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

Optimal. Leaf size=166 \[ \frac{625}{24} \sqrt{2 x^2-x+3} x^5+\frac{10075}{96} \sqrt{2 x^2-x+3} x^4+\frac{79425}{512} \sqrt{2 x^2-x+3} x^3-\frac{111315 \sqrt{2 x^2-x+3} x^2}{2048}-\frac{8992487 \sqrt{2 x^2-x+3} x}{16384}-\frac{31009685 \sqrt{2 x^2-x+3}}{65536}-\frac{14641 (79 x+101)}{1472 \sqrt{2 x^2-x+3}}-\frac{310445587 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{131072 \sqrt{2}} \]

[Out]

(-14641*(101 + 79*x))/(1472*Sqrt[3 - x + 2*x^2]) - (31009685*Sqrt[3 - x + 2*x^2])/65536 - (8992487*x*Sqrt[3 -
x + 2*x^2])/16384 - (111315*x^2*Sqrt[3 - x + 2*x^2])/2048 + (79425*x^3*Sqrt[3 - x + 2*x^2])/512 + (10075*x^4*S
qrt[3 - x + 2*x^2])/96 + (625*x^5*Sqrt[3 - x + 2*x^2])/24 - (310445587*ArcSinh[(1 - 4*x)/Sqrt[23]])/(131072*Sq
rt[2])

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Rubi [A]  time = 0.203946, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{625}{24} \sqrt{2 x^2-x+3} x^5+\frac{10075}{96} \sqrt{2 x^2-x+3} x^4+\frac{79425}{512} \sqrt{2 x^2-x+3} x^3-\frac{111315 \sqrt{2 x^2-x+3} x^2}{2048}-\frac{8992487 \sqrt{2 x^2-x+3} x}{16384}-\frac{31009685 \sqrt{2 x^2-x+3}}{65536}-\frac{14641 (79 x+101)}{1472 \sqrt{2 x^2-x+3}}-\frac{310445587 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{131072 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-14641*(101 + 79*x))/(1472*Sqrt[3 - x + 2*x^2]) - (31009685*Sqrt[3 - x + 2*x^2])/65536 - (8992487*x*Sqrt[3 -
x + 2*x^2])/16384 - (111315*x^2*Sqrt[3 - x + 2*x^2])/2048 + (79425*x^3*Sqrt[3 - x + 2*x^2])/512 + (10075*x^4*S
qrt[3 - x + 2*x^2])/96 + (625*x^5*Sqrt[3 - x + 2*x^2])/24 - (310445587*ArcSinh[(1 - 4*x)/Sqrt[23]])/(131072*Sq
rt[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 )^{3/2}} \, dx &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}+\frac{2}{23} \int \frac{\frac{2821893}{256}-\frac{661181 x}{128}-\frac{488267 x^2}{64}+\frac{143635 x^3}{32}+\frac{213325 x^4}{16}+\frac{83375 x^5}{8}+\frac{14375 x^6}{4}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{1}{138} \int \frac{\frac{8465679}{64}-\frac{1983543 x}{32}-\frac{1464801 x^2}{16}+\frac{430905 x^3}{8}+\frac{212175 x^4}{2}+\frac{1158625 x^5}{8}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{\int \frac{\frac{42328395}{32}-\frac{9917715 x}{16}-\frac{7324005 x^2}{8}-\frac{4797225 x^3}{4}+\frac{27401625 x^4}{16}}{\sqrt{3-x+2 x^2}} \, dx}{1380}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{\int \frac{\frac{42328395}{4}-\frac{9917715 x}{2}-\frac{363798705 x^2}{16}-\frac{115211025 x^3}{32}}{\sqrt{3-x+2 x^2}} \, dx}{11040}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}-\frac{111315 x^2 \sqrt{3-x+2 x^2}}{2048}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{\int \frac{\frac{126985185}{2}-\frac{130417245 x}{16}-\frac{9307224045 x^2}{64}}{\sqrt{3-x+2 x^2}} \, dx}{66240}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}-\frac{8992487 x \sqrt{3-x+2 x^2}}{16384}-\frac{111315 x^2 \sqrt{3-x+2 x^2}}{2048}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{\int \frac{\frac{44175775815}{64}-\frac{32095023975 x}{128}}{\sqrt{3-x+2 x^2}} \, dx}{264960}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}-\frac{31009685 \sqrt{3-x+2 x^2}}{65536}-\frac{8992487 x \sqrt{3-x+2 x^2}}{16384}-\frac{111315 x^2 \sqrt{3-x+2 x^2}}{2048}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{310445587 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{131072}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}-\frac{31009685 \sqrt{3-x+2 x^2}}{65536}-\frac{8992487 x \sqrt{3-x+2 x^2}}{16384}-\frac{111315 x^2 \sqrt{3-x+2 x^2}}{2048}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{310445587 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{131072 \sqrt{46}}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}-\frac{31009685 \sqrt{3-x+2 x^2}}{65536}-\frac{8992487 x \sqrt{3-x+2 x^2}}{16384}-\frac{111315 x^2 \sqrt{3-x+2 x^2}}{2048}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}-\frac{310445587 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{131072 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.376634, size = 95, normalized size = 0.57 \[ \sqrt{2 x^2-x+3} \left (\frac{625 x^5}{24}+\frac{10075 x^4}{96}+\frac{79425 x^3}{512}-\frac{111315 x^2}{2048}-\frac{14641 (79 x+101)}{1472 \left (2 x^2-x+3\right )}-\frac{8992487 x}{16384}-\frac{31009685}{65536}\right )+\frac{310445587 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{131072 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[3 - x + 2*x^2]*(-31009685/65536 - (8992487*x)/16384 - (111315*x^2)/2048 + (79425*x^3)/512 + (10075*x^4)/9
6 + (625*x^5)/24 - (14641*(101 + 79*x))/(1472*(3 - x + 2*x^2))) + (310445587*ArcSinh[(-1 + 4*x)/Sqrt[23]])/(13
1072*Sqrt[2])

