3.1.72 \(\int (3-x+2 x^2)^{5/2} (2+3 x+5 x^2)^4 \, dx\) [72]

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

[Out]

-9226119881/2147483648*(1-4*x)*(2*x^2-x+3)^(3/2)-401135647/335544320*(1-4*x)*(2*x^2-x+3)^(5/2)[phone]/572
5224960*(2*x^2-x+3)^(7/2)[phone]/122683392*x*(2*x^2-x+3)^(7/2)[phone]/19169280*x^2*(2*x^2-x+3)^(7/2)+234
60839/532480*x^3*(2*x^2-x+3)^(7/2)+3684995/39936*x^4*(2*x^2-x+3)^(7/2)+1046225/9984*x^5*(2*x^2-x+3)^(7/2)+1387
5/208*x^6*(2*x^2-x+3)^(7/2)+625/28*x^7*(2*x^2-x+3)^(7/2)-14641852251147/137438953472*arcsinh(1/23*(1-4*x)*23^(
1/2))*2^(1/2)-636602271789/34359738368*(1-4*x)*(2*x^2-x+3)^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 254, normalized size of antiderivative = 1.00, 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 = {1675, 654, 626, 633, 221} \begin {gather*} \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 {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 {14641852251147 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{68719476736 \sqrt {2}} \end {gather*}

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 221

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]

Rule 626

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 633

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 654

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 1675

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]

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 ) \text {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*}

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Mathematica [A]
time = 1.28, size = 115, normalized size = 0.45 \begin {gather*} \frac {4 \sqrt {3-x+2 x^2} \left (10820567498568669+12071614275862524 x+50064174038215008 x^2+142490931553577856 x^3+257786732552566784 x^4+405468382284161024 x^5+485091164642279424 x^6+530502956133122048 x^7+439064558846345216 x^8+363646430503501824 x^9+204932411660697600 x^{10}+137233466130432000 x^{11}+37398427729920000 x^{12}+25125558681600000 x^{13}\right )-59958384968446965 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{562812514467840} \end {gather*}

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]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/562812
514467840

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Maple [A]
time = 0.14, size = 204, normalized size = 0.80

method	result	size
risch	\(\frac {\left (25125558681600000 x^{13}+37398427729920000 x^{12}+137233466130432000 x^{11}+204932411660697600 x^{10}+363646430503501824 x^{9}+439064558846345216 x^{8}+530502956133122048 x^{7}+485091164642279424 x^{6}+405468382284161024 x^{5}+257786732552566784 x^{4}+142490931553577856 x^{3}+50064174038215008 x^{2}+12071614275862524 x +10820567498568669\right ) \sqrt {2 x^{2}-x +3}}{140703128616960}+\frac {14641852251147 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{137438953472}\)	\(95\)
trager	\(\left (\frac {1250}{7} x^{13}+\frac {48375}{182} x^{12}+\frac {1217225}{1248} x^{11}+\frac {50895515}{34944} x^{10}+\frac {172023939}{66560} x^{9}+\frac {52340574127}{16773120} x^{8}+\frac {2023708176167}{536739840} x^{7}+\frac {2467301252453}{715653120} x^{6}+\frac {49495652134297}{17175674880} x^{5}+\frac {17981775429169}{9814671360} x^{4}+\frac {371070134254109}{366414397440} x^{3}+\frac {24833419661813}{69793218560} x^{2}+\frac {335322618773959}{3908420239360} x +\frac {1202285277618741}{15633680957440}\right ) \sqrt {2 x^{2}-x +3}+\frac {14641852251147 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{137438953472}\)	\(121\)
default	\(\frac {25250178739 \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{5725224960}+\frac {401135647 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{335544320}+\frac {636602271789 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{34359738368}+\frac {14641852251147 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{137438953472}+\frac {9226119881 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{2147483648}+\frac {112244125 x \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{122683392}+\frac {122595067 x^{2} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{19169280}+\frac {23460839 x^{3} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{532480}+\frac {3684995 x^{4} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{39936}+\frac {1046225 x^{5} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{9984}+\frac {13875 x^{6} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{208}+\frac {625 x^{7} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{28}\)	\(204\)

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,method=_RETURNVERBOSE)

[Out]

25250178739/5725224960*(2*x^2-x+3)^(7/2)[phone]/335544320*(4*x-1)*(2*x^2-x+3)^(5/2)[phone]/3435973836
8*(4*x-1)*(2*x^2-x+3)^(1/2)[phone]/137438953472*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))[phone]/2147
483648*(4*x-1)*(2*x^2-x+3)^(3/2)[phone]/122683392*x*(2*x^2-x+3)^(7/2)[phone]/19169280*x^2*(2*x^2-x+3)^(7
/2)[phone]/532480*x^3*(2*x^2-x+3)^(7/2)+3684995/39936*x^4*(2*x^2-x+3)^(7/2)+1046225/9984*x^5*(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 = 0.51, size = 235, normalized size = 0.93 \begin {gather*} \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 {gather*}

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 = 2.91, size = 118, normalized size = 0.46 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (2 x^{2} - x + 3\right )^{\frac {5}{2}} \left (5 x^{2} + 3 x + 2\right )^{4}\, dx \end {gather*}

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 = 4.18, size = 113, normalized size = 0.44 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (2\,x^2-x+3\right )}^{5/2}\,{\left (5\,x^2+3\,x+2\right )}^4 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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