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

Optimal. Leaf size=254 \[ \frac{625}{28} \left (2 x^2-x+3\right )^{7/2} x^7+\frac{13875}{208} \left (2 x^2-x+3\right )^{7/2} x^6+\frac{1046225 \left (2 x^2-x+3\right )^{7/2} x^5}{9984}+\frac{3684995 \left (2 x^2-x+3\right )^{7/2} x^4}{39936}+\frac{23460839 \left (2 x^2-x+3\right )^{7/2} x^3}{532480}+\frac{122595067 \left (2 x^2-x+3\right )^{7/2} x^2}{19169280}+\frac{112244125 \left (2 x^2-x+3\right )^{7/2} x}{122683392}+\frac{25250178739 \left (2 x^2-x+3\right )^{7/2}}{5725224960}-\frac{401135647 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{335544320}-\frac{9226119881 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2147483648}-\frac{636602271789 (1-4 x) \sqrt{2 x^2-x+3}}{34359738368}-\frac{14641852251147 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{68719476736 \sqrt{2}} \]

[Out]

(-636602271789*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/34359738368 - (9226119881*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/21474
83648 - (401135647*(1 - 4*x)*(3 - x + 2*x^2)^(5/2))/335544320 + (25250178739*(3 - x + 2*x^2)^(7/2))/5725224960
+ (112244125*x*(3 - x + 2*x^2)^(7/2))/122683392 + (122595067*x^2*(3 - x + 2*x^2)^(7/2))/19169280 + (23460839*
x^3*(3 - x + 2*x^2)^(7/2))/532480 + (3684995*x^4*(3 - x + 2*x^2)^(7/2))/39936 + (1046225*x^5*(3 - x + 2*x^2)^(
7/2))/9984 + (13875*x^6*(3 - x + 2*x^2)^(7/2))/208 + (625*x^7*(3 - x + 2*x^2)^(7/2))/28 - (14641852251147*ArcS
inh[(1 - 4*x)/Sqrt[23]])/(68719476736*Sqrt[2])

