Integrand size = 24, antiderivative size = 89 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {7 (2-7 x) (3+2 x)^3}{6 \sqrt {2+3 x^2}}-\frac {151}{27} (3+2 x)^2 \sqrt {2+3 x^2}-\frac {10}{81} (185+207 x) \sqrt {2+3 x^2}+\frac {880 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \]
880/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)-7/6*(2-7*x)*(3+2*x)^3/(3*x^2+2)^(1/2) -151/27*(3+2*x)^2*(3*x^2+2)^(1/2)-10/81*(185+207*x)*(3*x^2+2)^(1/2)
Time = 0.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {33914+14715 x-15024 x^2+432 x^3+288 x^4}{162 \sqrt {2+3 x^2}}-\frac {880 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{3 \sqrt {3}} \]
-1/162*(33914 + 14715*x - 15024*x^2 + 432*x^3 + 288*x^4)/Sqrt[2 + 3*x^2] - (880*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(3*Sqrt[3])
Time = 0.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {684, 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.
\(\displaystyle \int \frac {(5-x) (2 x+3)^4}{\left (3 x^2+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 684 |
\(\displaystyle \frac {1}{6} \int \frac {2 (36-151 x) (2 x+3)^2}{\sqrt {3 x^2+2}}dx-\frac {7 (2-7 x) (2 x+3)^3}{6 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int \frac {(36-151 x) (2 x+3)^2}{\sqrt {3 x^2+2}}dx-\frac {7 (2-7 x) (2 x+3)^3}{6 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 687 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{9} \int \frac {10 (218-207 x) (2 x+3)}{\sqrt {3 x^2+2}}dx-\frac {151}{9} (2 x+3)^2 \sqrt {3 x^2+2}\right )-\frac {7 (2-7 x) (2 x+3)^3}{6 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (\frac {10}{9} \int \frac {(218-207 x) (2 x+3)}{\sqrt {3 x^2+2}}dx-\frac {151}{9} (2 x+3)^2 \sqrt {3 x^2+2}\right )-\frac {7 (2-7 x) (2 x+3)^3}{6 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 676 |
\(\displaystyle \frac {1}{3} \left (\frac {10}{9} \left (792 \int \frac {1}{\sqrt {3 x^2+2}}dx-69 \sqrt {3 x^2+2} x-\frac {185}{3} \sqrt {3 x^2+2}\right )-\frac {151}{9} (2 x+3)^2 \sqrt {3 x^2+2}\right )-\frac {7 (2-7 x) (2 x+3)^3}{6 \sqrt {3 x^2+2}}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{3} \left (\frac {10}{9} \left (264 \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-69 \sqrt {3 x^2+2} x-\frac {185}{3} \sqrt {3 x^2+2}\right )-\frac {151}{9} (2 x+3)^2 \sqrt {3 x^2+2}\right )-\frac {7 (2-7 x) (2 x+3)^3}{6 \sqrt {3 x^2+2}}\) |
(-7*(2 - 7*x)*(3 + 2*x)^3)/(6*Sqrt[2 + 3*x^2]) + ((-151*(3 + 2*x)^2*Sqrt[2 + 3*x^2])/9 + (10*((-185*Sqrt[2 + 3*x^2])/3 - 69*x*Sqrt[2 + 3*x^2] + 264* Sqrt[3]*ArcSinh[Sqrt[3/2]*x]))/9)/3
3.15.8.3.1 Defintions of rubi rules used
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])
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 = 0.32 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.51
method | result | size |
risch | \(-\frac {288 x^{4}+432 x^{3}-15024 x^{2}+14715 x +33914}{162 \sqrt {3 x^{2}+2}}+\frac {880 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{9}\) | \(45\) |
trager | \(-\frac {288 x^{4}+432 x^{3}-15024 x^{2}+14715 x +33914}{162 \sqrt {3 x^{2}+2}}-\frac {880 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{9}\) | \(63\) |
default | \(-\frac {545 x}{6 \sqrt {3 x^{2}+2}}-\frac {16957}{81 \sqrt {3 x^{2}+2}}-\frac {16 x^{4}}{9 \sqrt {3 x^{2}+2}}+\frac {2504 x^{2}}{27 \sqrt {3 x^{2}+2}}-\frac {8 x^{3}}{3 \sqrt {3 x^{2}+2}}+\frac {880 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{9}\) | \(79\) |
