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

Optimal. Leaf size=231 \[ \frac{625}{24} \left (2 x^2-x+3\right )^{5/2} x^7+\frac{7625}{96} \left (2 x^2-x+3\right )^{5/2} x^6+\frac{95165}{768} \left (2 x^2-x+3\right )^{5/2} x^5+\frac{941905 \left (2 x^2-x+3\right )^{5/2} x^4}{9216}+\frac{10444117 \left (2 x^2-x+3\right )^{5/2} x^3}{294912}-\frac{56422489 \left (2 x^2-x+3\right )^{5/2} x^2}{8257536}+\frac{48669967 \left (2 x^2-x+3\right )^{5/2} x}{22020096}+\frac{2124689283 \left (2 x^2-x+3\right )^{5/2}}{146800640}-\frac{382121949 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{134217728}-\frac{26366414481 (1-4 x) \sqrt{2 x^2-x+3}}{2147483648}-\frac{606427533063 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4294967296 \sqrt{2}} \]

[Out]

(-26366414481*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/2147483648 - (382121949*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/13421772
8 + (2124689283*(3 - x + 2*x^2)^(5/2))/146800640 + (48669967*x*(3 - x + 2*x^2)^(5/2))/22020096 - (56422489*x^2
*(3 - x + 2*x^2)^(5/2))/8257536 + (10444117*x^3*(3 - x + 2*x^2)^(5/2))/294912 + (941905*x^4*(3 - x + 2*x^2)^(5
/2))/9216 + (95165*x^5*(3 - x + 2*x^2)^(5/2))/768 + (7625*x^6*(3 - x + 2*x^2)^(5/2))/96 + (625*x^7*(3 - x + 2*
x^2)^(5/2))/24 - (606427533063*ArcSinh[(1 - 4*x)/Sqrt[23]])/(4294967296*Sqrt[2])

