Integrand size = 24, antiderivative size = 100 \[ \int (5-x) (3+2 x)^3 \sqrt {2+3 x^2} \, dx=\frac {511}{9} x \sqrt {2+3 x^2}+\frac {17}{30} (3+2 x)^2 \left (2+3 x^2\right )^{3/2}-\frac {1}{18} (3+2 x)^3 \left (2+3 x^2\right )^{3/2}+\frac {7}{270} (898+267 x) \left (2+3 x^2\right )^{3/2}+\frac {1022 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \] Output:
511/9*x*(3*x^2+2)^(1/2)+17/30*(3+2*x)^2*(3*x^2+2)^(3/2)-1/18*(3+2*x)^3*(3* x^2+2)^(3/2)+7/270*(898+267*x)*(3*x^2+2)^(3/2)+1022/27*arcsinh(1/2*x*6^(1/ 2))*3^(1/2)
Time = 0.46 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.71 \[ \int (5-x) (3+2 x)^3 \sqrt {2+3 x^2} \, dx=-\frac {1}{270} \sqrt {2+3 x^2} \left (-14516-21120 x-21918 x^2-8445 x^3-216 x^4+360 x^5\right )-\frac {1022 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{9 \sqrt {3}} \] Input:
Integrate[(5 - x)*(3 + 2*x)^3*Sqrt[2 + 3*x^2],x]
Output:
-1/270*(Sqrt[2 + 3*x^2]*(-14516 - 21120*x - 21918*x^2 - 8445*x^3 - 216*x^4 + 360*x^5)) - (1022*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(9*Sqrt[3])
Time = 0.24 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {687, 27, 687, 27, 676, 211, 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.
\(\displaystyle \int (5-x) (2 x+3)^3 \sqrt {3 x^2+2} \, dx\) |
\(\Big \downarrow \) 687 |
\(\displaystyle \frac {1}{18} \int 3 (2 x+3)^2 (51 x+94) \sqrt {3 x^2+2}dx-\frac {1}{18} (2 x+3)^3 \left (3 x^2+2\right )^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \int (2 x+3)^2 (51 x+94) \sqrt {3 x^2+2}dx-\frac {1}{18} (2 x+3)^3 \left (3 x^2+2\right )^{3/2}\) |
\(\Big \downarrow \) 687 |
\(\displaystyle \frac {1}{6} \left (\frac {1}{15} \int 42 (2 x+3) (89 x+91) \sqrt {3 x^2+2}dx+\frac {17}{5} \left (3 x^2+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{18} (2 x+3)^3 \left (3 x^2+2\right )^{3/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \left (\frac {14}{5} \int (2 x+3) (89 x+91) \sqrt {3 x^2+2}dx+\frac {17}{5} \left (3 x^2+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{18} (2 x+3)^3 \left (3 x^2+2\right )^{3/2}\) |
\(\Big \downarrow \) 676 |
\(\displaystyle \frac {1}{6} \left (\frac {14}{5} \left (\frac {730}{3} \int \sqrt {3 x^2+2}dx+\frac {89}{6} x \left (3 x^2+2\right )^{3/2}+\frac {449}{9} \left (3 x^2+2\right )^{3/2}\right )+\frac {17}{5} \left (3 x^2+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{18} (2 x+3)^3 \left (3 x^2+2\right )^{3/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {1}{6} \left (\frac {14}{5} \left (\frac {730}{3} \left (\int \frac {1}{\sqrt {3 x^2+2}}dx+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {89}{6} x \left (3 x^2+2\right )^{3/2}+\frac {449}{9} \left (3 x^2+2\right )^{3/2}\right )+\frac {17}{5} \left (3 x^2+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{18} (2 x+3)^3 \left (3 x^2+2\right )^{3/2}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{6} \left (\frac {14}{5} \left (\frac {730}{3} \left (\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {89}{6} x \left (3 x^2+2\right )^{3/2}+\frac {449}{9} \left (3 x^2+2\right )^{3/2}\right )+\frac {17}{5} \left (3 x^2+2\right )^{3/2} (2 x+3)^2\right )-\frac {1}{18} (2 x+3)^3 \left (3 x^2+2\right )^{3/2}\) |
