3.3.24 \(\int \frac {(2-x+3 x^2)^{5/2} (1+3 x+4 x^2)}{(1+2 x)^2} \, dx\) [224]

Optimal. Leaf size=154 \[ -\frac {11 (4727-3090 x) \sqrt {2-x+3 x^2}}{6912}-\frac {11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac {11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac {\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}-\frac {315623 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{13824 \sqrt {3}}+\frac {429}{128} \sqrt {13} \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right ) \]

[Out]

-11/864*(67-78*x)*(3*x^2-x+2)^(3/2)-11/2340*(37-60*x)*(3*x^2-x+2)^(5/2)-1/13*(3*x^2-x+2)^(7/2)/(1+2*x)-315623/
41472*arcsinh(1/23*(1-6*x)*23^(1/2))*3^(1/2)+429/128*arctanh(1/26*(9-8*x)*13^(1/2)/(3*x^2-x+2)^(1/2))*13^(1/2)
-11/6912*(4727-3090*x)*(3*x^2-x+2)^(1/2)

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Rubi [A]
time = 0.10, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {1664, 828, 857, 633, 221, 738, 212} \begin {gather*} -\frac {\left (3 x^2-x+2\right )^{7/2}}{13 (2 x+1)}-\frac {11 (37-60 x) \left (3 x^2-x+2\right )^{5/2}}{2340}-\frac {11}{864} (67-78 x) \left (3 x^2-x+2\right )^{3/2}-\frac {11 (4727-3090 x) \sqrt {3 x^2-x+2}}{6912}+\frac {429}{128} \sqrt {13} \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {3 x^2-x+2}}\right )-\frac {315623 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{13824 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2))/(1 + 2*x)^2,x]

[Out]

(-11*(4727 - 3090*x)*Sqrt[2 - x + 3*x^2])/6912 - (11*(67 - 78*x)*(2 - x + 3*x^2)^(3/2))/864 - (11*(37 - 60*x)*
(2 - x + 3*x^2)^(5/2))/2340 - (2 - x + 3*x^2)^(7/2)/(13*(1 + 2*x)) - (315623*ArcSinh[(1 - 6*x)/Sqrt[23]])/(138
24*Sqrt[3]) + (429*Sqrt[13]*ArcTanh[(9 - 8*x)/(2*Sqrt[13]*Sqrt[2 - x + 3*x^2])])/128

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 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 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 828

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^
2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1664

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomia
lQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*
x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^
(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Q + c*d*R*(m + 1) - b*e*R*(m + p + 2)
- c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (2-x+3 x^2\right )^{5/2} \left (1+3 x+4 x^2\right )}{(1+2 x)^2} \, dx &=-\frac {\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}-\frac {1}{13} \int \frac {\left (-\frac {11}{2}-44 x\right ) \left (2-x+3 x^2\right )^{5/2}}{1+2 x} \, dx\\ &=-\frac {11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac {\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac {\int \frac {(-286+14872 x) \left (2-x+3 x^2\right )^{3/2}}{1+2 x} \, dx}{1872}\\ &=-\frac {11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac {11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac {\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}-\frac {\int \frac {(641784-3534960 x) \sqrt {2-x+3 x^2}}{1+2 x} \, dx}{179712}\\ &=-\frac {11 (4727-3090 x) \sqrt {2-x+3 x^2}}{6912}-\frac {11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac {11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac {\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac {\int \frac {-178896432+393897504 x}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx}{8626176}\\ &=-\frac {11 (4727-3090 x) \sqrt {2-x+3 x^2}}{6912}-\frac {11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac {11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac {\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac {315623 \int \frac {1}{\sqrt {2-x+3 x^2}} \, dx}{13824}-\frac {5577}{128} \int \frac {1}{(1+2 x) \sqrt {2-x+3 x^2}} \, dx\\ &=-\frac {11 (4727-3090 x) \sqrt {2-x+3 x^2}}{6912}-\frac {11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac {11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac {\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}+\frac {5577}{64} \text {Subst}\left (\int \frac {1}{52-x^2} \, dx,x,\frac {9-8 x}{\sqrt {2-x+3 x^2}}\right )+\frac {315623 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{23}}} \, dx,x,-1+6 x\right )}{13824 \sqrt {69}}\\ &=-\frac {11 (4727-3090 x) \sqrt {2-x+3 x^2}}{6912}-\frac {11}{864} (67-78 x) \left (2-x+3 x^2\right )^{3/2}-\frac {11 (37-60 x) \left (2-x+3 x^2\right )^{5/2}}{2340}-\frac {\left (2-x+3 x^2\right )^{7/2}}{13 (1+2 x)}-\frac {315623 \sinh ^{-1}\left (\frac {1-6 x}{\sqrt {23}}\right )}{13824 \sqrt {3}}+\frac {429}{128} \sqrt {13} \tanh ^{-1}\left (\frac {9-8 x}{2 \sqrt {13} \sqrt {2-x+3 x^2}}\right )\\ \end {align*}

