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

Optimal. Leaf size=185 \[ \frac{625}{16} \sqrt{2 x^2-x+3} x^7+\frac{57375}{448} \sqrt{2 x^2-x+3} x^6+\frac{2116475 \sqrt{2 x^2-x+3} x^5}{10752}+\frac{686531 \sqrt{2 x^2-x+3} x^4}{6144}-\frac{19750457 \sqrt{2 x^2-x+3} x^3}{229376}-\frac{15428243 \sqrt{2 x^2-x+3} x^2}{131072}+\frac{1572007407 \sqrt{2 x^2-x+3} x}{7340032}+\frac{16493087661 \sqrt{2 x^2-x+3}}{29360128}+\frac{2899366573 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8388608 \sqrt{2}} \]

[Out]

(16493087661*Sqrt[3 - x + 2*x^2])/29360128 + (1572007407*x*Sqrt[3 - x + 2*x^2])/7340032 - (15428243*x^2*Sqrt[3
- x + 2*x^2])/131072 - (19750457*x^3*Sqrt[3 - x + 2*x^2])/229376 + (686531*x^4*Sqrt[3 - x + 2*x^2])/6144 + (2
116475*x^5*Sqrt[3 - x + 2*x^2])/10752 + (57375*x^6*Sqrt[3 - x + 2*x^2])/448 + (625*x^7*Sqrt[3 - x + 2*x^2])/16
+ (2899366573*ArcSinh[(1 - 4*x)/Sqrt[23]])/(8388608*Sqrt[2])

