3.1.59 \(\int \sqrt {3-x+2 x^2} (2+3 x+5 x^2)^3 \, dx\) [59]

Optimal. Leaf size=166 \[ -\frac {6766097 (1-4 x) \sqrt {3-x+2 x^2}}{2097152}-\frac {22548119 \left (3-x+2 x^2\right )^{3/2}}{4587520}-\frac {9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac {531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac {247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac {8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}-\frac {155620231 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4194304 \sqrt {2}} \]

[Out]

-22548119/4587520*(2*x^2-x+3)^(3/2)-9627393/1146880*x*(2*x^2-x+3)^(3/2)+531681/71680*x^2*(2*x^2-x+3)^(3/2)+247
435/10752*x^3*(2*x^2-x+3)^(3/2)+8825/448*x^4*(2*x^2-x+3)^(3/2)+125/16*x^5*(2*x^2-x+3)^(3/2)-155620231/8388608*
arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)-6766097/2097152*(1-4*x)*(2*x^2-x+3)^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 166, 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 {531681 \left (2 x^2-x+3\right )^{3/2} x^2}{71680}-\frac {9627393 \left (2 x^2-x+3\right )^{3/2} x}{1146880}-\frac {22548119 \left (2 x^2-x+3\right )^{3/2}}{4587520}-\frac {6766097 (1-4 x) \sqrt {2 x^2-x+3}}{2097152}+\frac {125}{16} \left (2 x^2-x+3\right )^{3/2} x^5+\frac {8825}{448} \left (2 x^2-x+3\right )^{3/2} x^4+\frac {247435 \left (2 x^2-x+3\right )^{3/2} x^3}{10752}-\frac {155620231 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4194304 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6766097*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/2097152 - (22548119*(3 - x + 2*x^2)^(3/2))/4587520 - (9627393*x*(3 -
x + 2*x^2)^(3/2))/1146880 + (531681*x^2*(3 - x + 2*x^2)^(3/2))/71680 + (247435*x^3*(3 - x + 2*x^2)^(3/2))/1075
2 + (8825*x^4*(3 - x + 2*x^2)^(3/2))/448 + (125*x^5*(3 - x + 2*x^2)^(3/2))/16 - (155620231*ArcSinh[(1 - 4*x)/S
qrt[23]])/(4194304*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 \sqrt {3-x+2 x^2} \left (2+3 x+5 x^2\right )^3 \, dx &=\frac {125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac {1}{16} \int \sqrt {3-x+2 x^2} \left (128+576 x+1824 x^2+3312 x^3+2685 x^4+\frac {8825 x^5}{2}\right ) \, dx\\ &=\frac {8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac {1}{224} \int \sqrt {3-x+2 x^2} \left (1792+8064 x+25536 x^2-6582 x^3+\frac {247435 x^4}{4}\right ) \, dx\\ &=\frac {247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac {8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac {\int \sqrt {3-x+2 x^2} \left (21504+96768 x-\frac {1001187 x^2}{4}+\frac {1595043 x^3}{8}\right ) \, dx}{2688}\\ &=\frac {531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac {247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac {8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac {\int \left (215040-\frac {914409 x}{4}-\frac {28882179 x^2}{16}\right ) \sqrt {3-x+2 x^2} \, dx}{26880}\\ &=-\frac {9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac {531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac {247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac {8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac {\int \left (\frac {114171657}{16}-\frac {202933071 x}{32}\right ) \sqrt {3-x+2 x^2} \, dx}{215040}\\ &=-\frac {22548119 \left (3-x+2 x^2\right )^{3/2}}{4587520}-\frac {9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac {531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac {247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac {8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac {6766097 \int \sqrt {3-x+2 x^2} \, dx}{262144}\\ &=-\frac {6766097 (1-4 x) \sqrt {3-x+2 x^2}}{2097152}-\frac {22548119 \left (3-x+2 x^2\right )^{3/2}}{4587520}-\frac {9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac {531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac {247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac {8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac {155620231 \int \frac {1}{\sqrt {3-x+2 x^2}} \, dx}{4194304}\\ &=-\frac {6766097 (1-4 x) \sqrt {3-x+2 x^2}}{2097152}-\frac {22548119 \left (3-x+2 x^2\right )^{3/2}}{4587520}-\frac {9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac {531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac {247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac {8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}+\frac {\left (6766097 \sqrt {\frac {23}{2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+4 x\right )}{4194304}\\ &=-\frac {6766097 (1-4 x) \sqrt {3-x+2 x^2}}{2097152}-\frac {22548119 \left (3-x+2 x^2\right )^{3/2}}{4587520}-\frac {9627393 x \left (3-x+2 x^2\right )^{3/2}}{1146880}+\frac {531681 x^2 \left (3-x+2 x^2\right )^{3/2}}{71680}+\frac {247435 x^3 \left (3-x+2 x^2\right )^{3/2}}{10752}+\frac {8825}{448} x^4 \left (3-x+2 x^2\right )^{3/2}+\frac {125}{16} x^5 \left (3-x+2 x^2\right )^{3/2}-\frac {155620231 \sinh ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{4194304 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.58, size = 85, normalized size = 0.51 \begin {gather*} \frac {4 \sqrt {3-x+2 x^2} \left (-3957369321-1621307916 x+4583812128 x^2+9872163456 x^3+11212171264 x^4+10958233600 x^5+6955008000 x^6+3440640000 x^7\right )-16340124255 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{880803840} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-3957369321 - 1621307916*x + 4583812128*x^2 + 9872163456*x^3 + 11212171264*x^4 + 10958
233600*x^5 + 6955008000*x^6 + 3440640000*x^7) - 16340124255*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/88
0803840