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Mathematica [A]
time = 0.64, size = 131, normalized size = 0.85 \begin {gather*} \frac {\frac {6 \sqrt {2-x+3 x^2} \left (-364257-322972 x+310660 x^2-115680 x^3+251424 x^4-65664 x^5+103680 x^6\right )}{1+2 x}-1389960 \sqrt {13} \tanh ^{-1}\left (\frac {\sqrt {3}+2 \sqrt {3} x-2 \sqrt {2-x+3 x^2}}{\sqrt {13}}\right )-1578115 \sqrt {3} \log \left (1-6 x+2 \sqrt {6-3 x+9 x^2}\right )}{207360} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 - x + 3*x^2)^(5/2)*(1 + 3*x + 4*x^2))/(1 + 2*x)^2,x]

[Out]

((6*Sqrt[2 - x + 3*x^2]*(-364257 - 322972*x + 310660*x^2 - 115680*x^3 + 251424*x^4 - 65664*x^5 + 103680*x^6))/
(1 + 2*x) - 1389960*Sqrt[13]*ArcTanh[(Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 - x + 3*x^2])/Sqrt[13]] - 1578115*Sqrt[
3]*Log[1 - 6*x + 2*Sqrt[6 - 3*x + 9*x^2]])/207360

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Maple [A]
time = 0.17, size = 235, normalized size = 1.53

method	result	size
risch	\(\frac {311040 x^{8}-300672 x^{7}+1027296 x^{6}-729792 x^{5}+1550508 x^{4}-1510936 x^{3}-148479 x^{2}-281687 x -728514}{34560 \left (2 x +1\right ) \sqrt {3 x^{2}-x +2}}+\frac {315623 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{41472}+\frac {429 \sqrt {13}\, \arctanh \left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{128}\)	\(107\)
trager	\(\frac {\left (103680 x^{6}-65664 x^{5}+251424 x^{4}-115680 x^{3}+310660 x^{2}-322972 x -364257\right ) \sqrt {3 x^{2}-x +2}}{69120 x +34560}-\frac {315623 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}-x +2}+\RootOf \left (\textit {\_Z}^{2}-3\right )\right )}{41472}-\frac {429 \RootOf \left (\textit {\_Z}^{2}-13\right ) \ln \left (\frac {8 \RootOf \left (\textit {\_Z}^{2}-13\right ) x +26 \sqrt {3 x^{2}-x +2}-9 \RootOf \left (\textit {\_Z}^{2}-13\right )}{2 x +1}\right )}{128}\)	\(141\)
default	\(\frac {\left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {5}{2}}}{36}+\frac {115 \left (6 x -1\right ) \left (3 x^{2}-x +2\right )^{\frac {3}{2}}}{1728}+\frac {2645 \left (6 x -1\right ) \sqrt {3 x^{2}-x +2}}{13824}+\frac {315623 \sqrt {3}\, \arcsinh \left (\frac {6 \sqrt {23}\, \left (x -\frac {1}{6}\right )}{23}\right )}{41472}-\frac {33 \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {5}{2}}}{260}+\frac {19 \left (6 x -1\right ) \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {3}{2}}}{192}+\frac {965 \left (6 x -1\right ) \sqrt {3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}}}{1536}-\frac {11 \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {3}{2}}}{16}-\frac {429 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}{128}+\frac {429 \sqrt {13}\, \arctanh \left (\frac {2 \left (\frac {9}{2}-4 x \right ) \sqrt {13}}{13 \sqrt {12 \left (x +\frac {1}{2}\right )^{2}-16 x +5}}\right )}{128}-\frac {\left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {7}{2}}}{26 \left (x +\frac {1}{2}\right )}+\frac {\left (6 x -1\right ) \left (3 \left (x +\frac {1}{2}\right )^{2}-4 x +\frac {5}{4}\right )^{\frac {5}{2}}}{52}\)	\(235\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2-x+2)^(5/2)*(4*x^2+3*x+1)/(2*x+1)^2,x,method=_RETURNVERBOSE)

