\(\int \frac {(1+2 x)^3 (1+3 x+4 x^2)}{\sqrt {2+3 x^2}} \, dx\) [118]

Optimal result

Integrand size = 29, antiderivative size = 106 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=-\frac {19}{540} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}-\frac {1}{810} (3937+2073 x) \sqrt {2+3 x^2}+\frac {5 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]

[Out]

5/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)-19/540*(1+2*x)^2*(3*x^2+2)^(1/2)+13/60*(1+2*x)^3*(3*x^2+2)^(1/2)+2/15*(1+2*
x)^4*(3*x^2+2)^(1/2)-1/810*(3937+2073*x)*(3*x^2+2)^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1668, 847, 794, 221} \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {5 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}}+\frac {2}{15} \sqrt {3 x^2+2} (2 x+1)^4+\frac {13}{60} \sqrt {3 x^2+2} (2 x+1)^3-\frac {19}{540} \sqrt {3 x^2+2} (2 x+1)^2-\frac {1}{810} (2073 x+3937) \sqrt {3 x^2+2} \]

[In]

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

[Out]

(-19*(1 + 2*x)^2*Sqrt[2 + 3*x^2])/540 + (13*(1 + 2*x)^3*Sqrt[2 + 3*x^2])/60 + (2*(1 + 2*x)^4*Sqrt[2 + 3*x^2])/
15 - ((3937 + 2073*x)*Sqrt[2 + 3*x^2])/810 + (5*ArcSinh[Sqrt[3/2]*x])/(3*Sqrt[3])

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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 847

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

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}+\frac {1}{60} \int \frac {(1+2 x)^3 (-68+156 x)}{\sqrt {2+3 x^2}} \, dx \\ & = \frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}+\frac {1}{720} \int \frac {(-2688-228 x) (1+2 x)^2}{\sqrt {2+3 x^2}} \, dx \\ & = -\frac {19}{540} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}+\frac {\int \frac {(-22368-49752 x) (1+2 x)}{\sqrt {2+3 x^2}} \, dx}{6480} \\ & = -\frac {19}{540} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}-\frac {1}{810} (3937+2073 x) \sqrt {2+3 x^2}+\frac {5}{3} \int \frac {1}{\sqrt {2+3 x^2}} \, dx \\ & = -\frac {19}{540} (1+2 x)^2 \sqrt {2+3 x^2}+\frac {13}{60} (1+2 x)^3 \sqrt {2+3 x^2}+\frac {2}{15} (1+2 x)^4 \sqrt {2+3 x^2}-\frac {1}{810} (3937+2073 x) \sqrt {2+3 x^2}+\frac {5 \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.62 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{405} \sqrt {2+3 x^2} \left (-1841-135 x+2292 x^2+2430 x^3+864 x^4\right )-\frac {5 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{3 \sqrt {3}} \]

[In]

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

[Out]

(Sqrt[2 + 3*x^2]*(-1841 - 135*x + 2292*x^2 + 2430*x^3 + 864*x^4))/405 - (5*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]]
)/(3*Sqrt[3])

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.42

method	result	size
risch	\(\frac {\left (864 x^{4}+2430 x^{3}+2292 x^{2}-135 x -1841\right ) \sqrt {3 x^{2}+2}}{405}+\frac {5 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{9}\)	\(45\)
trager	\(\left (\frac {32}{15} x^{4}+6 x^{3}+\frac {764}{135} x^{2}-\frac {1}{3} x -\frac {1841}{405}\right ) \sqrt {3 x^{2}+2}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{9}\)	\(61\)
default	\(\frac {5 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{9}-\frac {1841 \sqrt {3 x^{2}+2}}{405}+\frac {32 x^{4} \sqrt {3 x^{2}+2}}{15}+\frac {764 x^{2} \sqrt {3 x^{2}+2}}{135}+6 x^{3} \sqrt {3 x^{2}+2}-\frac {x \sqrt {3 x^{2}+2}}{3}\)	\(79\)
meijerg	\(\frac {\sqrt {3}\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{3}+\frac {34 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \sqrt {\frac {3 x^{2}}{2}+1}}{2}-\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )\right )}{9 \sqrt {\pi }}+\frac {3 \sqrt {2}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {\frac {3 x^{2}}{2}+1}\right )}{2 \sqrt {\pi }}+\frac {68 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{6}\right )}{9 \sqrt {\pi }}+\frac {16 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \left (-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{40}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{4}\right )}{3 \sqrt {\pi }}+\frac {64 \sqrt {2}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (\frac {27}{2} x^{4}-12 x^{2}+16\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}\right )}{27 \sqrt {\pi }}\)	\(217\)

