\(\int \frac {(5-x) (3+2 x)^5}{(2+3 x^2)^{5/2}} \, dx\) [253]

Optimal result

Integrand size = 24, antiderivative size = 94 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {7 (2-7 x) (3+2 x)^4}{18 \left (2+3 x^2\right )^{3/2}}-\frac {5 (16-421 x) (3+2 x)^2}{54 \sqrt {2+3 x^2}}-\frac {50}{81} (299+93 x) \sqrt {2+3 x^2}+\frac {1600 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{27 \sqrt {3}} \] Output:

-7/18*(2-7*x)*(3+2*x)^4/(3*x^2+2)^(3/2)-5/54*(16-421*x)*(3+2*x)^2/(3*x^2+2 
)^(1/2)-50/81*(299+93*x)*(3*x^2+2)^(1/2)+1600/81*arcsinh(1/2*x*6^(1/2))*3^ 
(1/2)
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.76 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {134126-79215 x+147600 x^2-183945 x^3+4320 x^4+864 x^5}{162 \left (2+3 x^2\right )^{3/2}}-\frac {1600 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{27 \sqrt {3}} \] Input:

Integrate[((5 - x)*(3 + 2*x)^5)/(2 + 3*x^2)^(5/2),x]
 

Output:

-1/162*(134126 - 79215*x + 147600*x^2 - 183945*x^3 + 4320*x^4 + 864*x^5)/( 
2 + 3*x^2)^(3/2) - (1600*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(27*Sqrt[3])
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {684, 27, 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.

\(\displaystyle \int \frac {(5-x) (2 x+3)^5}{\left (3 x^2+2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 684

\(\displaystyle \frac {1}{18} \int \frac {10 (37-22 x) (2 x+3)^3}{\left (3 x^2+2\right )^{3/2}}dx-\frac {7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{9} \int \frac {(37-22 x) (2 x+3)^3}{\left (3 x^2+2\right )^{3/2}}dx-\frac {7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 684

\(\displaystyle \frac {5}{9} \left (\frac {1}{6} \int -\frac {20 (2 x+3) (93 x+10)}{\sqrt {3 x^2+2}}dx-\frac {(16-421 x) (2 x+3)^2}{6 \sqrt {3 x^2+2}}\right )-\frac {7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{9} \left (-\frac {10}{3} \int \frac {(2 x+3) (93 x+10)}{\sqrt {3 x^2+2}}dx-\frac {(16-421 x) (2 x+3)^2}{6 \sqrt {3 x^2+2}}\right )-\frac {7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {5}{9} \left (-\frac {10}{3} \left (-32 \int \frac {1}{\sqrt {3 x^2+2}}dx+31 \sqrt {3 x^2+2} x+\frac {299}{3} \sqrt {3 x^2+2}\right )-\frac {(16-421 x) (2 x+3)^2}{6 \sqrt {3 x^2+2}}\right )-\frac {7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {5}{9} \left (-\frac {10}{3} \left (-\frac {32 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+31 \sqrt {3 x^2+2} x+\frac {299}{3} \sqrt {3 x^2+2}\right )-\frac {(16-421 x) (2 x+3)^2}{6 \sqrt {3 x^2+2}}\right )-\frac {7 (2-7 x) (2 x+3)^4}{18 \left (3 x^2+2\right )^{3/2}}\)

Input:

Int[((5 - x)*(3 + 2*x)^5)/(2 + 3*x^2)^(5/2),x]
 

Output:

(-7*(2 - 7*x)*(3 + 2*x)^4)/(18*(2 + 3*x^2)^(3/2)) + (5*(-1/6*((16 - 421*x) 
*(3 + 2*x)^2)/Sqrt[2 + 3*x^2] - (10*((299*Sqrt[2 + 3*x^2])/3 + 31*x*Sqrt[2 
 + 3*x^2] - (32*ArcSinh[Sqrt[3/2]*x])/Sqrt[3]))/3))/9
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 222

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]
 

rule 676

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]
 

rule 684

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])
 

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.53

method	result	size
risch	\(-\frac {864 x^{5}+4320 x^{4}-183945 x^{3}+147600 x^{2}-79215 x +134126}{162 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {1600 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{81}\)	\(50\)
trager	\(-\frac {864 x^{5}+4320 x^{4}-183945 x^{3}+147600 x^{2}-79215 x +134126}{162 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {1600 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{81}\)	\(68\)
default	\(-\frac {615 x}{2 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {21505 x}{54 \sqrt {3 x^{2}+2}}-\frac {67063}{81 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {8200 x^{2}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {1600 x^{3}}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {1600 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{81}-\frac {80 x^{4}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {16 x^{5}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\)	\(105\)
meijerg	\(\frac {405 \sqrt {2}\, x \left (3 x^{2}+3\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}-\frac {160 \sqrt {2}\, \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (\frac {27}{2} x^{4}+36 x^{2}+16\right )}{4 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{81 \sqrt {\pi }}+\frac {320 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (30 x^{2}+15\right )}{20 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{2}\right )}{27 \sqrt {\pi }}+\frac {280 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (18 x^{2}+8\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{3 \sqrt {\pi }}+\frac {765 \sqrt {2}\, x^{3}}{4 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {423 \sqrt {2}\, \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{2 \sqrt {\pi }}-\frac {128 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (\frac {189}{4} x^{4}+210 x^{2}+105\right )}{56 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}-\frac {15 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{4}\right )}{243 \sqrt {\pi }}\)	\(251\)

