorithm="fricas")

[Out]

1/812544*(11894565*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) - 3
2*x^2 + 16*x - 25) + 8*(3174000*x^5 + 16980900*x^4 - 29423976*x^3 + 101546529*x^2 - 62463282*x + 89784565)*sqr
t(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{\left (5 x^{2} + 3 x + 2\right )^{3}}{\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)**3/(2*x**2-x+3)**(5/2),x)

[Out]

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

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Giac [A]  time = 1.22387, size = 97, normalized size = 0.92 \begin{align*} -\frac{7495}{256} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{3 \,{\left ({\left (4 \,{\left (13225 \,{\left (20 \, x + 107\right )} x - 2451998\right )} x + 33848843\right )} x - 20821094\right )} x + 89784565}{101568 \,{\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)^3/(2*x^2-x+3)^(5/2),x, algorithm="giac")

[Out]

-7495/256*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/101568*(3*((4*(13225*(20*x + 107)*
x - 2451998)*x + 33848843)*x - 20821094)*x + 89784565)/(2*x^2 - x + 3)^(3/2)