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

Optimal. Leaf size=143 \[ -\frac {3387 \sqrt {2 x^2-x+3} x^2}{1024}-\frac {372783 \sqrt {2 x^2-x+3} x}{8192}-\frac {203373 \sqrt {2 x^2-x+3}}{32768}+\frac {125}{12} \sqrt {2 x^2-x+3} x^5+\frac {1355}{48} \sqrt {2 x^2-x+3} x^4+\frac {8185}{256} \sqrt {2 x^2-x+3} x^3-\frac {9267707 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}} \]

[Out]

-9267707/131072*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-203373/32768*(2*x^2-x+3)^(1/2)-372783/8192*x*(2*x^2-x+3
)^(1/2)-3387/1024*x^2*(2*x^2-x+3)^(1/2)+8185/256*x^3*(2*x^2-x+3)^(1/2)+1355/48*x^4*(2*x^2-x+3)^(1/2)+125/12*x^
5*(2*x^2-x+3)^(1/2)

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Rubi [A]  time = 0.17, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1661, 640, 619, 215} \[ \frac {125}{12} \sqrt {2 x^2-x+3} x^5+\frac {1355}{48} \sqrt {2 x^2-x+3} x^4+\frac {8185}{256} \sqrt {2 x^2-x+3} x^3-\frac {3387 \sqrt {2 x^2-x+3} x^2}{1024}-\frac {372783 \sqrt {2 x^2-x+3} x}{8192}-\frac {203373 \sqrt {2 x^2-x+3}}{32768}-\frac {9267707 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-203373*Sqrt[3 - x + 2*x^2])/32768 - (372783*x*Sqrt[3 - x + 2*x^2])/8192 - (3387*x^2*Sqrt[3 - x + 2*x^2])/102
4 + (8185*x^3*Sqrt[3 - x + 2*x^2])/256 + (1355*x^4*Sqrt[3 - x + 2*x^2])/48 + (125*x^5*Sqrt[3 - x + 2*x^2])/12
- (9267707*ArcSinh[(1 - 4*x)/Sqrt[23]])/(65536*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 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 )^3}{\sqrt {3-x+2 x^2}} \, dx &=\frac {125}{12} x^5 \sqrt {3-x+2 x^2}+\frac {1}{12} \int \frac {96+432 x+1368 x^2+2484 x^3+1545 x^4+\frac {6775 x^5}{2}}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {1355}{48} x^4 \sqrt {3-x+2 x^2}+\frac {125}{12} x^5 \sqrt {3-x+2 x^2}+\frac {1}{120} \int \frac {960+4320 x+13680 x^2-15810 x^3+\frac {122775 x^4}{4}}{\sqrt {3-x+2 x^2}} \, dx\\ &=\frac {8185}{256} x^3 \sqrt {3-x+2 x^2}+\frac {1355}{48} x^4 \sqrt {3-x+2 x^2}+\frac {125}{12} x^5 \sqrt {3-x+2 x^2}+\frac {1}{960} \int \frac {7680+34560 x-\frac {667215 x^2}{4}-\frac {152415 x^3}{8}}{\sqrt {3-x+2 x^2}} \, dx\\ &=-\frac {3387 x^2 \sqrt {3-x+2 x^2}}{1024}+\frac {8185}{256} x^3 \sqrt {3-x+2 x^2}+\frac {1355}{48} x^4 \sqrt {3-x+2 x^2}+\frac {125}{12} x^5 \sqrt {3-x+2 x^2}+\frac {\int \frac {46080+\frac {1286685 x}{4}-\frac {16775235 x^2}{16}}{\sqrt {3-x+2 x^2}} \, dx}{5760}\\ &=-\frac {372783 x \sqrt {3-x+2 x^2}}{8192}-\frac {3387 x^2 \sqrt {3-x+2 x^2}}{1024}+\frac {8185}{256} x^3 \sqrt {3-x+2 x^2}+\frac {1355}{48} x^4 \sqrt {3-x+2 x^2}+\frac {125}{12} x^5 \sqrt {3-x+2 x^2}+\frac {\int \frac {\frac {53274825}{16}-\frac {9151785 x}{32}}{\sqrt {3-x+2 x^2}} \, dx}{23040}\\ &=-\frac {203373 \sqrt {3-x+2 x^2}}{32768}-\frac {372783 x \sqrt {3-x+2 x^2}}{8192}-\frac {3387 x^2 \sqrt {3-x+2 x^2}}{1024}+\frac {8185}{256} x^3 \sqrt {3-x+2 x^2}+\frac {1355}{48} x^4 \sqrt {3-x+2 x^2}+\frac {125}{12} x^5 \sqrt {3-x+2 x^2}+\frac {9267707 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{65536}\\ &=-\frac {203373 \sqrt {3-x+2 x^2}}{32768}-\frac {372783 x \sqrt {3-x+2 x^2}}{8192}-\frac {3387 x^2 \sqrt {3-x+2 x^2}}{1024}+\frac {8185}{256} x^3 \sqrt {3-x+2 x^2}+\frac {1355}{48} x^4 \sqrt {3-x+2 x^2}+\frac {125}{12} x^5 \sqrt {3-x+2 x^2}+\frac {9267707 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{65536 \sqrt {46}}\\ &=-\frac {203373 \sqrt {3-x+2 x^2}}{32768}-\frac {372783 x \sqrt {3-x+2 x^2}}{8192}-\frac {3387 x^2 \sqrt {3-x+2 x^2}}{1024}+\frac {8185}{256} x^3 \sqrt {3-x+2 x^2}+\frac {1355}{48} x^4 \sqrt {3-x+2 x^2}+\frac {125}{12} x^5 \sqrt {3-x+2 x^2}-\frac {9267707 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 65, normalized size = 0.45 \[ \frac {4 \sqrt {2 x^2-x+3} \left (1024000 x^5+2775040 x^4+3143040 x^3-325152 x^2-4473396 x-610119\right )-27803121 \sqrt {2} \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{393216} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-610119 - 4473396*x - 325152*x^2 + 3143040*x^3 + 2775040*x^4 + 1024000*x^5) - 27803121
*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/393216