Input:
Int[(5 - x)*(3 + 2*x)^3*Sqrt[2 + 3*x^2],x]
Output:
-1/18*((3 + 2*x)^3*(2 + 3*x^2)^(3/2)) + ((17*(3 + 2*x)^2*(2 + 3*x^2)^(3/2) )/5 + (14*((449*(2 + 3*x^2)^(3/2))/9 + (89*x*(2 + 3*x^2)^(3/2))/6 + (730*( (x*Sqrt[2 + 3*x^2])/2 + ArcSinh[Sqrt[3/2]*x]/Sqrt[3]))/3))/5)/6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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])
Time = 0.78 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.50
method | result | size |
risch | \(-\frac {\left (360 x^{5}-216 x^{4}-8445 x^{3}-21918 x^{2}-21120 x -14516\right ) \sqrt {3 x^{2}+2}}{270}+\frac {1022 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{27}\) | \(50\) |
trager | \(\left (-\frac {4}{3} x^{5}+\frac {4}{5} x^{4}+\frac {563}{18} x^{3}+\frac {3653}{45} x^{2}+\frac {704}{9} x +\frac {7258}{135}\right ) \sqrt {3 x^{2}+2}-\frac {1022 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{27}\) | \(67\) |
default | \(\frac {511 x \sqrt {3 x^{2}+2}}{9}+\frac {1022 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{27}+\frac {3629 \left (3 x^{2}+2\right )^{\frac {3}{2}}}{135}+\frac {193 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{18}+\frac {4 x^{2} \left (3 x^{2}+2\right )^{\frac {3}{2}}}{15}-\frac {4 x^{3} \left (3 x^{2}+2\right )^{\frac {3}{2}}}{9}\) | \(77\) |
meijerg | \(-\frac {45 \sqrt {3}\, \left (-\sqrt {6}\, \sqrt {\pi }\, x \sqrt {\frac {3 x^{2}}{2}+1}-2 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{2 \sqrt {\pi }}-\frac {4 \sqrt {2}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}} \left (-\frac {9 x^{2}}{2}+2\right )}{15}\right )}{9 \sqrt {\pi }}-\frac {14 \sqrt {3}\, \left (-\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (9 x^{2}+3\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{12}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{2}\right )}{\sqrt {\pi }}-\frac {81 \sqrt {2}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (3 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{3}\right )}{2 \sqrt {\pi }}+\frac {16 \sqrt {3}\, \left (\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (-90 x^{4}-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{120}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{4}\right )}{27 \sqrt {\pi }}\) | \(217\) |
Input:
int((5-x)*(2*x+3)^3*(3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/270*(360*x^5-216*x^4-8445*x^3-21918*x^2-21120*x-14516)*(3*x^2+2)^(1/2)+ 1022/27*arcsinh(1/2*6^(1/2)*x)*3^(1/2)
Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.65 \[ \int (5-x) (3+2 x)^3 \sqrt {2+3 x^2} \, dx=-\frac {1}{270} \, {\left (360 \, x^{5} - 216 \, x^{4} - 8445 \, x^{3} - 21918 \, x^{2} - 21120 \, x - 14516\right )} \sqrt {3 \, x^{2} + 2} + \frac {511}{27} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \] Input:
integrate((5-x)*(3+2*x)^3*(3*x^2+2)^(1/2),x, algorithm="fricas")
Output:
-1/270*(360*x^5 - 216*x^4 - 8445*x^3 - 21918*x^2 - 21120*x - 14516)*sqrt(3 *x^2 + 2) + 511/27*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)