[In]

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

[Out]

1/405*(864*x^4+2430*x^3+2292*x^2-135*x-1841)*(3*x^2+2)^(1/2)+5/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.57 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{405} \, {\left (864 \, x^{4} + 2430 \, x^{3} + 2292 \, x^{2} - 135 \, x - 1841\right )} \sqrt {3 \, x^{2} + 2} + \frac {5}{18} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]

[In]

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

[Out]

1/405*(864*x^4 + 2430*x^3 + 2292*x^2 - 135*x - 1841)*sqrt(3*x^2 + 2) + 5/18*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 +
2)*x - 3*x^2 - 1)

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.89 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {32 x^{4} \sqrt {3 x^{2} + 2}}{15} + 6 x^{3} \sqrt {3 x^{2} + 2} + \frac {764 x^{2} \sqrt {3 x^{2} + 2}}{135} - \frac {x \sqrt {3 x^{2} + 2}}{3} - \frac {1841 \sqrt {3 x^{2} + 2}}{405} + \frac {5 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{9} \]

[In]

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

[Out]

32*x**4*sqrt(3*x**2 + 2)/15 + 6*x**3*sqrt(3*x**2 + 2) + 764*x**2*sqrt(3*x**2 + 2)/135 - x*sqrt(3*x**2 + 2)/3 -
1841*sqrt(3*x**2 + 2)/405 + 5*sqrt(3)*asinh(sqrt(6)*x/2)/9

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.74 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {32}{15} \, \sqrt {3 \, x^{2} + 2} x^{4} + 6 \, \sqrt {3 \, x^{2} + 2} x^{3} + \frac {764}{135} \, \sqrt {3 \, x^{2} + 2} x^{2} - \frac {1}{3} \, \sqrt {3 \, x^{2} + 2} x + \frac {5}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {1841}{405} \, \sqrt {3 \, x^{2} + 2} \]

[In]

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

[Out]

32/15*sqrt(3*x^2 + 2)*x^4 + 6*sqrt(3*x^2 + 2)*x^3 + 764/135*sqrt(3*x^2 + 2)*x^2 - 1/3*sqrt(3*x^2 + 2)*x + 5/9*
sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 1841/405*sqrt(3*x^2 + 2)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.51 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{405} \, {\left (3 \, {\left (2 \, {\left (9 \, {\left (16 \, x + 45\right )} x + 382\right )} x - 45\right )} x - 1841\right )} \sqrt {3 \, x^{2} + 2} - \frac {5}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]

[In]

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

[Out]

1/405*(3*(2*(9*(16*x + 45)*x + 382)*x - 45)*x - 1841)*sqrt(3*x^2 + 2) - 5/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^
2 + 2))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.42 \[ \int \frac {(1+2 x)^3 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {5\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{9}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {32\,x^4}{5}+18\,x^3+\frac {764\,x^2}{45}-x-\frac {1841}{135}\right )}{3} \]

[In]

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

[Out]

(5*3^(1/2)*asinh((6^(1/2)*x)/2))/9 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((764*x^2)/45 - x + 18*x^3 + (32*x^4)/5 - 1841
/135))/3