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fricas [A]  time = 0.68, size = 78, normalized size = 0.55 \[ \frac {1}{98304} \, {\left (1024000 \, x^{5} + 2775040 \, x^{4} + 3143040 \, x^{3} - 325152 \, x^{2} - 4473396 \, x - 610119\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {9267707}{262144} \, \sqrt {2} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+3*x+2)^3/(2*x^2-x+3)^(1/2),x, algorithm="fricas")

[Out]

1/98304*(1024000*x^5 + 2775040*x^4 + 3143040*x^3 - 325152*x^2 - 4473396*x - 610119)*sqrt(2*x^2 - x + 3) + 9267
707/262144*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)

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giac [A]  time = 0.52, size = 73, normalized size = 0.51 \[ \frac {1}{98304} \, {\left (4 \, {\left (8 \, {\left (20 \, {\left (16 \, {\left (100 \, x + 271\right )} x + 4911\right )} x - 10161\right )} x - 1118349\right )} x - 610119\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {9267707}{131072} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+3*x+2)^3/(2*x^2-x+3)^(1/2),x, algorithm="giac")

[Out]

1/98304*(4*(8*(20*(16*(100*x + 271)*x + 4911)*x - 10161)*x - 1118349)*x - 610119)*sqrt(2*x^2 - x + 3) - 926770
7/131072*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)

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maple [A]  time = 0.01, size = 113, normalized size = 0.79 \[ \frac {125 \sqrt {2 x^{2}-x +3}\, x^{5}}{12}+\frac {1355 \sqrt {2 x^{2}-x +3}\, x^{4}}{48}+\frac {8185 \sqrt {2 x^{2}-x +3}\, x^{3}}{256}-\frac {3387 \sqrt {2 x^{2}-x +3}\, x^{2}}{1024}-\frac {372783 \sqrt {2 x^{2}-x +3}\, x}{8192}+\frac {9267707 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{131072}-\frac {203373 \sqrt {2 x^{2}-x +3}}{32768} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^2+3*x+2)^3/(2*x^2-x+3)^(1/2),x)

[Out]

-203373/32768*(2*x^2-x+3)^(1/2)+9267707/131072*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+125/12*(2*x^2-x+3)^(1/2)
*x^5+1355/48*(2*x^2-x+3)^(1/2)*x^4+8185/256*(2*x^2-x+3)^(1/2)*x^3-3387/1024*(2*x^2-x+3)^(1/2)*x^2-372783/8192*
(2*x^2-x+3)^(1/2)*x

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maxima [A]  time = 0.98, size = 114, normalized size = 0.80 \[ \frac {125}{12} \, \sqrt {2 \, x^{2} - x + 3} x^{5} + \frac {1355}{48} \, \sqrt {2 \, x^{2} - x + 3} x^{4} + \frac {8185}{256} \, \sqrt {2 \, x^{2} - x + 3} x^{3} - \frac {3387}{1024} \, \sqrt {2 \, x^{2} - x + 3} x^{2} - \frac {372783}{8192} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {9267707}{131072} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {203373}{32768} \, \sqrt {2 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+3*x+2)^3/(2*x^2-x+3)^(1/2),x, algorithm="maxima")

[Out]

125/12*sqrt(2*x^2 - x + 3)*x^5 + 1355/48*sqrt(2*x^2 - x + 3)*x^4 + 8185/256*sqrt(2*x^2 - x + 3)*x^3 - 3387/102
4*sqrt(2*x^2 - x + 3)*x^2 - 372783/8192*sqrt(2*x^2 - x + 3)*x + 9267707/131072*sqrt(2)*arcsinh(1/23*sqrt(23)*(
4*x - 1)) - 203373/32768*sqrt(2*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 5*x^2 + 2)^3/(2*x^2 - x + 3)^(1/2),x)

[Out]

int((3*x + 5*x^2 + 2)^3/(2*x^2 - x + 3)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**2+3*x+2)**3/(2*x**2-x+3)**(1/2),x)

[Out]

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

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