Integrand size = 29, antiderivative size = 80 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=-\frac {35}{9} \sqrt {2+3 x^2}+2 x \sqrt {2+3 x^2}+\frac {4}{3} x^3 \sqrt {2+3 x^2}+\frac {28}{27} \left (2+3 x^2\right )^{3/2}-\sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ) \] Output:
-35/9*(3*x^2+2)^(1/2)+2*x*(3*x^2+2)^(1/2)+4/3*x^3*(3*x^2+2)^(1/2)+28/27*(3 *x^2+2)^(3/2)-arcsinh(1/2*x*6^(1/2))*3^(1/2)
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.72 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{27} \sqrt {2+3 x^2} \left (-49+54 x+84 x^2+36 x^3\right )+\sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right ) \] Input:
Integrate[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/Sqrt[2 + 3*x^2],x]
Output:
(Sqrt[2 + 3*x^2]*(-49 + 54*x + 84*x^2 + 36*x^3))/27 + Sqrt[3]*Log[-(Sqrt[3 ]*x) + Sqrt[2 + 3*x^2]]
Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.25, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2185, 27, 687, 27, 676, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(2 x+1)^2 \left (4 x^2+3 x+1\right )}{\sqrt {3 x^2+2}} \, dx\) |
\(\Big \downarrow \) 2185 |
\(\displaystyle \frac {1}{48} \int -\frac {24 (2-5 x) (2 x+1)^2}{\sqrt {3 x^2+2}}dx+\frac {1}{6} \sqrt {3 x^2+2} (2 x+1)^3\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} (2 x+1)^3 \sqrt {3 x^2+2}-\frac {1}{2} \int \frac {(2-5 x) (2 x+1)^2}{\sqrt {3 x^2+2}}dx\) |
\(\Big \downarrow \) 687 |
\(\displaystyle \frac {1}{2} \left (\frac {5}{9} (2 x+1)^2 \sqrt {3 x^2+2}-\frac {1}{9} \int \frac {2 (2 x+1) (3 x+29)}{\sqrt {3 x^2+2}}dx\right )+\frac {1}{6} \sqrt {3 x^2+2} (2 x+1)^3\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {5}{9} (2 x+1)^2 \sqrt {3 x^2+2}-\frac {2}{9} \int \frac {(2 x+1) (3 x+29)}{\sqrt {3 x^2+2}}dx\right )+\frac {1}{6} \sqrt {3 x^2+2} (2 x+1)^3\) |
\(\Big \downarrow \) 676 |
\(\displaystyle \frac {1}{2} \left (\frac {5}{9} (2 x+1)^2 \sqrt {3 x^2+2}-\frac {2}{9} \left (27 \int \frac {1}{\sqrt {3 x^2+2}}dx+\sqrt {3 x^2+2} x+\frac {61}{3} \sqrt {3 x^2+2}\right )\right )+\frac {1}{6} \sqrt {3 x^2+2} (2 x+1)^3\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{2} \left (\frac {5}{9} (2 x+1)^2 \sqrt {3 x^2+2}-\frac {2}{9} \left (9 \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )+\sqrt {3 x^2+2} x+\frac {61}{3} \sqrt {3 x^2+2}\right )\right )+\frac {1}{6} \sqrt {3 x^2+2} (2 x+1)^3\) |
Input:
Int[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/Sqrt[2 + 3*x^2],x]
Output:
((1 + 2*x)^3*Sqrt[2 + 3*x^2])/6 + ((5*(1 + 2*x)^2*Sqrt[2 + 3*x^2])/9 - (2* ((61*Sqrt[2 + 3*x^2])/3 + x*Sqrt[2 + 3*x^2] + 9*Sqrt[3]*ArcSinh[Sqrt[3/2]* x]))/9)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(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]
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] + Simp[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] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(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 + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 0.90 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.50
method | result | size |
risch | \(\frac {\left (36 x^{3}+84 x^{2}+54 x -49\right ) \sqrt {3 x^{2}+2}}{27}-\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}\) | \(40\) |
trager | \(\left (\frac {4}{3} x^{3}+\frac {28}{9} x^{2}+2 x -\frac {49}{27}\right ) \sqrt {3 x^{2}+2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )\) | \(56\) |
default | \(-\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}-\frac {49 \sqrt {3 x^{2}+2}}{27}+2 x \sqrt {3 x^{2}+2}+\frac {28 x^{2} \sqrt {3 x^{2}+2}}{9}+\frac {4 x^{3} \sqrt {3 x^{2}+2}}{3}\) | \(65\) |
