3.58 \(\int \sqrt{3-x+2 x^2} (2+3 x+5 x^2)^4 \, dx\)

Optimal. Leaf size=208 \[ \frac{125}{4} \left (2 x^2-x+3\right )^{3/2} x^7+\frac{14125}{144} \left (2 x^2-x+3\right )^{3/2} x^6+\frac{233225 \left (2 x^2-x+3\right )^{3/2} x^5}{1536}+\frac{4796405 \left (2 x^2-x+3\right )^{3/2} x^4}{43008}+\frac{8325631 \left (2 x^2-x+3\right )^{3/2} x^3}{1032192}-\frac{83948353 \left (2 x^2-x+3\right )^{3/2} x^2}{2293760}+\frac{804243809 \left (2 x^2-x+3\right )^{3/2} x}{36700160}+\frac{27185733541 \left (2 x^2-x+3\right )^{3/2}}{440401920}-\frac{359471503 (1-4 x) \sqrt{2 x^2-x+3}}{67108864}-\frac{8267844569 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{134217728 \sqrt{2}} \]

[Out]

(-359471503*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/67108864 + (27185733541*(3 - x + 2*x^2)^(3/2))/440401920 + (8042438
09*x*(3 - x + 2*x^2)^(3/2))/36700160 - (83948353*x^2*(3 - x + 2*x^2)^(3/2))/2293760 + (8325631*x^3*(3 - x + 2*
x^2)^(3/2))/1032192 + (4796405*x^4*(3 - x + 2*x^2)^(3/2))/43008 + (233225*x^5*(3 - x + 2*x^2)^(3/2))/1536 + (1
4125*x^6*(3 - x + 2*x^2)^(3/2))/144 + (125*x^7*(3 - x + 2*x^2)^(3/2))/4 - (8267844569*ArcSinh[(1 - 4*x)/Sqrt[2
3]])/(134217728*Sqrt[2])

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Rubi [A]  time = 0.309322, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1661, 640, 612, 619, 215} \[ \frac{125}{4} \left (2 x^2-x+3\right )^{3/2} x^7+\frac{14125}{144} \left (2 x^2-x+3\right )^{3/2} x^6+\frac{233225 \left (2 x^2-x+3\right )^{3/2} x^5}{1536}+\frac{4796405 \left (2 x^2-x+3\right )^{3/2} x^4}{43008}+\frac{8325631 \left (2 x^2-x+3\right )^{3/2} x^3}{1032192}-\frac{83948353 \left (2 x^2-x+3\right )^{3/2} x^2}{2293760}+\frac{804243809 \left (2 x^2-x+3\right )^{3/2} x}{36700160}+\frac{27185733541 \left (2 x^2-x+3\right )^{3/2}}{440401920}-\frac{359471503 (1-4 x) \sqrt{2 x^2-x+3}}{67108864}-\frac{8267844569 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{134217728 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-359471503*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/67108864 + (27185733541*(3 - x + 2*x^2)^(3/2))/440401920 + (8042438
09*x*(3 - x + 2*x^2)^(3/2))/36700160 - (83948353*x^2*(3 - x + 2*x^2)^(3/2))/2293760 + (8325631*x^3*(3 - x + 2*
x^2)^(3/2))/1032192 + (4796405*x^4*(3 - x + 2*x^2)^(3/2))/43008 + (233225*x^5*(3 - x + 2*x^2)^(3/2))/1536 + (1
4125*x^6*(3 - x + 2*x^2)^(3/2))/144 + (125*x^7*(3 - x + 2*x^2)^(3/2))/4 - (8267844569*ArcSinh[(1 - 4*x)/Sqrt[2
3]])/(134217728*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 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

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 \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx &=\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{20} \int \sqrt{3-x+2 x^2} \left (320+1920 x+7520 x^2+18720 x^3+35220 x^4+46800 x^5+33875 x^6+\frac{70625 x^7}{2}\right ) \, dx\\ &=\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{360} \int \sqrt{3-x+2 x^2} \left (5760+34560 x+135360 x^2+336960 x^3+633960 x^4+206775 x^5+\frac{3498375 x^6}{4}\right ) \, dx\\ &=\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \sqrt{3-x+2 x^2} \left (92160+552960 x+2165760 x^2+5391360 x^3-\frac{11902185 x^4}{4}+\frac{71946075 x^5}{8}\right ) \, dx}{5760}\\ &=\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \sqrt{3-x+2 x^2} \left (1290240+7741440 x+30320640 x^2-\frac{64880145 x^3}{2}+\frac{124884465 x^4}{16}\right ) \, dx}{80640}\\ &=\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \sqrt{3-x+2 x^2} \left (15482880+92897280 x+\frac{4697602695 x^2}{16}-\frac{11333027655 x^3}{32}\right ) \, dx}{967680}\\ &=-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \sqrt{3-x+2 x^2} \left (154828800+\frac{48862647765 x}{16}+\frac{108572914215 x^2}{64}\right ) \, dx}{9676800}\\ &=\frac{804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \left (-\frac{246446397045}{64}+\frac{3670074028035 x}{128}\right ) \sqrt{3-x+2 x^2} \, dx}{77414400}\\ &=\frac{27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac{804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{359471503 \int \sqrt{3-x+2 x^2} \, dx}{8388608}\\ &=-\frac{359471503 (1-4 x) \sqrt{3-x+2 x^2}}{67108864}+\frac{27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac{804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{8267844569 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{134217728}\\ &=-\frac{359471503 (1-4 x) \sqrt{3-x+2 x^2}}{67108864}+\frac{27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac{804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\left (359471503 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{134217728}\\ &=-\frac{359471503 (1-4 x) \sqrt{3-x+2 x^2}}{67108864}+\frac{27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac{804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}-\frac{8267844569 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{134217728 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.291108, size = 85, normalized size = 0.41 \[ \frac{4 \sqrt{2 x^2-x+3} \left (1321205760000 x^9+3486515200000 x^8+6327795712000 x^7+7725962035200 x^6+7612808028160 x^5+5354741991424 x^4+2211683657856 x^3-174418077792 x^2+537752185764 x+3801512106459\right )-2604371039235 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{84557168640} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(3801512106459 + 537752185764*x - 174418077792*x^2 + 2211683657856*x^3 + 5354741991424*
x^4 + 7612808028160*x^5 + 7725962035200*x^6 + 6327795712000*x^7 + 3486515200000*x^8 + 1321205760000*x^9) - 260
4371039235*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/84557168640

