Integrand size = 24, antiderivative size = 60 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {7 (2-7 x) (3+2 x)}{6 \sqrt {2+3 x^2}}-\frac {53}{9} \sqrt {2+3 x^2}+\frac {8 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \] Output:
-7/6*(2-7*x)*(3+2*x)/(3*x^2+2)^(1/2)-53/9*(3*x^2+2)^(1/2)+8/9*arcsinh(1/2* x*6^(1/2))*3^(1/2)
Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.93 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {338-357 x+24 x^2}{18 \sqrt {2+3 x^2}}-\frac {8 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{3 \sqrt {3}} \] Input:
Integrate[((5 - x)*(3 + 2*x)^2)/(2 + 3*x^2)^(3/2),x]
Output:
-1/18*(338 - 357*x + 24*x^2)/Sqrt[2 + 3*x^2] - (8*Log[-(Sqrt[3]*x) + Sqrt[ 2 + 3*x^2]])/(3*Sqrt[3])
Time = 0.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {684, 27, 455, 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.
\(\displaystyle \int \frac {(5-x) (2 x+3)^2}{\left (3 x^2+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 684 |
\(\displaystyle \frac {1}{6} \int \frac {2 (8-53 x)}{\sqrt {3 x^2+2}}dx-\frac {7 (2-7 x) (2 x+3)}{6 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {8-53 x}{\sqrt {3 x^2+2}}dx-\frac {7 (2-7 x) (2 x+3)}{6 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 455 |
\(\displaystyle \frac {1}{3} \left (8 \int \frac {1}{\sqrt {3 x^2+2}}dx-\frac {53}{3} \sqrt {3 x^2+2}\right )-\frac {7 (2-7 x) (2 x+3)}{6 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{3} \left (\frac {8 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}-\frac {53}{3} \sqrt {3 x^2+2}\right )-\frac {7 (2-7 x) (2 x+3)}{6 \sqrt {3 x^2+2}}\) |
Input:
Int[((5 - x)*(3 + 2*x)^2)/(2 + 3*x^2)^(3/2),x]
Output:
(-7*(2 - 7*x)*(3 + 2*x))/(6*Sqrt[2 + 3*x^2]) + ((-53*Sqrt[2 + 3*x^2])/3 + (8*ArcSinh[Sqrt[3/2]*x])/Sqrt[3])/3
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[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g ) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1)) Int[ (d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a , c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Time = 0.75 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.58
method | result | size |
risch | \(-\frac {24 x^{2}-357 x +338}{18 \sqrt {3 x^{2}+2}}+\frac {8 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{9}\) | \(35\) |
default | \(\frac {119 x}{6 \sqrt {3 x^{2}+2}}+\frac {8 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{9}-\frac {4 x^{2}}{3 \sqrt {3 x^{2}+2}}-\frac {169}{9 \sqrt {3 x^{2}+2}}\) | \(51\) |
trager | \(-\frac {24 x^{2}-357 x +338}{18 \sqrt {3 x^{2}+2}}+\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{9}\) | \(52\) |
meijerg | \(\frac {45 \sqrt {2}\, x}{4 \sqrt {\frac {3 x^{2}}{2}+1}}+\frac {8 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}}{2 \sqrt {\frac {3 x^{2}}{2}+1}}+\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{9 \sqrt {\pi }}+\frac {17 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {\frac {3 x^{2}}{2}+1}}\right )}{2 \sqrt {\pi }}-\frac {4 \sqrt {2}\, \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (6 x^{2}+8\right )}{4 \sqrt {\frac {3 x^{2}}{2}+1}}\right )}{9 \sqrt {\pi }}\) | \(122\) |
Input:
int((5-x)*(2*x+3)^2/(3*x^2+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/18*(24*x^2-357*x+338)/(3*x^2+2)^(1/2)+8/9*arcsinh(1/2*6^(1/2)*x)*3^(1/2 )
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.13 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {8 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - {\left (24 \, x^{2} - 357 \, x + 338\right )} \sqrt {3 \, x^{2} + 2}}{18 \, {\left (3 \, x^{2} + 2\right )}} \] Input:
integrate((5-x)*(3+2*x)^2/(3*x^2+2)^(3/2),x, algorithm="fricas")
Output:
1/18*(8*sqrt(3)*(3*x^2 + 2)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - (24*x^2 - 357*x + 338)*sqrt(3*x^2 + 2))/(3*x^2 + 2)
\[ \int \frac {(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{3/2}} \, dx=- \int \left (- \frac {51 x}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {8 x^{2}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {4 x^{3}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {45}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx \] Input:
integrate((5-x)*(3+2*x)**2/(3*x**2+2)**(3/2),x)
Output:
-Integral(-51*x/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integ ral(-8*x**2/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral( 4*x**3/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-45/( 3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x)
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {4 \, x^{2}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {8}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {119 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} - \frac {169}{9 \, \sqrt {3 \, x^{2} + 2}} \] Input:
integrate((5-x)*(3+2*x)^2/(3*x^2+2)^(3/2),x, algorithm="maxima")
Output:
-4/3*x^2/sqrt(3*x^2 + 2) + 8/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 119/6*x/sq rt(3*x^2 + 2) - 169/9/sqrt(3*x^2 + 2)
Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.73 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {8}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {3 \, {\left (8 \, x - 119\right )} x + 338}{18 \, \sqrt {3 \, x^{2} + 2}} \] Input:
integrate((5-x)*(3+2*x)^2/(3*x^2+2)^(3/2),x, algorithm="giac")
Output:
-8/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/18*(3*(8*x - 119)*x + 3 38)/sqrt(3*x^2 + 2)
Time = 5.98 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.67 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {8\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}-\frac {4\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-966+\sqrt {6}\,357{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{648\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (966+\sqrt {6}\,357{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{648\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:
int(-((2*x + 3)^2*(x - 5))/(3*x^2 + 2)^(3/2),x)
Output:
(8*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/9 - (4*3^(1/2)*(x^2 + 2/3)^(1/2)) /9 - (3^(1/2)*6^(1/2)*(6^(1/2)*357i - 966)*(x^2 + 2/3)^(1/2)*1i)/(648*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*357i + 966)*(x^2 + 2/3)^(1/2 )*1i)/(648*(x + (6^(1/2)*1i)/3))
Time = 0.30 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.75 \[ \int \frac {(5-x) (3+2 x)^2}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {-24 \sqrt {3 x^{2}+2}\, x^{2}+357 \sqrt {3 x^{2}+2}\, x -338 \sqrt {3 x^{2}+2}+48 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right ) x^{2}+32 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )+357 \sqrt {3}\, x^{2}+238 \sqrt {3}}{54 x^{2}+36} \] Input:
int((5-x)*(3+2*x)^2/(3*x^2+2)^(3/2),x)
Output:
( - 24*sqrt(3*x**2 + 2)*x**2 + 357*sqrt(3*x**2 + 2)*x - 338*sqrt(3*x**2 + 2) + 48*sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2))*x**2 + 32*sqrt (3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2)) + 357*sqrt(3)*x**2 + 238*s qrt(3))/(18*(3*x**2 + 2))