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Rubi [A]  time = 0.372556, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 13, 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} \[ \frac{625}{28} \left (2 x^2-x+3\right )^{7/2} x^7+\frac{13875}{208} \left (2 x^2-x+3\right )^{7/2} x^6+\frac{1046225 \left (2 x^2-x+3\right )^{7/2} x^5}{9984}+\frac{3684995 \left (2 x^2-x+3\right )^{7/2} x^4}{39936}+\frac{23460839 \left (2 x^2-x+3\right )^{7/2} x^3}{532480}+\frac{122595067 \left (2 x^2-x+3\right )^{7/2} x^2}{19169280}+\frac{112244125 \left (2 x^2-x+3\right )^{7/2} x}{122683392}+\frac{25250178739 \left (2 x^2-x+3\right )^{7/2}}{5725224960}-\frac{401135647 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{335544320}-\frac{9226119881 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2147483648}-\frac{636602271789 (1-4 x) \sqrt{2 x^2-x+3}}{34359738368}-\frac{14641852251147 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{68719476736 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-636602271789*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/34359738368 - (9226119881*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/21474
83648 - (401135647*(1 - 4*x)*(3 - x + 2*x^2)^(5/2))/335544320 + (25250178739*(3 - x + 2*x^2)^(7/2))/5725224960
+ (112244125*x*(3 - x + 2*x^2)^(7/2))/122683392 + (122595067*x^2*(3 - x + 2*x^2)^(7/2))/19169280 + (23460839*
x^3*(3 - x + 2*x^2)^(7/2))/532480 + (3684995*x^4*(3 - x + 2*x^2)^(7/2))/39936 + (1046225*x^5*(3 - x + 2*x^2)^(
7/2))/9984 + (13875*x^6*(3 - x + 2*x^2)^(7/2))/208 + (625*x^7*(3 - x + 2*x^2)^(7/2))/28 - (14641852251147*ArcS
inh[(1 - 4*x)/Sqrt[23]])/(68719476736*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 )^{5/2} \left (2+3 x+5 x^2\right )^4 \, dx &=\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{1}{28} \int \left (3-x+2 x^2\right )^{5/2} \left (448+2688 x+10528 x^2+26208 x^3+49308 x^4+65520 x^5+52675 x^6+\frac{97125 x^7}{2}\right ) \, dx\\ &=\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{1}{728} \int \left (3-x+2 x^2\right )^{5/2} \left (11648+69888 x+273728 x^2+681408 x^3+1282008 x^4+829395 x^5+\frac{7323575 x^6}{4}\right ) \, dx\\ &=\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (3-x+2 x^2\right )^{5/2} \left (279552+1677312 x+6569472 x^2+16353792 x^3+\frac{13219143 x^4}{4}+\frac{283744615 x^5}{8}\right ) \, dx}{17472}\\ &=\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (3-x+2 x^2\right )^{5/2} \left (6150144+36900864 x+144528384 x^2-\frac{131666997 x^3}{2}+\frac{5419453809 x^4}{16}\right ) \, dx}{384384}\\ &=\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (3-x+2 x^2\right )^{5/2} \left (123002880+738017280 x-\frac{2526001401 x^2}{16}+\frac{28319460477 x^3}{32}\right ) \, dx}{7687680}\\ &=\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (3-x+2 x^2\right )^{5/2} \left (2214051840+\frac{127590595209 x}{16}+\frac{129641964375 x^2}{64}\right ) \, dx}{138378240}\\ &=\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (\frac{1878263191035}{64}+\frac{17498373866127 x}{128}\right ) \left (3-x+2 x^2\right )^{5/2} \, dx}{2214051840}\\ &=\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{1203406941 \int \left (3-x+2 x^2\right )^{5/2} \, dx}{41943040}\\ &=-\frac{401135647 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{335544320}+\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{9226119881 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{134217728}\\ &=-\frac{9226119881 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2147483648}-\frac{401135647 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{335544320}+\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{636602271789 \int \sqrt{3-x+2 x^2} \, dx}{4294967296}\\ &=-\frac{636602271789 (1-4 x) \sqrt{3-x+2 x^2}}{34359738368}-\frac{9226119881 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2147483648}-\frac{401135647 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{335544320}+\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{14641852251147 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{68719476736}\\ &=-\frac{636602271789 (1-4 x) \sqrt{3-x+2 x^2}}{34359738368}-\frac{9226119881 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2147483648}-\frac{401135647 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{335544320}+\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\left (636602271789 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{68719476736}\\ &=-\frac{636602271789 (1-4 x) \sqrt{3-x+2 x^2}}{34359738368}-\frac{9226119881 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2147483648}-\frac{401135647 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{335544320}+\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}-\frac{14641852251147 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{68719476736 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.453735, size = 105, normalized size = 0.41 \[ \frac{4 \sqrt{2 x^2-x+3} \left (25125558681600000 x^{13}+37398427729920000 x^{12}+137233466130432000 x^{11}+204932411660697600 x^{10}+363646430503501824 x^9+439064558846345216 x^8+530502956133122048 x^7+485091164642279424 x^6+405468382284161024 x^5+257786732552566784 x^4+142490931553577856 x^3+50064174038215008 x^2+12071614275862524 x+10820567498568669\right )-59958384968446965 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{562812514467840} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(10820567498568669 + 12071614275862524*x + 50064174038215008*x^2 + 142490931553577856*x
^3 + 257786732552566784*x^4 + 405468382284161024*x^5 + 485091164642279424*x^6 + 530502956133122048*x^7 + 43906
4558846345216*x^8 + 363646430503501824*x^9 + 204932411660697600*x^10 + 137233466130432000*x^11 + 3739842772992
0000*x^12 + 25125558681600000*x^13) - 59958384968446965*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/562812514467840

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Maple [A]  time = 0.08, size = 204, normalized size = 0.8 \begin{align*}{\frac{1046225\,{x}^{5}}{9984} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{3684995\,{x}^{4}}{39936} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{23460839\,{x}^{3}}{532480} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{122595067\,{x}^{2}}{19169280} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{112244125\,x}{122683392} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{-636602271789+2546409087156\,x}{34359738368}\sqrt{2\,{x}^{2}-x+3}}+{\frac{14641852251147\,\sqrt{2}}{137438953472}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-401135647+1604542588\,x}{335544320} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-9226119881+36904479524\,x}{2147483648} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{25250178739}{5725224960} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{13875\,{x}^{6}}{208} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{625\,{x}^{7}}{28} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1046225/9984*x^5*(2*x^2-x+3)^(7/2)+3684995/39936*x^4*(2*x^2-x+3)^(7/2)[phone]/532480*x^3*(2*x^2-x+3)^(7/2)+1
22595067/19169280*x^2*(2*x^2-x+3)^(7/2)[phone]/122683392*x*(2*x^2-x+3)^(7/2)[phone]/34359738368*(-1+4
*x)*(2*x^2-x+3)^(1/2)[phone]/137438953472*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))[phone]/335544320*(
-1+4*x)*(2*x^2-x+3)^(5/2)[phone]/2147483648*(-1+4*x)*(2*x^2-x+3)^(3/2)[phone]/5725224960*(2*x^2-x+3)^
(7/2)+13875/208*x^6*(2*x^2-x+3)^(7/2)+625/28*x^7*(2*x^2-x+3)^(7/2)