[Out]

1/36*(6*x-1)*(3*x^2-x+2)^(5/2)+115/1728*(6*x-1)*(3*x^2-x+2)^(3/2)+2645/13824*(6*x-1)*(3*x^2-x+2)^(1/2)+315623/
41472*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))-33/260*(3*(x+1/2)^2-4*x+5/4)^(5/2)+19/192*(6*x-1)*(3*(x+1/2)^2-4*
x+5/4)^(3/2)+965/1536*(6*x-1)*(3*(x+1/2)^2-4*x+5/4)^(1/2)-11/16*(3*(x+1/2)^2-4*x+5/4)^(3/2)-429/128*(12*(x+1/2
)^2-16*x+5)^(1/2)+429/128*13^(1/2)*arctanh(2/13*(9/2-4*x)*13^(1/2)/(12*(x+1/2)^2-16*x+5)^(1/2))-1/26/(x+1/2)*(
3*(x+1/2)^2-4*x+5/4)^(7/2)+1/52*(6*x-1)*(3*(x+1/2)^2-4*x+5/4)^(5/2)

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Maxima [A]
time = 0.51, size = 161, normalized size = 1.05 \begin {gather*} \frac {1}{6} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} x - \frac {7}{90} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}} + \frac {143}{144} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} x - \frac {737}{864} \, {\left (3 \, x^{2} - x + 2\right )}^{\frac {3}{2}} - \frac {{\left (3 \, x^{2} - x + 2\right )}^{\frac {5}{2}}}{4 \, {\left (2 \, x + 1\right )}} + \frac {5665}{1152} \, \sqrt {3 \, x^{2} - x + 2} x + \frac {315623}{41472} \, \sqrt {3} \operatorname {arsinh}\left (\frac {6}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {429}{128} \, \sqrt {13} \operatorname {arsinh}\left (\frac {8 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 1 \right |}} - \frac {9 \, \sqrt {23}}{23 \, {\left | 2 \, x + 1 \right |}}\right ) - \frac {51997}{6912} \, \sqrt {3 \, x^{2} - x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-x+2)^(5/2)*(4*x^2+3*x+1)/(1+2*x)^2,x, algorithm="maxima")

[Out]

1/6*(3*x^2 - x + 2)^(5/2)*x - 7/90*(3*x^2 - x + 2)^(5/2) + 143/144*(3*x^2 - x + 2)^(3/2)*x - 737/864*(3*x^2 -
x + 2)^(3/2) - 1/4*(3*x^2 - x + 2)^(5/2)/(2*x + 1) + 5665/1152*sqrt(3*x^2 - x + 2)*x + 315623/41472*sqrt(3)*ar
csinh(6/23*sqrt(23)*x - 1/23*sqrt(23)) - 429/128*sqrt(13)*arcsinh(8/23*sqrt(23)*x/abs(2*x + 1) - 9/23*sqrt(23)
/abs(2*x + 1)) - 51997/6912*sqrt(3*x^2 - x + 2)

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Fricas [A]
time = 0.37, size = 153, normalized size = 0.99 \begin {gather*} \frac {1578115 \, \sqrt {3} {\left (2 \, x + 1\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} - x + 2} {\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) + 694980 \, \sqrt {13} {\left (2 \, x + 1\right )} \log \left (\frac {4 \, \sqrt {13} \sqrt {3 \, x^{2} - x + 2} {\left (8 \, x - 9\right )} - 220 \, x^{2} + 196 \, x - 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 12 \, {\left (103680 \, x^{6} - 65664 \, x^{5} + 251424 \, x^{4} - 115680 \, x^{3} + 310660 \, x^{2} - 322972 \, x - 364257\right )} \sqrt {3 \, x^{2} - x + 2}}{414720 \, {\left (2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-x+2)^(5/2)*(4*x^2+3*x+1)/(1+2*x)^2,x, algorithm="fricas")

[Out]

