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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.168162, antiderivative size = 143, normalized size of antiderivative = 1., 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 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}{\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*}

Mathematica [A]  time = 0.13242, 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

________________________________________________________________________________________

Maple [A]  time = 0.054, size = 113, normalized size = 0.8 \begin{align*}{\frac{125\,{x}^{5}}{12}\sqrt{2\,{x}^{2}-x+3}}+{\frac{1355\,{x}^{4}}{48}\sqrt{2\,{x}^{2}-x+3}}+{\frac{9267707\,\sqrt{2}}{131072}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{8185\,{x}^{3}}{256}\sqrt{2\,{x}^{2}-x+3}}-{\frac{3387\,{x}^{2}}{1024}\sqrt{2\,{x}^{2}-x+3}}-{\frac{372783\,x}{8192}\sqrt{2\,{x}^{2}-x+3}}-{\frac{203373}{32768}\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)^3/(2*x^2-x+3)^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.44931, size = 154, normalized size = 1.08 \begin{align*} \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} \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)^(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)

________________________________________________________________________________________

Fricas [A]  time = 1.34021, size = 266, normalized size = 1.86 \begin{align*} \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 ) \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)^(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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (5 x^{2} + 3 x + 2\right )^{3}}{\sqrt{2 x^{2} - x + 3}}\, 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)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.19468, size = 99, normalized size = 0.69 \begin{align*} \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 ) \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)^(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)