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

Optimal. Leaf size=147 \[ \frac{625}{32} \sqrt{2 x^2-x+3} x^3+\frac{38375}{384} \sqrt{2 x^2-x+3} x^2+\frac{526075 \sqrt{2 x^2-x+3} x}{3072}-\frac{1308645 \sqrt{2 x^2-x+3}}{4096}+\frac{1331 (116368 x+7409)}{101568 \sqrt{2 x^2-x+3}}-\frac{14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}+\frac{16955197 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8192 \sqrt{2}} \]

[Out]

(-14641*(101 + 79*x))/(4416*(3 - x + 2*x^2)^(3/2)) + (1331*(7409 + 116368*x))/(101568*Sqrt[3 - x + 2*x^2]) - (
1308645*Sqrt[3 - x + 2*x^2])/4096 + (526075*x*Sqrt[3 - x + 2*x^2])/3072 + (38375*x^2*Sqrt[3 - x + 2*x^2])/384
+ (625*x^3*Sqrt[3 - x + 2*x^2])/32 + (16955197*ArcSinh[(1 - 4*x)/Sqrt[23]])/(8192*Sqrt[2])

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Rubi [A]  time = 0.167403, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{625}{32} \sqrt{2 x^2-x+3} x^3+\frac{38375}{384} \sqrt{2 x^2-x+3} x^2+\frac{526075 \sqrt{2 x^2-x+3} x}{3072}-\frac{1308645 \sqrt{2 x^2-x+3}}{4096}+\frac{1331 (116368 x+7409)}{101568 \sqrt{2 x^2-x+3}}-\frac{14641 (79 x+101)}{4416 \left (2 x^2-x+3\right )^{3/2}}+\frac{16955197 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8192 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-14641*(101 + 79*x))/(4416*(3 - x + 2*x^2)^(3/2)) + (1331*(7409 + 116368*x))/(101568*Sqrt[3 - x + 2*x^2]) - (
1308645*Sqrt[3 - x + 2*x^2])/4096 + (526075*x*Sqrt[3 - x + 2*x^2])/3072 + (38375*x^2*Sqrt[3 - x + 2*x^2])/384
+ (625*x^3*Sqrt[3 - x + 2*x^2])/32 + (16955197*ArcSinh[(1 - 4*x)/Sqrt[23]])/(8192*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 )^4}{\left (3-x+2 x^2\right )^{5/2}} \, dx &=-\frac{14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac{2}{69} \int \frac{\frac{3839123}{256}-\frac{1983543 x}{128}-\frac{1464801 x^2}{64}+\frac{430905 x^3}{32}+\frac{639975 x^4}{16}+\frac{250125 x^5}{8}+\frac{43125 x^6}{4}}{\left (3-x+2 x^2\right )^{3/2}} \, dx\\ &=-\frac{14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac{1331 (7409+116368 x)}{101568 \sqrt{3-x+2 x^2}}+\frac{4 \int \frac{-\frac{141812733}{256}-\frac{1880595 x}{16}+\frac{15512925 x^2}{64}+\frac{3372375 x^3}{16}+\frac{991875 x^4}{16}}{\sqrt{3-x+2 x^2}} \, dx}{1587}\\ &=-\frac{14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac{1331 (7409+116368 x)}{101568 \sqrt{3-x+2 x^2}}+\frac{625}{32} x^3 \sqrt{3-x+2 x^2}+\frac{\int \frac{-\frac{141812733}{32}-\frac{1880595 x}{2}+\frac{22098975 x^2}{16}+\frac{60901125 x^3}{32}}{\sqrt{3-x+2 x^2}} \, dx}{3174}\\ &=-\frac{14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac{1331 (7409+116368 x)}{101568 \sqrt{3-x+2 x^2}}+\frac{38375}{384} x^2 \sqrt{3-x+2 x^2}+\frac{625}{32} x^3 \sqrt{3-x+2 x^2}+\frac{\int \frac{-\frac{425438199}{16}-\frac{272971935 x}{16}+\frac{834881025 x^2}{64}}{\sqrt{3-x+2 x^2}} \, dx}{19044}\\ &=-\frac{14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac{1331 (7409+116368 x)}{101568 \sqrt{3-x+2 x^2}}+\frac{526075 x \sqrt{3-x+2 x^2}}{3072}+\frac{38375}{384} x^2 \sqrt{3-x+2 x^2}+\frac{625}{32} x^3 \sqrt{3-x+2 x^2}+\frac{\int \frac{-\frac{9311654259}{64}-\frac{6230458845 x}{128}}{\sqrt{3-x+2 x^2}} \, dx}{76176}\\ &=-\frac{14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac{1331 (7409+116368 x)}{101568 \sqrt{3-x+2 x^2}}-\frac{1308645 \sqrt{3-x+2 x^2}}{4096}+\frac{526075 x \sqrt{3-x+2 x^2}}{3072}+\frac{38375}{384} x^2 \sqrt{3-x+2 x^2}+\frac{625}{32} x^3 \sqrt{3-x+2 x^2}-\frac{16955197 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{8192}\\ &=-\frac{14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac{1331 (7409+116368 x)}{101568 \sqrt{3-x+2 x^2}}-\frac{1308645 \sqrt{3-x+2 x^2}}{4096}+\frac{526075 x \sqrt{3-x+2 x^2}}{3072}+\frac{38375}{384} x^2 \sqrt{3-x+2 x^2}+\frac{625}{32} x^3 \sqrt{3-x+2 x^2}-\frac{16955197 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{8192 \sqrt{46}}\\ &=-\frac{14641 (101+79 x)}{4416 \left (3-x+2 x^2\right )^{3/2}}+\frac{1331 (7409+116368 x)}{101568 \sqrt{3-x+2 x^2}}-\frac{1308645 \sqrt{3-x+2 x^2}}{4096}+\frac{526075 x \sqrt{3-x+2 x^2}}{3072}+\frac{38375}{384} x^2 \sqrt{3-x+2 x^2}+\frac{625}{32} x^3 \sqrt{3-x+2 x^2}+\frac{16955197 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8192 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.517884, size = 75, normalized size = 0.51 \[ \frac{507840000 x^7+2090608000 x^6+3504730800 x^5-5076781260 x^4+39848900984 x^3-36481630395 x^2+49883864262 x-18974698519}{6500352 \left (2 x^2-x+3\right )^{3/2}}-\frac{16955197 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{8192 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-18974698519 + 49883864262*x - 36481630395*x^2 + 39848900984*x^3 - 5076781260*x^4 + 3504730800*x^5 + 20906080
00*x^6 + 507840000*x^7)/(6500352*(3 - x + 2*x^2)^(3/2)) - (16955197*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(8192*Sqrt[2
])

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Maple [A]  time = 0.071, size = 214, normalized size = 1.5 \begin{align*}{\frac{138025\,{x}^{5}}{256} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{16955197\,{x}^{3}}{12288} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{67488035\,{x}^{2}}{16384} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{16955197\,\sqrt{2}}{16384}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-992926033+3971704132\,x}{13000704}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{-5141612725+20566450900\,x}{36175872} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{55167267\,x}{131072} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}-{\frac{799745\,{x}^{4}}{1024} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{30875\,{x}^{6}}{96} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{16955197\,x}{8192}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{16955197}{32768}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{2149616639}{524288} \left ( 2\,{x}^{2}-x+3 \right ) ^{-{\frac{3}{2}}}}+{\frac{625\,{x}^{7}}{8} \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)^4/(2*x^2-x+3)^(5/2),x)

[Out]