1/414720*(1578115*sqrt(3)*(2*x + 1)*log(-4*sqrt(3)*sqrt(3*x^2 - x + 2)*(6*x - 1) - 72*x^2 + 24*x - 25) + 69498
0*sqrt(13)*(2*x + 1)*log((4*sqrt(13)*sqrt(3*x^2 - x + 2)*(8*x - 9) - 220*x^2 + 196*x - 185)/(4*x^2 + 4*x + 1))
+ 12*(103680*x^6 - 65664*x^5 + 251424*x^4 - 115680*x^3 + 310660*x^2 - 322972*x - 364257)*sqrt(3*x^2 - x + 2))
/(2*x + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x^{2} - x + 2\right )^{\frac {5}{2}} \cdot \left (4 x^{2} + 3 x + 1\right )}{\left (2 x + 1\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2-x+2)**(5/2)*(4*x**2+3*x+1)/(1+2*x)**2,x)

[Out]

Integral((3*x**2 - x + 2)**(5/2)*(4*x**2 + 3*x + 1)/(2*x + 1)**2, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (123) = 246\).
time = 6.41, size = 760, normalized size = 4.94 \begin {gather*} \frac {429}{128} \, \sqrt {13} \log \left (\sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )} - 4\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - \frac {315623}{41472} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {2 \, \sqrt {13}}{2 \, x + 1} \right |}}{2 \, {\left (\sqrt {3} + \sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - \frac {169}{128} \, \sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + \frac {5154065 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{11} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - 7837020 \, \sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{10} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 39468815 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{9} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - 14445540 \, \sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{8} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 460893402 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{7} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - 343084680 \, \sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{6} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 944150094 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - 22871160 \, \sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 1397032245 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) - 683367516 \, \sqrt {13} {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 392684355 \, {\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right ) + 197538588 \, \sqrt {13} \mathrm {sgn}\left (\frac {1}{2 \, x + 1}\right )}{34560 \, {\left ({\left (\sqrt {-\frac {8}{2 \, x + 1} + \frac {13}{{\left (2 \, x + 1\right )}^{2}} + 3} + \frac {\sqrt {13}}{2 \, x + 1}\right )}^{2} - 3\right )}^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2-x+2)^(5/2)*(4*x^2+3*x+1)/(1+2*x)^2,x, algorithm="giac")

[Out]

429/128*sqrt(13)*log(sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1)) - 4)*sgn(1/(2*x +
1)) - 315623/41472*sqrt(3)*log(1/2*abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + 2*sqrt(13)/(2
*x + 1))/(sqrt(3) + sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1)))*sgn(1/(2*x + 1)) - 169/128*
sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3)*sgn(1/(2*x + 1)) + 1/34560*(5154065*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^
2 + 3) + sqrt(13)/(2*x + 1))^11*sgn(1/(2*x + 1)) - 7837020*sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) +
sqrt(13)/(2*x + 1))^10*sgn(1/(2*x + 1)) + 39468815*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x +
1))^9*sgn(1/(2*x + 1)) - 14445540*sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^8*s
gn(1/(2*x + 1)) + 460893402*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^7*sgn(1/(2*x + 1))
- 343084680*sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^6*sgn(1/(2*x + 1)) + 94415
0094*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^5*sgn(1/(2*x + 1)) - 22871160*sqrt(13)*(sq
rt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^4*sgn(1/(2*x + 1)) + 1397032245*(sqrt(-8/(2*x + 1)
+ 13/(2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^3*sgn(1/(2*x + 1)) - 683367516*sqrt(13)*(sqrt(-8/(2*x + 1) + 13/(
2*x + 1)^2 + 3) + sqrt(13)/(2*x + 1))^2*sgn(1/(2*x + 1)) + 392684355*(sqrt(-8/(2*x + 1) + 13/(2*x + 1)^2 + 3)
+ sqrt(13)/(2*x + 1))*sgn(1/(2*x + 1)) + 197538588*sqrt(13)*sgn(1/(2*x + 1)))/((sqrt(-8/(2*x + 1) + 13/(2*x +
1)^2 + 3) + sqrt(13)/(2*x + 1))^2 - 3)^6

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (3\,x^2-x+2\right )}^{5/2}\,\left (4\,x^2+3\,x+1\right )}{{\left (2\,x+1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2 - x + 2)^(5/2)*(3*x + 4*x^2 + 1))/(2*x + 1)^2,x)

[Out]

int(((3*x^2 - x + 2)^(5/2)*(3*x + 4*x^2 + 1))/(2*x + 1)^2, x)

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