3.1.74 \(\int (3-x+2 x^2)^{5/2} (2+3 x+5 x^2)^2 \, dx\) [74]

Optimal. Leaf size=170 \[ -\frac {4091815 (1-4 x) \sqrt {3-x+2 x^2}}{16777216}-\frac {177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}-\frac {94111745 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{33554432 \sqrt {2}} \]

[Out]

-177905/3145728*(1-4*x)*(2*x^2-x+3)^(3/2)-1547/98304*(1-4*x)*(2*x^2-x+3)^(5/2)+23225/43008*(2*x^2-x+3)^(7/2)+8
467/4608*x*(2*x^2-x+3)^(7/2)+305/144*x^2*(2*x^2-x+3)^(7/2)+5/4*x^3*(2*x^2-x+3)^(7/2)-94111745/67108864*arcsinh
(1/23*(1-4*x)*23^(1/2))*2^(1/2)-4091815/16777216*(1-4*x)*(2*x^2-x+3)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {305}{144} x^2 \left (2 x^2-x+3\right )^{7/2}+\frac {8467 x \left (2 x^2-x+3\right )^{7/2}}{4608}+\frac {23225 \left (2 x^2-x+3\right )^{7/2}}{43008}-\frac {1547 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{98304}-\frac {177905 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{3145728}-\frac {4091815 (1-4 x) \sqrt {2 x^2-x+3}}{16777216}+\frac {5}{4} x^3 \left (2 x^2-x+3\right )^{7/2}-\frac {94111745 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{33554432 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4091815*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/16777216 - (177905*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/3145728 - (1547*(
1 - 4*x)*(3 - x + 2*x^2)^(5/2))/98304 + (23225*(3 - x + 2*x^2)^(7/2))/43008 + (8467*x*(3 - x + 2*x^2)^(7/2))/4
608 + (305*x^2*(3 - x + 2*x^2)^(7/2))/144 + (5*x^3*(3 - x + 2*x^2)^(7/2))/4 - (94111745*ArcSinh[(1 - 4*x)/Sqrt
[23]])/(33554432*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 )^2 \, dx &=\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1}{20} \int \left (3-x+2 x^2\right )^{5/2} \left (80+240 x+355 x^2+\frac {1525 x^3}{2}\right ) \, dx\\ &=\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1}{360} \int \left (3-x+2 x^2\right )^{5/2} \left (1440-255 x+\frac {42335 x^2}{4}\right ) \, dx\\ &=\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {\int \left (-\frac {34845}{4}+\frac {348375 x}{8}\right ) \left (3-x+2 x^2\right )^{5/2} \, dx}{5760}\\ &=\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {1547 \int \left (3-x+2 x^2\right )^{5/2} \, dx}{4096}\\ &=-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {177905 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{196608}\\ &=-\frac {177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {4091815 \int \sqrt {3-x+2 x^2} \, dx}{2097152}\\ &=-\frac {4091815 (1-4 x) \sqrt {3-x+2 x^2}}{16777216}-\frac {177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {94111745 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{33554432}\\ &=-\frac {4091815 (1-4 x) \sqrt {3-x+2 x^2}}{16777216}-\frac {177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac {\left (4091815 \sqrt {\frac {23}{2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{33554432}\\ &=-\frac {4091815 (1-4 x) \sqrt {3-x+2 x^2}}{16777216}-\frac {177905 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{3145728}-\frac {1547 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{98304}+\frac {23225 \left (3-x+2 x^2\right )^{7/2}}{43008}+\frac {8467 x \left (3-x+2 x^2\right )^{7/2}}{4608}+\frac {305}{144} x^2 \left (3-x+2 x^2\right )^{7/2}+\frac {5}{4} x^3 \left (3-x+2 x^2\right )^{7/2}-\frac {94111745 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{33554432 \sqrt {2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.78, size = 95, normalized size = 0.56 \begin {gather*} \frac {4 \sqrt {3-x+2 x^2} \left (14824182519+39533249652 x+42992644128 x^2+77872272000 x^3+57147467776 x^4+75389820928 x^5+26401898496 x^6+44163137536 x^7+2055208960 x^8+10569646080 x^9\right )-5929039935 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{4227858432} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(14824182519 + 39533249652*x + 42992644128*x^2 + 77872272000*x^3 + 57147467776*x^4 + 75
389820928*x^5 + 26401898496*x^6 + 44163137536*x^7 + 2055208960*x^8 + 10569646080*x^9) - 5929039935*Sqrt[2]*Log
[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/4227858432