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Maple [A]  time = 0.066, size = 166, normalized size = 1. \begin{align*}{\frac{8825\,{x}^{6}}{48}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{217675\,{x}^{5}}{768}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{52235\,{x}^{4}}{1024}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{310445587\,\sqrt{2}}{262144}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-1234044515+4936178060\,x}{12058624}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{625\,{x}^{7}}{12}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{310445587\,x}{131072}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{4734827\,{x}^{3}}{8192}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{18367831\,{x}^{2}}{32768}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{1217267299}{524288}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}} \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)^(3/2),x)

[Out]

8825/48*x^6/(2*x^2-x+3)^(1/2)+217675/768*x^5/(2*x^2-x+3)^(1/2)+52235/1024*x^4/(2*x^2-x+3)^(1/2)[phone]/2621
44*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))[phone]/12058624*(-1+4*x)/(2*x^2-x+3)^(1/2)+625/12*x^7/(2*x^2-x+3
)^(1/2)-310445587/131072*x/(2*x^2-x+3)^(1/2)-4734827/8192*x^3/(2*x^2-x+3)^(1/2)-18367831/32768*x^2/(2*x^2-x+3)
^(1/2)-1217267299/524288/(2*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.49527, size = 200, normalized size = 1.2 \begin{align*} \frac{625 \, x^{7}}{12 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{8825 \, x^{6}}{48 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{217675 \, x^{5}}{768 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{52235 \, x^{4}}{1024 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{4734827 \, x^{3}}{8192 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{18367831 \, x^{2}}{32768 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{310445587}{262144} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{2953101993 \, x}{1507328 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{3653899049}{1507328 \, \sqrt{2 \, x^{2} - x + 3}} \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)^(3/2),x, algorithm="maxima")

[Out]

625/12*x^7/sqrt(2*x^2 - x + 3) + 8825/48*x^6/sqrt(2*x^2 - x + 3) + 217675/768*x^5/sqrt(2*x^2 - x + 3) + 52235/
1024*x^4/sqrt(2*x^2 - x + 3) - 4734827/8192*x^3/sqrt(2*x^2 - x + 3) - 18367831/32768*x^2/sqrt(2*x^2 - x + 3) +
310445587/262144*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 2953101993/1507328*x/sqrt(2*x^2 - x + 3) - 365389
9049/1507328/sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.41592, size = 385, normalized size = 2.32 \begin{align*} \frac{21420745503 \, \sqrt{2}{\left (2 \, x^{2} - x + 3\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 (235520000 \, x^{7} + 831385600 \, x^{6} + 1281670400 \, x^{5} + 230669760 \, x^{4} - 2613624504 \, x^{3} - 2534760678 \, x^{2} - 8859305979 \, x - 10961697147\right )} \sqrt{2 \, x^{2} - x + 3}}{36175872 \,{\left (2 \, x^{2} - x + 3\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)^(3/2),x, algorithm="fricas")

[Out]

1/36175872*(21420745503*sqrt(2)*(2*x^2 - x + 3)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x -
25) + 8*(235520000*x^7 + 831385600*x^6 + 1281670400*x^5 + 230669760*x^4 - 2613624504*x^3 - 2534760678*x^2 - 8
859305979*x - 10961697147)*sqrt(2*x^2 - x + 3))/(2*x^2 - x + 3)

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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{3}{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)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.1978, size = 111, normalized size = 0.67 \begin{align*} -\frac{310445587}{262144} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{{\left (46 \,{\left (4 \,{\left (40 \,{\left (20 \,{\left (16 \,{\left (100 \, x + 353\right )} x + 8707\right )} x + 31341\right )} x - 14204481\right )} x - 55103493\right )} x - 8859305979\right )} x - 10961697147}{4521984 \, \sqrt{2 \, x^{2} - x + 3}} \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)^(3/2),x, algorithm="giac")

[Out]

-310445587/262144*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1/4521984*((46*(4*(40*(20*(1
6*(100*x + 353)*x + 8707)*x + 31341)*x - 14204481)*x - 55103493)*x - 8859305979)*x - 10961697147)/sqrt(2*x^2 -
x + 3)