________________________________________________________________________________________

Rubi [A]  time = 0.342448, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{625}{24} \left (2 x^2-x+3\right )^{5/2} x^7+\frac{7625}{96} \left (2 x^2-x+3\right )^{5/2} x^6+\frac{95165}{768} \left (2 x^2-x+3\right )^{5/2} x^5+\frac{941905 \left (2 x^2-x+3\right )^{5/2} x^4}{9216}+\frac{10444117 \left (2 x^2-x+3\right )^{5/2} x^3}{294912}-\frac{56422489 \left (2 x^2-x+3\right )^{5/2} x^2}{8257536}+\frac{48669967 \left (2 x^2-x+3\right )^{5/2} x}{22020096}+\frac{2124689283 \left (2 x^2-x+3\right )^{5/2}}{146800640}-\frac{382121949 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{134217728}-\frac{26366414481 (1-4 x) \sqrt{2 x^2-x+3}}{2147483648}-\frac{606427533063 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4294967296 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-26366414481*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/2147483648 - (382121949*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/13421772
8 + (2124689283*(3 - x + 2*x^2)^(5/2))/146800640 + (48669967*x*(3 - x + 2*x^2)^(5/2))/22020096 - (56422489*x^2
*(3 - x + 2*x^2)^(5/2))/8257536 + (10444117*x^3*(3 - x + 2*x^2)^(5/2))/294912 + (941905*x^4*(3 - x + 2*x^2)^(5
/2))/9216 + (95165*x^5*(3 - x + 2*x^2)^(5/2))/768 + (7625*x^6*(3 - x + 2*x^2)^(5/2))/96 + (625*x^7*(3 - x + 2*
x^2)^(5/2))/24 - (606427533063*ArcSinh[(1 - 4*x)/Sqrt[23]])/(4294967296*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 \left (3-x+2 x^2\right )^{3/2} \left (2+3 x+5 x^2\right )^4 \, dx &=\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{24} \int \left (3-x+2 x^2\right )^{3/2} \left (384+2304 x+9024 x^2+22464 x^3+42264 x^4+56160 x^5+43275 x^6+\frac{83875 x^7}{2}\right ) \, dx\\ &=\frac{7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{528} \int \left (3-x+2 x^2\right )^{3/2} \left (8448+50688 x+198528 x^2+494208 x^3+929808 x^4+480645 x^5+\frac{5234075 x^6}{4}\right ) \, dx\\ &=\frac{95165}{768} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (3-x+2 x^2\right )^{3/2} \left (168960+1013760 x+3970560 x^2+9884160 x^3-\frac{4126485 x^4}{4}+\frac{155414325 x^5}{8}\right ) \, dx}{10560}\\ &=\frac{941905 x^4 \left (3-x+2 x^2\right )^{5/2}}{9216}+\frac{95165}{768} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (3-x+2 x^2\right )^{3/2} \left (3041280+18247680 x+71470080 x^2-\frac{110413215 x^3}{2}+\frac{1723279305 x^4}{16}\right ) \, dx}{190080}\\ &=\frac{10444117 x^3 \left (3-x+2 x^2\right )^{5/2}}{294912}+\frac{941905 x^4 \left (3-x+2 x^2\right )^{5/2}}{9216}+\frac{95165}{768} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (3-x+2 x^2\right )^{3/2} \left (48660480+291962880 x+\frac{2786826735 x^2}{16}-\frac{9309710685 x^3}{32}\right ) \, dx}{3041280}\\ &=-\frac{56422489 x^2 \left (3-x+2 x^2\right )^{5/2}}{8257536}+\frac{10444117 x^3 \left (3-x+2 x^2\right )^{5/2}}{294912}+\frac{941905 x^4 \left (3-x+2 x^2\right )^{5/2}}{9216}+\frac{95165}{768} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (3-x+2 x^2\right )^{3/2} \left (681246720+\frac{93328817175 x}{16}+\frac{72274900995 x^2}{64}\right ) \, dx}{42577920}\\ &=\frac{48669967 x \left (3-x+2 x^2\right )^{5/2}}{22020096}-\frac{56422489 x^2 \left (3-x+2 x^2\right )^{5/2}}{8257536}+\frac{10444117 x^3 \left (3-x+2 x^2\right )^{5/2}}{294912}+\frac{941905 x^4 \left (3-x+2 x^2\right )^{5/2}}{9216}+\frac{95165}{768} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (\frac{306372777975}{64}+\frac{9465490755765 x}{128}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx}{510935040}\\ &=\frac{2124689283 \left (3-x+2 x^2\right )^{5/2}}{146800640}+\frac{48669967 x \left (3-x+2 x^2\right )^{5/2}}{22020096}-\frac{56422489 x^2 \left (3-x+2 x^2\right )^{5/2}}{8257536}+\frac{10444117 x^3 \left (3-x+2 x^2\right )^{5/2}}{294912}+\frac{941905 x^4 \left (3-x+2 x^2\right )^{5/2}}{9216}+\frac{95165}{768} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}+\frac{382121949 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{8388608}\\ &=-\frac{382121949 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{134217728}+\frac{2124689283 \left (3-x+2 x^2\right )^{5/2}}{146800640}+\frac{48669967 x \left (3-x+2 x^2\right )^{5/2}}{22020096}-\frac{56422489 x^2 \left (3-x+2 x^2\right )^{5/2}}{8257536}+\frac{10444117 x^3 \left (3-x+2 x^2\right )^{5/2}}{294912}+\frac{941905 x^4 \left (3-x+2 x^2\right )^{5/2}}{9216}+\frac{95165}{768} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}+\frac{26366414481 \int \sqrt{3-x+2 x^2} \, dx}{268435456}\\ &=-\frac{26366414481 (1-4 x) \sqrt{3-x+2 x^2}}{2147483648}-\frac{382121949 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{134217728}+\frac{2124689283 \left (3-x+2 x^2\right )^{5/2}}{146800640}+\frac{48669967 x \left (3-x+2 x^2\right )^{5/2}}{22020096}-\frac{56422489 x^2 \left (3-x+2 x^2\right )^{5/2}}{8257536}+\frac{10444117 x^3 \left (3-x+2 x^2\right )^{5/2}}{294912}+\frac{941905 x^4 \left (3-x+2 x^2\right )^{5/2}}{9216}+\frac{95165}{768} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}+\frac{606427533063 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{4294967296}\\ &=-\frac{26366414481 (1-4 x) \sqrt{3-x+2 x^2}}{2147483648}-\frac{382121949 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{134217728}+\frac{2124689283 \left (3-x+2 x^2\right )^{5/2}}{146800640}+\frac{48669967 x \left (3-x+2 x^2\right )^{5/2}}{22020096}-\frac{56422489 x^2 \left (3-x+2 x^2\right )^{5/2}}{8257536}+\frac{10444117 x^3 \left (3-x+2 x^2\right )^{5/2}}{294912}+\frac{941905 x^4 \left (3-x+2 x^2\right )^{5/2}}{9216}+\frac{95165}{768} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}+\frac{\left (26366414481 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{4294967296}\\ &=-\frac{26366414481 (1-4 x) \sqrt{3-x+2 x^2}}{2147483648}-\frac{382121949 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{134217728}+\frac{2124689283 \left (3-x+2 x^2\right )^{5/2}}{146800640}+\frac{48669967 x \left (3-x+2 x^2\right )^{5/2}}{22020096}-\frac{56422489 x^2 \left (3-x+2 x^2\right )^{5/2}}{8257536}+\frac{10444117 x^3 \left (3-x+2 x^2\right )^{5/2}}{294912}+\frac{941905 x^4 \left (3-x+2 x^2\right )^{5/2}}{9216}+\frac{95165}{768} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{7625}{96} x^6 \left (3-x+2 x^2\right )^{5/2}+\frac{625}{24} x^7 \left (3-x+2 x^2\right )^{5/2}-\frac{606427533063 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{4294967296 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.366561, size = 95, normalized size = 0.41 \[ \frac{4 \sqrt{2 x^2-x+3} \left (70464307200000 x^{11}[phone] x^{10}[phone] x^9+534038708224000 x^8+745133229998080 x^7+765087080448000 x^6+675479464714240 x^5+451581382260736 x^4+239021184223104 x^3+65151998063712 x^2+12971175524316 x+74032009514181\right )-191024672914845 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2705829396480} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(74032009514181 + 12971175524316*x + 65151998063712*x^2 + 239021184223104*x^3 + 4515813
82260736*x^4 + 675479464714240*x^5 + 765087080448000*x^6 + 745133229998080*x^7 + 534038708224000*x^8 + 3493796
51174400*x^9 + 144451829760000*x^10 + 70464307200000*x^11) - 191024672914845*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23
]])/2705829396480