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 136, normalized size = 0.80

method	result	size
risch	\(\frac {\left (10569646080 x^{9}[phone] x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {2 x^{2}-x +3}}{1056964608}+\frac {94111745 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{67108864}\)	\(75\)
trager	\(\left (10 x^{9}+\frac {35}{18} x^{8}+\frac {24067}{576} x^{7}+\frac {134287}{5376} x^{6}+\frac {9202859}{129024} x^{5}+\frac {3986291}{73728} x^{4}+\frac {202792375}{2752512} x^{3}+\frac {63977149}{1572864} x^{2}+\frac {3294437471}{88080384} x +\frac {1647131391}{117440512}\right ) \sqrt {2 x^{2}-x +3}-\frac {94111745 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {2 x^{2}-x +3}+\RootOf \left (\textit {\_Z}^{2}-2\right )\right )}{67108864}\)	\(99\)
default	\(\frac {23225 \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{43008}+\frac {1547 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {5}{2}}}{98304}+\frac {4091815 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{16777216}+\frac {94111745 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{67108864}+\frac {177905 \left (4 x -1\right ) \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{3145728}+\frac {8467 x \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{4608}+\frac {305 x^{2} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{144}+\frac {5 x^{3} \left (2 x^{2}-x +3\right )^{\frac {7}{2}}}{4}\)	\(136\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

23225/43008*(2*x^2-x+3)^(7/2)+1547/98304*(4*x-1)*(2*x^2-x+3)^(5/2)+4091815/16777216*(4*x-1)*(2*x^2-x+3)^(1/2)+
94111745/67108864*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+177905/3145728*(4*x-1)*(2*x^2-x+3)^(3/2)+8467/4608*x*
(2*x^2-x+3)^(7/2)+305/144*x^2*(2*x^2-x+3)^(7/2)+5/4*x^3*(2*x^2-x+3)^(7/2)

________________________________________________________________________________________

Maxima [A]
time = 0.55, size = 167, normalized size = 0.98 \begin {gather*} \frac {5}{4} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{3} + \frac {305}{144} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{2} + \frac {8467}{4608} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x + \frac {23225}{43008} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {1547}{24576} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {1547}{98304} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {177905}{786432} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {177905}{3145728} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {4091815}{4194304} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {94111745}{67108864} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {4091815}{16777216} \, \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)^2,x, algorithm="maxima")

[Out]

5/4*(2*x^2 - x + 3)^(7/2)*x^3 + 305/144*(2*x^2 - x + 3)^(7/2)*x^2 + 8467/4608*(2*x^2 - x + 3)^(7/2)*x + 23225/
43008*(2*x^2 - x + 3)^(7/2) + 1547/24576*(2*x^2 - x + 3)^(5/2)*x - 1547/98304*(2*x^2 - x + 3)^(5/2) + 177905/7
86432*(2*x^2 - x + 3)^(3/2)*x - 177905/3145728*(2*x^2 - x + 3)^(3/2) + 4091815/4194304*sqrt(2*x^2 - x + 3)*x +
94111745/67108864*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 4091815/16777216*sqrt(2*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]
time = 2.64, size = 98, normalized size = 0.58 \begin {gather*} \frac {1}{1056964608} \, {\left (10569646080 \, x^{9} + 2055208960 \, x^{8} + 44163137536 \, x^{7} + 26401898496 \, x^{6} + 75389820928 \, x^{5} + 57147467776 \, x^{4} + 77872272000 \, x^{3} + 42992644128 \, x^{2} + 39533249652 \, x + 14824182519\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {94111745}{134217728} \, \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)^2,x, algorithm="fricas")

[Out]

1/1056964608*(10569646080*x^9 + 2055208960*x^8 + 44163137536*x^7 + 26401898496*x^6 + 75389820928*x^5 + 5714746
7776*x^4 + 77872272000*x^3 + 42992644128*x^2 + 39533249652*x + 14824182519)*sqrt(2*x^2 - x + 3) + 94111745/134
217728*sqrt(2)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)

________________________________________________________________________________________

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 )^{2}\, 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)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 4.18, size = 93, normalized size = 0.55 \begin {gather*} \frac {1}{1056964608} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (8 \, {\left (28 \, {\left (160 \, {\left (36 \, x + 7\right )} x + 24067\right )} x + 402861\right )} x + 9202859\right )} x + 27904037\right )} x + 608377125\right )} x + 1343520129\right )} x + 9883312413\right )} x + 14824182519\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {94111745}{67108864} \, \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)^2,x, algorithm="giac")

[Out]

1/1056964608*(4*(8*(4*(16*(4*(8*(28*(160*(36*x + 7)*x + 24067)*x + 402861)*x + 9202859)*x + 27904037)*x + 6083
77125)*x + 1343520129)*x + 9883312413)*x + 14824182519)*sqrt(2*x^2 - x + 3) - 94111745/67108864*sqrt(2)*log(-2
*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (2\,x^2-x+3\right )}^{5/2}\,{\left (5\,x^2+3\,x+2\right )}^2 \,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)^2,x)

[Out]

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

________________________________________________________________________________________