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Rubi [A]  time = 0.312208, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1661, 640, 619, 215} \[ \frac{625}{16} \sqrt{2 x^2-x+3} x^7+\frac{57375}{448} \sqrt{2 x^2-x+3} x^6+\frac{2116475 \sqrt{2 x^2-x+3} x^5}{10752}+\frac{686531 \sqrt{2 x^2-x+3} x^4}{6144}-\frac{19750457 \sqrt{2 x^2-x+3} x^3}{229376}-\frac{15428243 \sqrt{2 x^2-x+3} x^2}{131072}+\frac{1572007407 \sqrt{2 x^2-x+3} x}{7340032}+\frac{16493087661 \sqrt{2 x^2-x+3}}{29360128}+\frac{2899366573 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8388608 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16493087661*Sqrt[3 - x + 2*x^2])/29360128 + (1572007407*x*Sqrt[3 - x + 2*x^2])/7340032 - (15428243*x^2*Sqrt[3
- x + 2*x^2])/131072 - (19750457*x^3*Sqrt[3 - x + 2*x^2])/229376 + (686531*x^4*Sqrt[3 - x + 2*x^2])/6144 + (2
116475*x^5*Sqrt[3 - x + 2*x^2])/10752 + (57375*x^6*Sqrt[3 - x + 2*x^2])/448 + (625*x^7*Sqrt[3 - x + 2*x^2])/16
+ (2899366573*ArcSinh[(1 - 4*x)/Sqrt[23]])/(8388608*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 )^4}{\sqrt{3-x+2 x^2}} \, dx &=\frac{625}{16} x^7 \sqrt{3-x+2 x^2}+\frac{1}{16} \int \frac{256+1536 x+6016 x^2+14976 x^3+28176 x^4+37440 x^5+24475 x^6+\frac{57375 x^7}{2}}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{57375}{448} x^6 \sqrt{3-x+2 x^2}+\frac{625}{16} x^7 \sqrt{3-x+2 x^2}+\frac{1}{224} \int \frac{3584+21504 x+84224 x^2+209664 x^3+394464 x^4+7785 x^5+\frac{2116475 x^6}{4}}{\sqrt{3-x+2 x^2}} \, dx\\ &=\frac{2116475 x^5 \sqrt{3-x+2 x^2}}{10752}+\frac{57375}{448} x^6 \sqrt{3-x+2 x^2}+\frac{625}{16} x^7 \sqrt{3-x+2 x^2}+\frac{\int \frac{43008+258048 x+1010688 x^2+2515968 x^3-\frac{12812853 x^4}{4}+\frac{24028585 x^5}{8}}{\sqrt{3-x+2 x^2}} \, dx}{2688}\\ &=\frac{686531 x^4 \sqrt{3-x+2 x^2}}{6144}+\frac{2116475 x^5 \sqrt{3-x+2 x^2}}{10752}+\frac{57375}{448} x^6 \sqrt{3-x+2 x^2}+\frac{625}{16} x^7 \sqrt{3-x+2 x^2}+\frac{\int \frac{430080+2580480 x+10106880 x^2-\frac{21766395 x^3}{2}-\frac{296256855 x^4}{16}}{\sqrt{3-x+2 x^2}} \, dx}{26880}\\ &=-\frac{19750457 x^3 \sqrt{3-x+2 x^2}}{229376}+\frac{686531 x^4 \sqrt{3-x+2 x^2}}{6144}+\frac{2116475 x^5 \sqrt{3-x+2 x^2}}{10752}+\frac{57375}{448} x^6 \sqrt{3-x+2 x^2}+\frac{625}{16} x^7 \sqrt{3-x+2 x^2}+\frac{\int \frac{3440640+20643840 x+\frac{3959992335 x^2}{16}-\frac{4859896545 x^3}{32}}{\sqrt{3-x+2 x^2}} \, dx}{215040}\\ &=-\frac{15428243 x^2 \sqrt{3-x+2 x^2}}{131072}-\frac{19750457 x^3 \sqrt{3-x+2 x^2}}{229376}+\frac{686531 x^4 \sqrt{3-x+2 x^2}}{6144}+\frac{2116475 x^5 \sqrt{3-x+2 x^2}}{10752}+\frac{57375}{448} x^6 \sqrt{3-x+2 x^2}+\frac{625}{16} x^7 \sqrt{3-x+2 x^2}+\frac{\int \frac{20643840+\frac{16561498275 x}{16}+\frac{70740333315 x^2}{64}}{\sqrt{3-x+2 x^2}} \, dx}{1290240}\\ &=\frac{1572007407 x \sqrt{3-x+2 x^2}}{7340032}-\frac{15428243 x^2 \sqrt{3-x+2 x^2}}{131072}-\frac{19750457 x^3 \sqrt{3-x+2 x^2}}{229376}+\frac{686531 x^4 \sqrt{3-x+2 x^2}}{6144}+\frac{2116475 x^5 \sqrt{3-x+2 x^2}}{10752}+\frac{57375}{448} x^6 \sqrt{3-x+2 x^2}+\frac{625}{16} x^7 \sqrt{3-x+2 x^2}+\frac{\int \frac{-\frac{206936176905}{64}+\frac{742188944745 x}{128}}{\sqrt{3-x+2 x^2}} \, dx}{5160960}\\ &=\frac{16493087661 \sqrt{3-x+2 x^2}}{29360128}+\frac{1572007407 x \sqrt{3-x+2 x^2}}{7340032}-\frac{15428243 x^2 \sqrt{3-x+2 x^2}}{131072}-\frac{19750457 x^3 \sqrt{3-x+2 x^2}}{229376}+\frac{686531 x^4 \sqrt{3-x+2 x^2}}{6144}+\frac{2116475 x^5 \sqrt{3-x+2 x^2}}{10752}+\frac{57375}{448} x^6 \sqrt{3-x+2 x^2}+\frac{625}{16} x^7 \sqrt{3-x+2 x^2}-\frac{2899366573 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{8388608}\\ &=\frac{16493087661 \sqrt{3-x+2 x^2}}{29360128}+\frac{1572007407 x \sqrt{3-x+2 x^2}}{7340032}-\frac{15428243 x^2 \sqrt{3-x+2 x^2}}{131072}-\frac{19750457 x^3 \sqrt{3-x+2 x^2}}{229376}+\frac{686531 x^4 \sqrt{3-x+2 x^2}}{6144}+\frac{2116475 x^5 \sqrt{3-x+2 x^2}}{10752}+\frac{57375}{448} x^6 \sqrt{3-x+2 x^2}+\frac{625}{16} x^7 \sqrt{3-x+2 x^2}-\frac{2899366573 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{8388608 \sqrt{46}}\\ &=\frac{16493087661 \sqrt{3-x+2 x^2}}{29360128}+\frac{1572007407 x \sqrt{3-x+2 x^2}}{7340032}-\frac{15428243 x^2 \sqrt{3-x+2 x^2}}{131072}-\frac{19750457 x^3 \sqrt{3-x+2 x^2}}{229376}+\frac{686531 x^4 \sqrt{3-x+2 x^2}}{6144}+\frac{2116475 x^5 \sqrt{3-x+2 x^2}}{10752}+\frac{57375}{448} x^6 \sqrt{3-x+2 x^2}+\frac{625}{16} x^7 \sqrt{3-x+2 x^2}+\frac{2899366573 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8388608 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.246325, size = 75, normalized size = 0.41 \[ \frac{4 \sqrt{2 x^2-x+3} \left (3440640000 x^7+11280384000 x^6+17338163200 x^5+9842108416 x^4-7584175488 x^3-10367779296 x^2+18864088884 x+49479262983\right )[phone] \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{352321536} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(49479262983 + 18864088884*x - 10367779296*x^2 - 7584175488*x^3 + 9842108416*x^4 + 1733
8163200*x^5 + 11280384000*x^6 + 3440640000*x^7) + 60886698033*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/352321536

