Integrand size = 24, antiderivative size = 67 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {7 (2-7 x) (3+2 x)^2}{6 \sqrt {2+3 x^2}}-\frac {2}{9} (131+51 x) \sqrt {2+3 x^2}+\frac {134 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \] Output:
-7/6*(2-7*x)*(3+2*x)^2/(3*x^2+2)^(1/2)-2/9*(131+51*x)*(3*x^2+2)^(1/2)+134/ 9*arcsinh(1/2*x*6^(1/2))*3^(1/2)
Time = 0.34 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {1426-411 x-24 x^2+24 x^3}{18 \sqrt {2+3 x^2}}-\frac {134 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{3 \sqrt {3}} \] Input:
Integrate[((5 - x)*(3 + 2*x)^3)/(2 + 3*x^2)^(3/2),x]
Output:
-1/18*(1426 - 411*x - 24*x^2 + 24*x^3)/Sqrt[2 + 3*x^2] - (134*Log[-(Sqrt[3 ]*x) + Sqrt[2 + 3*x^2]])/(3*Sqrt[3])
Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {684, 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.
\(\displaystyle \int \frac {(5-x) (2 x+3)^3}{\left (3 x^2+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 684 |
\(\displaystyle \frac {1}{6} \int \frac {4 (11-51 x) (2 x+3)}{\sqrt {3 x^2+2}}dx-\frac {7 (2-7 x) (2 x+3)^2}{6 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{3} \int \frac {(11-51 x) (2 x+3)}{\sqrt {3 x^2+2}}dx-\frac {7 (2-7 x) (2 x+3)^2}{6 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 676 |
\(\displaystyle \frac {2}{3} \left (67 \int \frac {1}{\sqrt {3 x^2+2}}dx-17 \sqrt {3 x^2+2} x-\frac {131}{3} \sqrt {3 x^2+2}\right )-\frac {7 (2-7 x) (2 x+3)^2}{6 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {2}{3} \left (\frac {67 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}-17 \sqrt {3 x^2+2} x-\frac {131}{3} \sqrt {3 x^2+2}\right )-\frac {7 (2-7 x) (2 x+3)^2}{6 \sqrt {3 x^2+2}}\) |
Input:
Int[((5 - x)*(3 + 2*x)^3)/(2 + 3*x^2)^(3/2),x]
Output:
(-7*(2 - 7*x)*(3 + 2*x)^2)/(6*Sqrt[2 + 3*x^2]) + (2*((-131*Sqrt[2 + 3*x^2] )/3 - 17*x*Sqrt[2 + 3*x^2] + (67*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[((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[(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 = 0.78 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.60
method | result | size |
risch | \(-\frac {24 x^{3}-24 x^{2}-411 x +1426}{18 \sqrt {3 x^{2}+2}}+\frac {134 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{9}\) | \(40\) |
trager | \(-\frac {24 x^{3}-24 x^{2}-411 x +1426}{18 \sqrt {3 x^{2}+2}}+\frac {134 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{9}\) | \(57\) |
default | \(\frac {137 x}{6 \sqrt {3 x^{2}+2}}+\frac {134 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{9}+\frac {4 x^{2}}{3 \sqrt {3 x^{2}+2}}-\frac {713}{9 \sqrt {3 x^{2}+2}}-\frac {4 x^{3}}{3 \sqrt {3 x^{2}+2}}\) | \(65\) |
meijerg | \(\frac {135 \sqrt {2}\, x}{4 \sqrt {\frac {3 x^{2}}{2}+1}}+\frac {4 \sqrt {2}\, \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (6 x^{2}+8\right )}{4 \sqrt {\frac {3 x^{2}}{2}+1}}\right )}{9 \sqrt {\pi }}+\frac {14 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}}{2 \sqrt {\frac {3 x^{2}}{2}+1}}+\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{\sqrt {\pi }}+\frac {81 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {\frac {3 x^{2}}{2}+1}}\right )}{2 \sqrt {\pi }}-\frac {16 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (\frac {15 x^{2}}{2}+15\right )}{20 \sqrt {\frac {3 x^{2}}{2}+1}}-\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{2}\right )}{27 \sqrt {\pi }}\) | \(174\) |
Input:
int((5-x)*(2*x+3)^3/(3*x^2+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/18*(24*x^3-24*x^2-411*x+1426)/(3*x^2+2)^(1/2)+134/9*arcsinh(1/2*6^(1/2) *x)*3^(1/2)
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.09 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {134 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - {\left (24 \, x^{3} - 24 \, x^{2} - 411 \, x + 1426\right )} \sqrt {3 \, x^{2} + 2}}{18 \, {\left (3 \, x^{2} + 2\right )}} \] Input:
integrate((5-x)*(3+2*x)^3/(3*x^2+2)^(3/2),x, algorithm="fricas")
Output:
1/18*(134*sqrt(3)*(3*x^2 + 2)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) - (24*x^3 - 24*x^2 - 411*x + 1426)*sqrt(3*x^2 + 2))/(3*x^2 + 2)
\[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx=- \int \left (- \frac {243 x}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {126 x^{2}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {4 x^{3}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {8 x^{4}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {135}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx \] Input:
integrate((5-x)*(3+2*x)**3/(3*x**2+2)**(3/2),x)
Output:
-Integral(-243*x/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Inte gral(-126*x**2/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integr al(-4*x**3/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(8 *x**4/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-135/( 3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x)
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {4 \, x^{3}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {4 \, x^{2}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {134}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {137 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} - \frac {713}{9 \, \sqrt {3 \, x^{2} + 2}} \] Input:
integrate((5-x)*(3+2*x)^3/(3*x^2+2)^(3/2),x, algorithm="maxima")
Output:
-4/3*x^3/sqrt(3*x^2 + 2) + 4/3*x^2/sqrt(3*x^2 + 2) + 134/9*sqrt(3)*arcsinh (1/2*sqrt(6)*x) + 137/6*x/sqrt(3*x^2 + 2) - 713/9/sqrt(3*x^2 + 2)
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.70 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {134}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {3 \, {\left (8 \, {\left (x - 1\right )} x - 137\right )} x + 1426}{18 \, \sqrt {3 \, x^{2} + 2}} \] Input:
integrate((5-x)*(3+2*x)^3/(3*x^2+2)^(3/2),x, algorithm="giac")
Output:
-134/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/18*(3*(8*(x - 1)*x - 137)*x + 1426)/sqrt(3*x^2 + 2)
Time = 0.02 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.57 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {134\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}-\frac {\sqrt {3}\,\left (\frac {4\,x}{3}-\frac {4}{3}\right )\,\sqrt {x^2+\frac {2}{3}}}{3}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-12978+\sqrt {6}\,1281{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (12978+\sqrt {6}\,1281{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:
int(-((2*x + 3)^3*(x - 5))/(3*x^2 + 2)^(3/2),x)
Output:
(134*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/9 - (3^(1/2)*((4*x)/3 - 4/3)*(x ^2 + 2/3)^(1/2))/3 - (3^(1/2)*6^(1/2)*(6^(1/2)*1281i - 12978)*(x^2 + 2/3)^ (1/2)*1i)/(1944*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*1281i + 12978)*(x^2 + 2/3)^(1/2)*1i)/(1944*(x + (6^(1/2)*1i)/3))
Time = 0.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.76 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {-24 \sqrt {3 x^{2}+2}\, x^{3}+24 \sqrt {3 x^{2}+2}\, x^{2}+411 \sqrt {3 x^{2}+2}\, x -1426 \sqrt {3 x^{2}+2}+804 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right ) x^{2}+536 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )+423 \sqrt {3}\, x^{2}+282 \sqrt {3}}{54 x^{2}+36} \] Input:
int((5-x)*(3+2*x)^3/(3*x^2+2)^(3/2),x)
Output:
( - 24*sqrt(3*x**2 + 2)*x**3 + 24*sqrt(3*x**2 + 2)*x**2 + 411*sqrt(3*x**2 + 2)*x - 1426*sqrt(3*x**2 + 2) + 804*sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt( 3)*x)/sqrt(2))*x**2 + 536*sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt( 2)) + 423*sqrt(3)*x**2 + 282*sqrt(3))/(18*(3*x**2 + 2))