138025/256*x^5/(2*x^2-x+3)^(3/2)[phone]/12288*x^3/(2*x^2-x+3)^(3/2)-67488035/16384*x^2/(2*x^2-x+3)^(3/2)-169
55197/16384*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))[phone]/13000704*(-1+4*x)/(2*x^2-x+3)^(1/2)[phone]/36
175872*(-1+4*x)/(2*x^2-x+3)^(3/2)[phone]/131072*x/(2*x^2-x+3)^(3/2)-799745/1024*x^4/(2*x^2-x+3)^(3/2)+30875/
96*x^6/(2*x^2-x+3)^(3/2)[phone]/8192*x/(2*x^2-x+3)^(1/2)[phone]/32768/(2*x^2-x+3)^(1/2)-2149616639/524288/
(2*x^2-x+3)^(3/2)+625/8*x^7/(2*x^2-x+3)^(3/2)

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Maxima [B]  time = 1.91995, size = 342, normalized size = 2.33 \begin{align*} \frac{625 \, x^{7}}{8 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{30875 \, x^{6}}{96 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{138025 \, x^{5}}{256 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{799745 \, x^{4}}{1024 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{16955197}{13000704} \, 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{16955197}{16384} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{1203818987}{6500352} \, \sqrt{2 \, x^{2} - x + 3} + \frac{3536205583 \, x}{3250176 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{2638851 \, x^{2}}{512 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{257773037}{1083392 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{29484067 \, x}{23552 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{374445479}{70656 \,{\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)^4/(2*x^2-x+3)^(5/2),x, algorithm="maxima")

[Out]

625/8*x^7/(2*x^2 - x + 3)^(3/2) + 30875/96*x^6/(2*x^2 - x + 3)^(3/2) + 138025/256*x^5/(2*x^2 - x + 3)^(3/2) -
799745/1024*x^4/(2*x^2 - x + 3)^(3/2) - 16955197/13000704*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)) - 16955197/1638
4*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) + 1203818987/6500352*sqrt(2*x^2 - x + 3) + 3536205583/3250176*x/sqr
t(2*x^2 - x + 3) - 2638851/512*x^2/(2*x^2 - x + 3)^(3/2) + 257773037/1083392/sqrt(2*x^2 - x + 3) + 29484067/23
552*x/(2*x^2 - x + 3)^(3/2) - 374445479/70656/(2*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.37283, size = 441, normalized size = 3. \begin{align*} \frac{26907897639 \, \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 (507840000 \, x^{7} + 2090608000 \, x^{6} + 3504730800 \, x^{5} - 5076781260 \, x^{4} + 39848900984 \, x^{3} - 36481630395 \, x^{2} + 49883864262 \, x - 18974698519\right )} \sqrt{2 \, x^{2} - x + 3}}{52002816 \,{\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)^4/(2*x^2-x+3)^(5/2),x, algorithm="fricas")

[Out]

1/52002816*(26907897639*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*(507840000*x^7 + 2090608000*x^6 + 3504730800*x^5 - 5076781260*x^4 + 39848900984*x^3
- 36481630395*x^2 + 49883864262*x - 18974698519)*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{\left (5 x^{2} + 3 x + 2\right )^{4}}{\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)**4/(2*x**2-x+3)**(5/2),x)

[Out]

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

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Giac [A]  time = 1.15042, size = 109, normalized size = 0.74 \begin{align*} \frac{16955197}{16384} \, \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 (2645 \,{\left (20 \,{\left (40 \,{\left (60 \, x + 247\right )} x + 16563\right )} x - 479847\right )} x + 9962225246\right )} x - 36481630395\right )} x + 49883864262\right )} x - 18974698519}{6500352 \,{\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)^4/(2*x^2-x+3)^(5/2),x, algorithm="giac")

[Out]

16955197/16384*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/6500352*(((4*(2645*(20*(40*(6
0*x + 247)*x + 16563)*x - 479847)*x + 9962225246)*x - 36481630395)*x + 49883864262)*x - 18974698519)/(2*x^2 -
x + 3)^(3/2)