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Maple [A]  time = 0.072, size = 185, normalized size = 0.8 \begin{align*}{\frac{7625\,{x}^{6}}{96} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{95165\,{x}^{5}}{768} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{941905\,{x}^{4}}{9216} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{10444117\,{x}^{3}}{294912} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{56422489\,{x}^{2}}{8257536} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{48669967\,x}{22020096} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-26366414481+105465657924\,x}{2147483648}\sqrt{2\,{x}^{2}-x+3}}+{\frac{606427533063\,\sqrt{2}}{8589934592}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-382121949+1528487796\,x}{134217728} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{2124689283}{146800640} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{625\,{x}^{7}}{24} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7625/96*x^6*(2*x^2-x+3)^(5/2)+95165/768*x^5*(2*x^2-x+3)^(5/2)+941905/9216*x^4*(2*x^2-x+3)^(5/2)[phone]/29491
2*x^3*(2*x^2-x+3)^(5/2)-56422489/8257536*x^2*(2*x^2-x+3)^(5/2)[phone]/22020096*x*(2*x^2-x+3)^(5/2)[phone]
81/2147483648*(-1+4*x)*(2*x^2-x+3)^(1/2)[phone]/8589934592*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+382121
949/134217728*(-1+4*x)*(2*x^2-x+3)^(3/2)[phone]/146800640*(2*x^2-x+3)^(5/2)+625/24*x^7*(2*x^2-x+3)^(5/2)