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Maxima [A]  time = 1.53405, size = 317, normalized size = 1.25 \begin{align*} \frac{625}{28} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{7} + \frac{13875}{208} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{6} + \frac{1046225}{9984} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{5} + \frac{3684995}{39936} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{4} + \frac{23460839}{532480} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{3} + \frac{122595067}{19169280} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{2} + \frac{112244125}{122683392} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x + \frac{25250178739}{5725224960} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{401135647}{83886080} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{401135647}{335544320} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{9226119881}{536870912} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{9226119881}{2147483648} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{636602271789}{8589934592} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{14641852251147}{137438953472} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{636602271789}{34359738368} \, \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)^(5/2)*(5*x^2+3*x+2)^4,x, algorithm="maxima")

[Out]

625/28*(2*x^2 - x + 3)^(7/2)*x^7 + 13875/208*(2*x^2 - x + 3)^(7/2)*x^6 + 1046225/9984*(2*x^2 - x + 3)^(7/2)*x^
5 + 3684995/39936*(2*x^2 - x + 3)^(7/2)*x^4 + 23460839/532480*(2*x^2 - x + 3)^(7/2)*x^3 + 122595067/19169280*(
2*x^2 - x + 3)^(7/2)*x^2 + 112244125/122683392*(2*x^2 - x + 3)^(7/2)*x + 25250178739/5725224960*(2*x^2 - x + 3
)^(7/2) + 401135647/83886080*(2*x^2 - x + 3)^(5/2)*x - 401135647/335544320*(2*x^2 - x + 3)^(5/2) + 9226119881/
536870912*(2*x^2 - x + 3)^(3/2)*x - 9226119881/2147483648*(2*x^2 - x + 3)^(3/2) + 636602271789/8589934592*sqrt
(2*x^2 - x + 3)*x + 14641852251147/137438953472*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 636602271789/343597
38368*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.40691, size = 657, normalized size = 2.59 \begin{align*} \frac{1}{140703128616960} \,{\left (25125558681600000 \, x^{13} + 37398427729920000 \, x^{12} + 137233466130432000 \, x^{11} + 204932411660697600 \, x^{10} + 363646430503501824 \, x^{9} + 439064558846345216 \, x^{8} + 530502956133122048 \, x^{7} + 485091164642279424 \, x^{6} + 405468382284161024 \, x^{5} + 257786732552566784 \, x^{4} + 142490931553577856 \, x^{3} + 50064174038215008 \, x^{2} + 12071614275862524 \, x + 10820567498568669\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{14641852251147}{274877906944} \, \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)^(5/2)*(5*x^2+3*x+2)^4,x, algorithm="fricas")

[Out]

1/140703128616960*(25125558681600000*x^13 + 37398427729920000*x^12 + 137233466130432000*x^11 + 204932411660697
600*x^10 + 363646430503501824*x^9 + 439064558846345216*x^8 + 530502956133122048*x^7 + 485091164642279424*x^6 +
405468382284161024*x^5 + 257786732552566784*x^4 + 142490931553577856*x^3 + 50064174038215008*x^2 + 1207161427
5862524*x + 10820567498568669)*sqrt(2*x^2 - x + 3) + 14641852251147/274877906944*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{5}{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)**(5/2)*(5*x**2+3*x+2)**4,x)

[Out]

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

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Giac [A]  time = 1.1553, size = 153, normalized size = 0.6 \begin{align*} \frac{1}{140703128616960} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (32 \,{\left (12 \,{\left (200 \,{\left (20 \,{\left (240 \,{\left (260 \, x + 387\right )} x + 340823\right )} x + 10179103\right )} x + 3612502719\right )} x + 52340574127\right )} x + 2023708176167\right )} x + 7401903757359\right )} x + 49495652134297\right )} x + 125872428004183\right )} x + 1113210402762327\right )} x + 1564505438694219\right )} x + 3017903568965631\right )} x + 10820567498568669\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{14641852251147}{137438953472} \, \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)^(5/2)*(5*x^2+3*x+2)^4,x, algorithm="giac")

[Out]

1/140703128616960*(4*(8*(4*(16*(4*(8*(4*(32*(12*(200*(20*(240*(260*x + 387)*x + 340823)*x + 10179103)*x + 3612
502719)*x + 52340574127)*x + 2023708176167)*x + 7401903757359)*x + 49495652134297)*x + 125872428004183)*x + 11
13210402762327)*x + 1564505438694219)*x + 3017903568965631)*x + 10820567498568669)*sqrt(2*x^2 - x + 3) - 14641
852251147/137438953472*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)