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

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

[Out]

-310445587/262144*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-14641/1472*(101+79*x)/(2*x^2-x+3)^(1/2)-31009685/6553
6*(2*x^2-x+3)^(1/2)-8992487/16384*x*(2*x^2-x+3)^(1/2)-111315/2048*x^2*(2*x^2-x+3)^(1/2)+79425/512*x^3*(2*x^2-x
+3)^(1/2)+10075/96*x^4*(2*x^2-x+3)^(1/2)+625/24*x^5*(2*x^2-x+3)^(1/2)

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Rubi [A]  time = 0.20, antiderivative size = 166, normalized size of antiderivative = 1.00, 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 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
]

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]

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*}

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Mathematica [A]  time = 0.37, 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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fricas [A]  time = 0.91, size = 112, normalized size = 0.67 \[ \frac {21420745503 \, \sqrt {2} {\left (2 \, x^{2} - x + 3\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8 \, {\left (235520000 \, x^{7} + 831385600 \, x^{6} + 1281670400 \, x^{5} + 230669760 \, x^{4} - 2613624504 \, x^{3} - 2534760678 \, x^{2} - 8859305979 \, x - 10961697147\right )} \sqrt {2 \, x^{2} - x + 3}}{36175872 \, {\left (2 \, x^{2} - x + 3\right )}} \]

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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giac [A]  time = 0.25, size = 82, normalized size = 0.49 \[ -\frac {310445587}{262144} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {{\left (46 \, {\left (4 \, {\left (40 \, {\left (20 \, {\left (16 \, {\left (100 \, x + 353\right )} x + 8707\right )} x + 31341\right )} x - 14204481\right )} x - 55103493\right )} x - 8859305979\right )} x - 10961697147}{4521984 \, \sqrt {2 \, x^{2} - x + 3}} \]

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)

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maple [A]  time = 0.03, size = 166, normalized size = 1.00 \[ \frac {625 x^{7}}{12 \sqrt {2 x^{2}-x +3}}+\frac {8825 x^{6}}{48 \sqrt {2 x^{2}-x +3}}+\frac {217675 x^{5}}{768 \sqrt {2 x^{2}-x +3}}+\frac {52235 x^{4}}{1024 \sqrt {2 x^{2}-x +3}}-\frac {4734827 x^{3}}{8192 \sqrt {2 x^{2}-x +3}}-\frac {18367831 x^{2}}{32768 \sqrt {2 x^{2}-x +3}}-\frac {310445587 x}{131072 \sqrt {2 x^{2}-x +3}}+\frac {310445587 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{262144}-\frac {1217267299}{524288 \sqrt {2 x^{2}-x +3}}+\frac {\frac {1234044515 x}{3014656}-\frac {1234044515}{12058624}}{\sqrt {2 x^{2}-x +3}} \]

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]

-1217267299/524288/(2*x^2-x+3)^(1/2)[phone]/12058624*(4*x-1)/(2*x^2-x+3)^(1/2)[phone]/262144*2^(1/2)*ar
csinh(4/23*23^(1/2)*(x-1/4))+217675/768*x^5/(2*x^2-x+3)^(1/2)+52235/1024*x^4/(2*x^2-x+3)^(1/2)-4734827/8192*x^
3/(2*x^2-x+3)^(1/2)+625/12*x^7/(2*x^2-x+3)^(1/2)+8825/48*x^6/(2*x^2-x+3)^(1/2)-18367831/32768*x^2/(2*x^2-x+3)^
(1/2)-310445587/131072*x/(2*x^2-x+3)^(1/2)

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maxima [A]  time = 0.98, size = 148, normalized size = 0.89 \[ \frac {625 \, x^{7}}{12 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {8825 \, x^{6}}{48 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {217675 \, x^{5}}{768 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {52235 \, x^{4}}{1024 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {4734827 \, x^{3}}{8192 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {18367831 \, x^{2}}{32768 \, \sqrt {2 \, x^{2} - x + 3}} + \frac {310445587}{262144} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {2953101993 \, x}{1507328 \, \sqrt {2 \, x^{2} - x + 3}} - \frac {3653899049}{1507328 \, \sqrt {2 \, x^{2} - x + 3}} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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