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Maple [A]  time = 0.069, size = 166, normalized size = 0.8 \begin{align*}{\frac{125\,{x}^{7}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{14125\,{x}^{6}}{144} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{233225\,{x}^{5}}{1536} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{4796405\,{x}^{4}}{43008} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{8325631\,{x}^{3}}{1032192} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{83948353\,{x}^{2}}{2293760} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{804243809\,x}{36700160} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{-359471503+1437886012\,x}{67108864}\sqrt{2\,{x}^{2}-x+3}}+{\frac{8267844569\,\sqrt{2}}{268435456}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{27185733541}{440401920} \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)^(1/2),x)

[Out]

125/4*x^7*(2*x^2-x+3)^(3/2)+14125/144*x^6*(2*x^2-x+3)^(3/2)+233225/1536*x^5*(2*x^2-x+3)^(3/2)+4796405/43008*x^
4*(2*x^2-x+3)^(3/2)+8325631/1032192*x^3*(2*x^2-x+3)^(3/2)-83948353/2293760*x^2*(2*x^2-x+3)^(3/2)[phone]/367
00160*x*(2*x^2-x+3)^(3/2)[phone]/67108864*(-1+4*x)*(2*x^2-x+3)^(1/2)[phone]/268435456*2^(1/2)*arcsinh(4
/23*23^(1/2)*(x-1/4))[phone]/440401920*(2*x^2-x+3)^(3/2)

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Maxima [A]  time = 1.48403, size = 239, normalized size = 1.15 \begin{align*} \frac{125}{4} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{7} + \frac{14125}{144} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{6} + \frac{233225}{1536} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{5} + \frac{4796405}{43008} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + \frac{8325631}{1032192} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{83948353}{2293760} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{804243809}{36700160} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{27185733541}{440401920} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{359471503}{16777216} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{8267844569}{268435456} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{359471503}{67108864} \, \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)^(1/2),x, algorithm="maxima")

[Out]

125/4*(2*x^2 - x + 3)^(3/2)*x^7 + 14125/144*(2*x^2 - x + 3)^(3/2)*x^6 + 233225/1536*(2*x^2 - x + 3)^(3/2)*x^5
+ 4796405/43008*(2*x^2 - x + 3)^(3/2)*x^4 + 8325631/1032192*(2*x^2 - x + 3)^(3/2)*x^3 - 83948353/2293760*(2*x^
2 - x + 3)^(3/2)*x^2 + 804243809/36700160*(2*x^2 - x + 3)^(3/2)*x + 27185733541/440401920*(2*x^2 - x + 3)^(3/2
) + 359471503/16777216*sqrt(2*x^2 - x + 3)*x + 8267844569/268435456*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) -
359471503/67108864*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.40846, size = 439, normalized size = 2.11 \begin{align*} \frac{1}{21139292160} \,{\left (1321205760000 \, x^{9} + 3486515200000 \, x^{8} + 6327795712000 \, x^{7} + 7725962035200 \, x^{6} + 7612808028160 \, x^{5} + 5354741991424 \, x^{4} + 2211683657856 \, x^{3} - 174418077792 \, x^{2} + 537752185764 \, x + 3801512106459\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{8267844569}{536870912} \, \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)^4*(2*x^2-x+3)^(1/2),x, algorithm="fricas")

[Out]

1/21139292160*(1321205760000*x^9 + 3486515200000*x^8 + 6327795712000*x^7 + 7725962035200*x^6 + 7612808028160*x
^5 + 5354741991424*x^4 + 2211683657856*x^3 - 174418077792*x^2 + 537752185764*x + 3801512106459)*sqrt(2*x^2 - x
+ 3) + 8267844569/536870912*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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{4}\, 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)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.13665, size = 126, normalized size = 0.61 \begin{align*} \frac{1}{21139292160} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (20 \,{\left (40 \,{\left (140 \,{\left (160 \,{\left (36 \, x + 95\right )} x + 27587\right )} x + 4715553\right )} x + 185859571\right )} x + 2614620113\right )} x + 17278778577\right )} x - 5450564931\right )} x + 134438046441\right )} x + 3801512106459\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{8267844569}{268435456} \, \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)^4*(2*x^2-x+3)^(1/2),x, algorithm="giac")

[Out]

1/21139292160*(4*(8*(4*(16*(20*(40*(140*(160*(36*x + 95)*x + 27587)*x + 4715553)*x + 185859571)*x + 2614620113
)*x + 17278778577)*x - 5450564931)*x + 134438046441)*x + 3801512106459)*sqrt(2*x^2 - x + 3) - 8267844569/26843
5456*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)