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Maple [A]  time = 0.065, size = 147, normalized size = 0.8 \begin{align*}{\frac{625\,{x}^{7}}{16}\sqrt{2\,{x}^{2}-x+3}}+{\frac{2116475\,{x}^{5}}{10752}\sqrt{2\,{x}^{2}-x+3}}+{\frac{686531\,{x}^{4}}{6144}\sqrt{2\,{x}^{2}-x+3}}-{\frac{2899366573\,\sqrt{2}}{16777216}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{19750457\,{x}^{3}}{229376}\sqrt{2\,{x}^{2}-x+3}}-{\frac{15428243\,{x}^{2}}{131072}\sqrt{2\,{x}^{2}-x+3}}+{\frac{57375\,{x}^{6}}{448}\sqrt{2\,{x}^{2}-x+3}}+{\frac{1572007407\,x}{7340032}\sqrt{2\,{x}^{2}-x+3}}+{\frac{16493087661}{29360128}\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)^(1/2),x)

[Out]

625/16*x^7*(2*x^2-x+3)^(1/2)+2116475/10752*x^5*(2*x^2-x+3)^(1/2)+686531/6144*x^4*(2*x^2-x+3)^(1/2)-2899366573/
16777216*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))-19750457/229376*x^3*(2*x^2-x+3)^(1/2)-15428243/131072*x^2*(2*x
^2-x+3)^(1/2)+57375/448*x^6*(2*x^2-x+3)^(1/2)[phone]/7340032*x*(2*x^2-x+3)^(1/2)[phone]/29360128*(2*x
^2-x+3)^(1/2)

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Maxima [A]  time = 1.53155, size = 200, normalized size = 1.08 \begin{align*} \frac{625}{16} \, \sqrt{2 \, x^{2} - x + 3} x^{7} + \frac{57375}{448} \, \sqrt{2 \, x^{2} - x + 3} x^{6} + \frac{2116475}{10752} \, \sqrt{2 \, x^{2} - x + 3} x^{5} + \frac{686531}{6144} \, \sqrt{2 \, x^{2} - x + 3} x^{4} - \frac{19750457}{229376} \, \sqrt{2 \, x^{2} - x + 3} x^{3} - \frac{15428243}{131072} \, \sqrt{2 \, x^{2} - x + 3} x^{2} + \frac{1572007407}{7340032} \, \sqrt{2 \, x^{2} - x + 3} x - \frac{2899366573}{16777216} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{16493087661}{29360128} \, \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]

625/16*sqrt(2*x^2 - x + 3)*x^7 + 57375/448*sqrt(2*x^2 - x + 3)*x^6 + 2116475/10752*sqrt(2*x^2 - x + 3)*x^5 + 6
86531/6144*sqrt(2*x^2 - x + 3)*x^4 - 19750457/229376*sqrt(2*x^2 - x + 3)*x^3 - 15428243/131072*sqrt(2*x^2 - x
+ 3)*x^2 + 1572007407/7340032*sqrt(2*x^2 - x + 3)*x - 2899366573/16777216*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x -
1)) + 16493087661/29360128*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.36022, size = 355, normalized size = 1.92 \begin{align*} \frac{1}{88080384} \,{\left (3440640000 \, x^{7} + 11280384000 \, x^{6} + 17338163200 \, x^{5} + 9842108416 \, x^{4} - 7584175488 \, x^{3} - 10367779296 \, x^{2} + 18864088884 \, x + 49479262983\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{2899366573}{33554432} \, \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]

1/88080384*(3440640000*x^7 + 11280384000*x^6 + 17338163200*x^5 + 9842108416*x^4 - 7584175488*x^3 - 10367779296
*x^2 + 18864088884*x + 49479262983)*sqrt(2*x^2 - x + 3) + 2899366573/33554432*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 \frac{\left (5 x^{2} + 3 x + 2\right )^{4}}{\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)**4/(2*x**2-x+3)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.17971, size = 112, normalized size = 0.61 \begin{align*} \frac{1}{88080384} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \,{\left (120 \,{\left (140 \, x + 459\right )} x + 84659\right )} x + 4805717\right )} x - 59251371\right )} x - 323993103\right )} x + 4716022221\right )} x + 49479262983\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{2899366573}{16777216} \, \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]

1/88080384*(4*(8*(4*(16*(100*(120*(140*x + 459)*x + 84659)*x + 4805717)*x - 59251371)*x - 323993103)*x + 47160
22221)*x + 49479262983)*sqrt(2*x^2 - x + 3) + 2899366573/16777216*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x
^2 - x + 3)) + 1)