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Maxima [A]  time = 1.54572, size = 278, normalized size = 1.2 \begin{align*} \frac{625}{24} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{7} + \frac{7625}{96} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{6} + \frac{95165}{768} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{5} + \frac{941905}{9216} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{4} + \frac{10444117}{294912} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} - \frac{56422489}{8257536} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} + \frac{48669967}{22020096} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{2124689283}{146800640} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{382121949}{33554432} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{382121949}{134217728} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{26366414481}{536870912} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{606427533063}{8589934592} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{26366414481}{2147483648} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^4,x, algorithm="maxima")

[Out]

625/24*(2*x^2 - x + 3)^(5/2)*x^7 + 7625/96*(2*x^2 - x + 3)^(5/2)*x^6 + 95165/768*(2*x^2 - x + 3)^(5/2)*x^5 + 9
41905/9216*(2*x^2 - x + 3)^(5/2)*x^4 + 10444117/294912*(2*x^2 - x + 3)^(5/2)*x^3 - 56422489/8257536*(2*x^2 - x
+ 3)^(5/2)*x^2 + 48669967/22020096*(2*x^2 - x + 3)^(5/2)*x + 2124689283/146800640*(2*x^2 - x + 3)^(5/2) + 382
121949/33554432*(2*x^2 - x + 3)^(3/2)*x - 382121949/134217728*(2*x^2 - x + 3)^(3/2) + 26366414481/536870912*sq
rt(2*x^2 - x + 3)*x + 606427533063/8589934592*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 26366414481/214748364
8*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.58455, size = 532, normalized size = 2.3 \begin{align*} \frac{1}{676457349120} \,{\left (70464307200000 \, x^{11} + 144451829760000 \, x^{10} + 349379651174400 \, x^{9} + 534038708224000 \, x^{8} + 745133229998080 \, x^{7} + 765087080448000 \, x^{6} + 675479464714240 \, x^{5} + 451581382260736 \, x^{4} + 239021184223104 \, x^{3} + 65151998063712 \, x^{2} + 12971175524316 \, x + 74032009514181\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{606427533063}{17179869184} \, \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((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^4,x, algorithm="fricas")

[Out]

1/676457349120*(70464307200000*x^11 + 144451829760000*x^10 + 349379651174400*x^9 + 534038708224000*x^8 + 74513
3229998080*x^7 + 765087080448000*x^6 + 675479464714240*x^5 + 451581382260736*x^4 + 239021184223104*x^3 + 65151
998063712*x^2 + 12971175524316*x + 74032009514181)*sqrt(2*x^2 - x + 3) + 606427533063/17179869184*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 \left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17427, size = 139, normalized size = 0.6 \begin{align*} \frac{1}{676457349120} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (20 \,{\left (8 \,{\left (28 \,{\left (160 \,{\left (12 \,{\left (200 \,{\left (20 \, x + 41\right )} x + 19833\right )} x + 363785\right )} x + 81213077\right )} x + 2334860475\right )} x + 16491197869\right )} x + 220498721807\right )} x + 1867353001743\right )} x + 2035999939491\right )} x + 3242793881079\right )} x + 74032009514181\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{606427533063}{8589934592} \, \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((2*x^2-x+3)^(3/2)*(5*x^2+3*x+2)^4,x, algorithm="giac")

[Out]

1/676457349120*(4*(8*(4*(16*(20*(8*(28*(160*(12*(200*(20*x + 41)*x + 19833)*x + 363785)*x + 81213077)*x + 2334
860475)*x + 16491197869)*x + 220498721807)*x + 1867353001743)*x + 2035999939491)*x + 3242793881079)*x + 740320
09514181)*sqrt(2*x^2 - x + 3) - 606427533063/8589934592*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3
)) + 1)