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Maple [A]
time = 0.13, size = 132, normalized size = 0.80

method	result	size
risch	\(\frac {\left (3440640000 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-1621307916 x -3957369321\right ) \sqrt {2 x^{2}-x +3}}{220200960}+\frac {155620231 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8388608}\)	\(65\)
trager	\(\left (\frac {125}{8} x^{7}+\frac {7075}{224} x^{6}+\frac {267535}{5376} x^{5}+\frac {782099}{15360} x^{4}+\frac {25708759}{573440} x^{3}+\frac {6821149}{327680} x^{2}-\frac {135108993}{18350080} x -\frac {1319123107}{73400320}\right ) \sqrt {2 x^{2}-x +3}+\frac {155620231 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x -\RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {2 x^{2}-x +3}\right )}{8388608}\)	\(91\)
default	\(-\frac {22548119 \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{4587520}+\frac {6766097 \left (4 x -1\right ) \sqrt {2 x^{2}-x +3}}{2097152}+\frac {155620231 \sqrt {2}\, \arcsinh \left (\frac {4 \sqrt {23}\, \left (x -\frac {1}{4}\right )}{23}\right )}{8388608}-\frac {9627393 x \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{1146880}+\frac {531681 x^{2} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{71680}+\frac {247435 x^{3} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{10752}+\frac {8825 x^{4} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{448}+\frac {125 x^{5} \left (2 x^{2}-x +3\right )^{\frac {3}{2}}}{16}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^2+3*x+2)^3*(2*x^2-x+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-22548119/4587520*(2*x^2-x+3)^(3/2)+6766097/2097152*(4*x-1)*(2*x^2-x+3)^(1/2)[phone]/8388608*2^(1/2)*arcsin
h(4/23*23^(1/2)*(x-1/4))-9627393/1146880*x*(2*x^2-x+3)^(3/2)+531681/71680*x^2*(2*x^2-x+3)^(3/2)+247435/10752*x
^3*(2*x^2-x+3)^(3/2)+8825/448*x^4*(2*x^2-x+3)^(3/2)+125/16*x^5*(2*x^2-x+3)^(3/2)

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Maxima [A]
time = 0.52, size = 143, normalized size = 0.86 \begin {gather*} \frac {125}{16} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{5} + \frac {8825}{448} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + \frac {247435}{10752} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {531681}{71680} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {9627393}{1146880} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {22548119}{4587520} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {6766097}{524288} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {155620231}{8388608} \, \sqrt {2} \operatorname {arsinh}\left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {6766097}{2097152} \, \sqrt {2 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+3*x+2)^3*(2*x^2-x+3)^(1/2),x, algorithm="maxima")

[Out]

125/16*(2*x^2 - x + 3)^(3/2)*x^5 + 8825/448*(2*x^2 - x + 3)^(3/2)*x^4 + 247435/10752*(2*x^2 - x + 3)^(3/2)*x^3
+ 531681/71680*(2*x^2 - x + 3)^(3/2)*x^2 - 9627393/1146880*(2*x^2 - x + 3)^(3/2)*x - 22548119/4587520*(2*x^2
- x + 3)^(3/2) + 6766097/524288*sqrt(2*x^2 - x + 3)*x + 155620231/8388608*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x -
1)) - 6766097/2097152*sqrt(2*x^2 - x + 3)