Input:

int((5-x)*(2*x+3)^5/(3*x^2+2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/162*(864*x^5+4320*x^4-183945*x^3+147600*x^2-79215*x+134126)/(3*x^2+2)^( 
3/2)+1600/81*arcsinh(1/2*6^(1/2)*x)*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {1600 \, \sqrt {3} {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - {\left (864 \, x^{5} + 4320 \, x^{4} - 183945 \, x^{3} + 147600 \, x^{2} - 79215 \, x + 134126\right )} \sqrt {3 \, x^{2} + 2}}{162 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \] Input:

integrate((5-x)*(3+2*x)^5/(3*x^2+2)^(5/2),x, algorithm="fricas")
 

Output:

1/162*(1600*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 
3*x^2 - 1) - (864*x^5 + 4320*x^4 - 183945*x^3 + 147600*x^2 - 79215*x + 134 
126)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x^2 + 4)
 

Sympy [F]

\[ \int \frac {(5-x) (3+2 x)^5}{\left (2+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {3807 x}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {4590 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {2520 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {480 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {80 x^{5}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \frac {32 x^{6}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {1215}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx \] Input:

integrate((5-x)*(3+2*x)**5/(3*x**2+2)**(5/2),x)
 

Output:

-Integral(-3807*x/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4* 
sqrt(3*x**2 + 2)), x) - Integral(-4590*x**2/(9*x**4*sqrt(3*x**2 + 2) + 12* 
x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(-2520*x**3/(9*x 
**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) 
- Integral(-480*x**4/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 
 4*sqrt(3*x**2 + 2)), x) - Integral(80*x**5/(9*x**4*sqrt(3*x**2 + 2) + 12* 
x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Integral(32*x**6/(9*x**4 
*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - I 
ntegral(-1215/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt 
(3*x**2 + 2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.27 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {16 \, x^{5}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {80 \, x^{4}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {1600}{81} \, x {\left (\frac {9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {4}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}\right )} + \frac {1600}{81} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {70915 \, x}{162 \, \sqrt {3 \, x^{2} + 2}} - \frac {8200 \, x^{2}}{9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {615 \, x}{2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {67063}{81 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

integrate((5-x)*(3+2*x)^5/(3*x^2+2)^(5/2),x, algorithm="maxima")
 

Output:

-16/3*x^5/(3*x^2 + 2)^(3/2) - 80/3*x^4/(3*x^2 + 2)^(3/2) - 1600/81*x*(9*x^ 
2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(3/2)) + 1600/81*sqrt(3)*arcsinh(1/2*s 
qrt(6)*x) + 70915/162*x/sqrt(3*x^2 + 2) - 8200/9*x^2/(3*x^2 + 2)^(3/2) - 6 
15/2*x/(3*x^2 + 2)^(3/2) - 67063/81/(3*x^2 + 2)^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.59 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {1600}{81} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {3 \, {\left ({\left ({\left (288 \, {\left (x + 5\right )} x - 61315\right )} x + 49200\right )} x - 26405\right )} x + 134126}{162 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

integrate((5-x)*(3+2*x)^5/(3*x^2+2)^(5/2),x, algorithm="giac")
 

Output:

-1600/81*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/162*(3*(((288*(x + 
5)*x - 61315)*x + 49200)*x - 26405)*x + 134126)/(3*x^2 + 2)^(3/2)
 

Mupad [B] (verification not implemented)

Time = 5.96 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.31 \[ \int \frac {(5-x) (3+2 x)^5}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {1600\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{81}-\frac {\sqrt {3}\,\left (\frac {16\,x}{9}+\frac {80}{9}\right )\,\sqrt {x^2+\frac {2}{3}}}{3}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {43799}{144}+\frac {\sqrt {6}\,18823{}\mathrm {i}}{144}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {43799}{216}+\frac {\sqrt {6}\,18823{}\mathrm {i}}{216}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {43799}{144}+\frac {\sqrt {6}\,18823{}\mathrm {i}}{144}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {43799}{216}+\frac {\sqrt {6}\,18823{}\mathrm {i}}{216}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-567360+\sqrt {6}\,290595{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{23328\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (567360+\sqrt {6}\,290595{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{23328\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:

int(-((2*x + 3)^5*(x - 5))/(3*x^2 + 2)^(5/2),x)
 

Output:

(1600*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/81 - (3^(1/2)*((16*x)/9 + 80/9 
)*(x^2 + 2/3)^(1/2))/3 + (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*18823i)/144 
 - 43799/144)/(x - (6^(1/2)*1i)/3) - (6^(1/2)*((6^(1/2)*18823i)/216 - 4379 
9/216)*1i)/(2*(x - (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(( 
(6^(1/2)*18823i)/144 + 43799/144)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2 
)*18823i)/216 + 43799/216)*1i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)* 
6^(1/2)*(6^(1/2)*290595i - 567360)*(x^2 + 2/3)^(1/2)*1i)/(23328*(x - (6^(1 
/2)*1i)/3)) - (3^(1/2)*6^(1/2)*(6^(1/2)*290595i + 567360)*(x^2 + 2/3)^(1/2 
)*1i)/(23328*(x + (6^(1/2)*1i)/3))
 

Reduce [F]

\[ \int \frac {(5-x) (3+2 x)^5}{\left (2+3 x^2\right )^{5/2}} \, dx=\int \frac {\left (5-x \right ) \left (2 x +3\right )^{5}}{\left (3 x^{2}+2\right )^{\frac {5}{2}}}d x \] Input:

int((5-x)*(3+2*x)^5/(3*x^2+2)^(5/2),x)
 

Output:

int((5-x)*(3+2*x)^5/(3*x^2+2)^(5/2),x)