Time = 1.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.63 \[ \int (5-x) (3+2 x)^3 \sqrt {2+3 x^2} \, dx=\sqrt {3 x^{2} + 2} \left (- \frac {4 x^{5}}{3} + \frac {4 x^{4}}{5} + \frac {563 x^{3}}{18} + \frac {3653 x^{2}}{45} + \frac {704 x}{9} + \frac {7258}{135}\right ) + \frac {1022 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{27} \] Input:
integrate((5-x)*(3+2*x)**3*(3*x**2+2)**(1/2),x)
Output:
sqrt(3*x**2 + 2)*(-4*x**5/3 + 4*x**4/5 + 563*x**3/18 + 3653*x**2/45 + 704* x/9 + 7258/135) + 1022*sqrt(3)*asinh(sqrt(6)*x/2)/27
Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.76 \[ \int (5-x) (3+2 x)^3 \sqrt {2+3 x^2} \, dx=-\frac {4}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{3} + \frac {4}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + \frac {193}{18} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {3629}{135} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} + \frac {511}{9} \, \sqrt {3 \, x^{2} + 2} x + \frac {1022}{27} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \] Input:
integrate((5-x)*(3+2*x)^3*(3*x^2+2)^(1/2),x, algorithm="maxima")
Output:
-4/9*(3*x^2 + 2)^(3/2)*x^3 + 4/15*(3*x^2 + 2)^(3/2)*x^2 + 193/18*(3*x^2 + 2)^(3/2)*x + 3629/135*(3*x^2 + 2)^(3/2) + 511/9*sqrt(3*x^2 + 2)*x + 1022/2 7*sqrt(3)*arcsinh(1/2*sqrt(6)*x)
Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.57 \[ \int (5-x) (3+2 x)^3 \sqrt {2+3 x^2} \, dx=-\frac {1}{270} \, {\left (3 \, {\left ({\left ({\left (24 \, {\left (5 \, x - 3\right )} x - 2815\right )} x - 7306\right )} x - 7040\right )} x - 14516\right )} \sqrt {3 \, x^{2} + 2} - \frac {1022}{27} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \] Input:
integrate((5-x)*(3+2*x)^3*(3*x^2+2)^(1/2),x, algorithm="giac")
Output:
-1/270*(3*(((24*(5*x - 3)*x - 2815)*x - 7306)*x - 7040)*x - 14516)*sqrt(3* x^2 + 2) - 1022/27*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))
Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.50 \[ \int (5-x) (3+2 x)^3 \sqrt {2+3 x^2} \, dx=\frac {1022\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-4\,x^5+\frac {12\,x^4}{5}+\frac {563\,x^3}{6}+\frac {3653\,x^2}{15}+\frac {704\,x}{3}+\frac {7258}{45}\right )}{3} \] Input:
int(-(2*x + 3)^3*(3*x^2 + 2)^(1/2)*(x - 5),x)
Output:
(1022*3^(1/2)*asinh((6^(1/2)*x)/2))/27 + (3^(1/2)*(x^2 + 2/3)^(1/2)*((704* x)/3 + (3653*x^2)/15 + (563*x^3)/6 + (12*x^4)/5 - 4*x^5 + 7258/45))/3
Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.97 \[ \int (5-x) (3+2 x)^3 \sqrt {2+3 x^2} \, dx=-\frac {4 \sqrt {3 x^{2}+2}\, x^{5}}{3}+\frac {4 \sqrt {3 x^{2}+2}\, x^{4}}{5}+\frac {563 \sqrt {3 x^{2}+2}\, x^{3}}{18}+\frac {3653 \sqrt {3 x^{2}+2}\, x^{2}}{45}+\frac {704 \sqrt {3 x^{2}+2}\, x}{9}+\frac {7258 \sqrt {3 x^{2}+2}}{135}+\frac {1022 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )}{27} \] Input:
int((5-x)*(3+2*x)^3*(3*x^2+2)^(1/2),x)
Output:
( - 360*sqrt(3*x**2 + 2)*x**5 + 216*sqrt(3*x**2 + 2)*x**4 + 8445*sqrt(3*x* *2 + 2)*x**3 + 21918*sqrt(3*x**2 + 2)*x**2 + 21120*sqrt(3*x**2 + 2)*x + 14 516*sqrt(3*x**2 + 2) + 10220*sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sq rt(2)))/270