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Fricas [A]
time = 1.85, size = 88, normalized size = 0.53 \begin {gather*} \frac {1}{220200960} \, {\left (3440640000 \, x^{7} + 6955008000 \, x^{6} + 10958233600 \, x^{5} + 11212171264 \, x^{4} + 9872163456 \, x^{3} + 4583812128 \, x^{2} - 1621307916 \, x - 3957369321\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {155620231}{16777216} \, \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((5*x^2+3*x+2)^3*(2*x^2-x+3)^(1/2),x, algorithm="fricas")

[Out]

1/220200960*(3440640000*x^7 + 6955008000*x^6 + 10958233600*x^5 + 11212171264*x^4 + 9872163456*x^3 + 4583812128
*x^2 - 1621307916*x - 3957369321)*sqrt(2*x^2 - x + 3) + 155620231/16777216*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 \sqrt {2 x^{2} - x + 3} \left (5 x^{2} + 3 x + 2\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**2+3*x+2)**3*(2*x**2-x+3)**(1/2),x)

[Out]

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

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Giac [A]
time = 0.96, size = 83, normalized size = 0.50 \begin {gather*} \frac {1}{220200960} \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, {\left (120 \, {\left (140 \, x + 283\right )} x + 53507\right )} x + 5474693\right )} x + 77126277\right )} x + 143244129\right )} x - 405326979\right )} x - 3957369321\right )} \sqrt {2 \, x^{2} - x + 3} - \frac {155620231}{8388608} \, \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((5*x^2+3*x+2)^3*(2*x^2-x+3)^(1/2),x, algorithm="giac")

[Out]

1/220200960*(4*(8*(4*(16*(100*(120*(140*x + 283)*x + 53507)*x + 5474693)*x + 77126277)*x + 143244129)*x - 4053
26979)*x - 3957369321)*sqrt(2*x^2 - x + 3) - 155620231/8388608*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2
- x + 3)) + 1)

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Mupad [B]
time = 4.69, size = 187, normalized size = 1.13 \begin {gather*} \frac {531681\,x^2\,{\left (2\,x^2-x+3\right )}^{3/2}}{71680}+\frac {247435\,x^3\,{\left (2\,x^2-x+3\right )}^{3/2}}{10752}+\frac {8825\,x^4\,{\left (2\,x^2-x+3\right )}^{3/2}}{448}+\frac {125\,x^5\,{\left (2\,x^2-x+3\right )}^{3/2}}{16}+\frac {875316037\,\sqrt {2}\,\ln \left (\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (2\,x-\frac {1}{2}\right )}{2}\right )}{36700160}+\frac {38057219\,\left (\frac {x}{2}-\frac {1}{8}\right )\,\sqrt {2\,x^2-x+3}}{1146880}-\frac {22548119\,\sqrt {2\,x^2-x+3}\,\left (32\,x^2-4\,x+45\right )}{73400320}-\frac {9627393\,x\,{\left (2\,x^2-x+3\right )}^{3/2}}{1146880}-\frac {1555820211\,\sqrt {2}\,\ln \left (2\,\sqrt {2\,x^2-x+3}+\frac {\sqrt {2}\,\left (4\,x-1\right )}{2}\right )}{293601280} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 - x + 3)^(1/2)*(3*x + 5*x^2 + 2)^3,x)

[Out]

(531681*x^2*(2*x^2 - x + 3)^(3/2))/71680 + (247435*x^3*(2*x^2 - x + 3)^(3/2))/10752 + (8825*x^4*(2*x^2 - x + 3
)^(3/2))/448 + (125*x^5*(2*x^2 - x + 3)^(3/2))/16 + (875316037*2^(1/2)*log((2*x^2 - x + 3)^(1/2) + (2^(1/2)*(2
*x - 1/2))/2))/36700160 + (38057219*(x/2 - 1/8)*(2*x^2 - x + 3)^(1/2))/1146880 - (22548119*(2*x^2 - x + 3)^(1/
2)*(32*x^2 - 4*x + 45))/73400320 - (9627393*x*(2*x^2 - x + 3)^(3/2))/1146880 - (1555820211*2^(1/2)*log(2*(2*x^
2 - x + 3)^(1/2) + (2^(1/2)*(4*x - 1))/2))/293601280

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