meijerg | \(\frac {\sqrt {3}\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{3}+\frac {20 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \sqrt {\frac {3 x^{2}}{2}+1}}{2}-\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{9 \sqrt {\pi }}+\frac {7 \sqrt {2}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {\frac {3 x^{2}}{2}+1}\right )}{6 \sqrt {\pi }}+\frac {28 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{6}\right )}{9 \sqrt {\pi }}+\frac {32 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{40}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{4}\right )}{27 \sqrt {\pi }}\) | \(177\) |
Input:
int((1+2*x)^2*(4*x^2+3*x+1)/(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/27*(36*x^3+84*x^2+54*x-49)*(3*x^2+2)^(1/2)-arcsinh(1/2*6^(1/2)*x)*3^(1/2 )
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.68 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{27} \, {\left (36 \, x^{3} + 84 \, x^{2} + 54 \, x - 49\right )} \sqrt {3 \, x^{2} + 2} + \frac {1}{2} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \] Input:
integrate((1+2*x)^2*(4*x^2+3*x+1)/(3*x^2+2)^(1/2),x, algorithm="fricas")
Output:
1/27*(36*x^3 + 84*x^2 + 54*x - 49)*sqrt(3*x^2 + 2) + 1/2*sqrt(3)*log(sqrt( 3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)
Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {4 x^{3} \sqrt {3 x^{2} + 2}}{3} + \frac {28 x^{2} \sqrt {3 x^{2} + 2}}{9} + 2 x \sqrt {3 x^{2} + 2} - \frac {49 \sqrt {3 x^{2} + 2}}{27} - \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )} \] Input:
integrate((1+2*x)**2*(4*x**2+3*x+1)/(3*x**2+2)**(1/2),x)
Output:
4*x**3*sqrt(3*x**2 + 2)/3 + 28*x**2*sqrt(3*x**2 + 2)/9 + 2*x*sqrt(3*x**2 + 2) - 49*sqrt(3*x**2 + 2)/27 - sqrt(3)*asinh(sqrt(6)*x/2)
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {4}{3} \, \sqrt {3 \, x^{2} + 2} x^{3} + \frac {28}{9} \, \sqrt {3 \, x^{2} + 2} x^{2} + 2 \, \sqrt {3 \, x^{2} + 2} x - \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {49}{27} \, \sqrt {3 \, x^{2} + 2} \] Input:
integrate((1+2*x)^2*(4*x^2+3*x+1)/(3*x^2+2)^(1/2),x, algorithm="maxima")
Output:
4/3*sqrt(3*x^2 + 2)*x^3 + 28/9*sqrt(3*x^2 + 2)*x^2 + 2*sqrt(3*x^2 + 2)*x - sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 49/27*sqrt(3*x^2 + 2)
Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.60 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {1}{27} \, {\left (6 \, {\left (2 \, {\left (3 \, x + 7\right )} x + 9\right )} x - 49\right )} \sqrt {3 \, x^{2} + 2} + \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \] Input:
integrate((1+2*x)^2*(4*x^2+3*x+1)/(3*x^2+2)^(1/2),x, algorithm="giac")
Output:
1/27*(6*(2*(3*x + 7)*x + 9)*x - 49)*sqrt(3*x^2 + 2) + sqrt(3)*log(-sqrt(3) *x + sqrt(3*x^2 + 2))
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.50 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (4\,x^3+\frac {28\,x^2}{3}+6\,x-\frac {49}{9}\right )}{3}-\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right ) \] Input:
int(((2*x + 1)^2*(3*x + 4*x^2 + 1))/(3*x^2 + 2)^(1/2),x)
Output:
(3^(1/2)*(x^2 + 2/3)^(1/2)*(6*x + (28*x^2)/3 + 4*x^3 - 49/9))/3 - 3^(1/2)* asinh((6^(1/2)*x)/2)
Time = 0.23 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int \frac {(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt {2+3 x^2}} \, dx=\frac {4 \sqrt {3 x^{2}+2}\, x^{3}}{3}+\frac {28 \sqrt {3 x^{2}+2}\, x^{2}}{9}+2 \sqrt {3 x^{2}+2}\, x -\frac {49 \sqrt {3 x^{2}+2}}{27}-\sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right ) \] Input:
int((1+2*x)^2*(4*x^2+3*x+1)/(3*x^2+2)^(1/2),x)
Output:
(36*sqrt(3*x**2 + 2)*x**3 + 84*sqrt(3*x**2 + 2)*x**2 + 54*sqrt(3*x**2 + 2) *x - 49*sqrt(3*x**2 + 2) - 27*sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/s qrt(2)))/27