meijerg | \(\frac {405 \sqrt {2}\, x}{4 \sqrt {\frac {3 x^{2}}{2}+1}}+\frac {333 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {\frac {3 x^{2}}{2}+1}}\right )}{2 \sqrt {\pi }}+\frac {96 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}}{2 \sqrt {\frac {3 x^{2}}{2}+1}}+\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )\right )}{\sqrt {\pi }}+\frac {88 \sqrt {2}\, \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (6 x^{2}+8\right )}{4 \sqrt {\frac {3 x^{2}}{2}+1}}\right )}{3 \sqrt {\pi }}-\frac {32 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \left (\frac {15 x^{2}}{2}+15\right )}{20 \sqrt {\frac {3 x^{2}}{2}+1}}-\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{2}\right )}{27 \sqrt {\pi }}-\frac {32 \sqrt {2}\, \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {9}{2} x^{4}+12 x^{2}+16\right )}{6 \sqrt {\frac {3 x^{2}}{2}+1}}\right )}{27 \sqrt {\pi }}\) | \(214\) |
-1/162*(288*x^4+432*x^3-15024*x^2+14715*x+33914)/(3*x^2+2)^(1/2)+880/9*arc sinh(1/2*x*6^(1/2))*3^(1/2)
Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {7920 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - {\left (288 \, x^{4} + 432 \, x^{3} - 15024 \, x^{2} + 14715 \, x + 33914\right )} \sqrt {3 \, x^{2} + 2}}{162 \, {\left (3 \, x^{2} + 2\right )}} \]
1/162*(7920*sqrt(3)*(3*x^2 + 2)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1 ) - (288*x^4 + 432*x^3 - 15024*x^2 + 14715*x + 33914)*sqrt(3*x^2 + 2))/(3* x^2 + 2)
\[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=- \int \left (- \frac {999 x}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {864 x^{2}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {264 x^{3}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {16 x^{4}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\, dx - \int \frac {16 x^{5}}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {405}{3 x^{2} \sqrt {3 x^{2} + 2} + 2 \sqrt {3 x^{2} + 2}}\right )\, dx \]
-Integral(-999*x/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Inte gral(-864*x**2/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integr al(-264*x**3/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral (16*x**4/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(16* x**5/(3*x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x) - Integral(-405/(3 *x**2*sqrt(3*x**2 + 2) + 2*sqrt(3*x**2 + 2)), x)
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {16 \, x^{4}}{9 \, \sqrt {3 \, x^{2} + 2}} - \frac {8 \, x^{3}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {2504 \, x^{2}}{27 \, \sqrt {3 \, x^{2} + 2}} + \frac {880}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {545 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} - \frac {16957}{81 \, \sqrt {3 \, x^{2} + 2}} \]
-16/9*x^4/sqrt(3*x^2 + 2) - 8/3*x^3/sqrt(3*x^2 + 2) + 2504/27*x^2/sqrt(3*x ^2 + 2) + 880/9*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 545/6*x/sqrt(3*x^2 + 2) - 16957/81/sqrt(3*x^2 + 2)
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.61 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {880}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {3 \, {\left (16 \, {\left (3 \, {\left (2 \, x + 3\right )} x - 313\right )} x + 4905\right )} x + 33914}{162 \, \sqrt {3 \, x^{2} + 2}} \]
-880/9*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/162*(3*(16*(3*(2*x + 3)*x - 313)*x + 4905)*x + 33914)/sqrt(3*x^2 + 2)
Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {880\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {16\,x^2}{9}+\frac {8\,x}{3}-\frac {2536}{27}\right )}{3}+\frac {\sqrt {3}\,\sqrt {6}\,\left (-44058+\sqrt {6}\,4809{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\left (44058+\sqrt {6}\,4809{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]
(880*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/9 - (3^(1/2)*(x^2 + 2/3)^(1/2)* ((8*x)/3 + (16*x^2)/9 - 2536/27))/3 + (3^(1/2)*6^(1/2)*(6^(1/2)*4809i - 44 058)*(x^2 + 2/3)^(1/2)*1i)/(1944*(x + (6^(1/2)*1i)/3)) + (3^(1/2)*6^(1/2)* (6^(1/2)*4809i + 44058)*(x^2 + 2/3)^(1/2)*1i)/(1944*(x - (6^(1/2)*1i)/3))