\(\int \frac {(5-x) (3+2 x)^4}{(2+3 x^2)^{5/2}} \, dx\) [254]

Optimal result

Integrand size = 24, antiderivative size = 87 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {7 (2-7 x) (3+2 x)^3}{18 \left (2+3 x^2\right )^{3/2}}-\frac {(318-1783 x) (3+2 x)}{54 \sqrt {2+3 x^2}}-\frac {2027}{81} \sqrt {2+3 x^2}-\frac {16 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \] Output:

-7/18*(2-7*x)*(3+2*x)^3/(3*x^2+2)^(3/2)-1/54*(318-1783*x)*(3+2*x)/(3*x^2+2 
)^(1/2)-2027/81*(3*x^2+2)^(1/2)-16/27*arcsinh(1/2*x*6^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.76 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {25342-33381 x+16560 x^2-57285 x^3+864 x^4}{162 \left (2+3 x^2\right )^{3/2}}+\frac {16 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{9 \sqrt {3}} \] Input:

Integrate[((5 - x)*(3 + 2*x)^4)/(2 + 3*x^2)^(5/2),x]
 

Output:

-1/162*(25342 - 33381*x + 16560*x^2 - 57285*x^3 + 864*x^4)/(2 + 3*x^2)^(3/ 
2) + (16*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(9*Sqrt[3])
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.09, 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, 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.

\(\displaystyle \int \frac {(5-x) (2 x+3)^4}{\left (3 x^2+2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 684

\(\displaystyle \frac {1}{18} \int \frac {2 (171-61 x) (2 x+3)^2}{\left (3 x^2+2\right )^{3/2}}dx-\frac {7 (2-7 x) (2 x+3)^3}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int \frac {(171-61 x) (2 x+3)^2}{\left (3 x^2+2\right )^{3/2}}dx-\frac {7 (2-7 x) (2 x+3)^3}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 684

\(\displaystyle \frac {1}{9} \left (\frac {1}{6} \int -\frac {2 (2027 x+48)}{\sqrt {3 x^2+2}}dx-\frac {(318-1783 x) (2 x+3)}{6 \sqrt {3 x^2+2}}\right )-\frac {7 (2-7 x) (2 x+3)^3}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (-\frac {1}{3} \int \frac {2027 x+48}{\sqrt {3 x^2+2}}dx-\frac {(318-1783 x) (2 x+3)}{6 \sqrt {3 x^2+2}}\right )-\frac {7 (2-7 x) (2 x+3)^3}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (-48 \int \frac {1}{\sqrt {3 x^2+2}}dx-\frac {2027}{3} \sqrt {3 x^2+2}\right )-\frac {(318-1783 x) (2 x+3)}{6 \sqrt {3 x^2+2}}\right )-\frac {7 (2-7 x) (2 x+3)^3}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{9} \left (\frac {1}{3} \left (-16 \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {2027}{3} \sqrt {3 x^2+2}\right )-\frac {(318-1783 x) (2 x+3)}{6 \sqrt {3 x^2+2}}\right )-\frac {7 (2-7 x) (2 x+3)^3}{18 \left (3 x^2+2\right )^{3/2}}\)

Input:

Int[((5 - x)*(3 + 2*x)^4)/(2 + 3*x^2)^(5/2),x]
 

Output:

(-7*(2 - 7*x)*(3 + 2*x)^3)/(18*(2 + 3*x^2)^(3/2)) + (-1/6*((318 - 1783*x)* 
(3 + 2*x))/Sqrt[2 + 3*x^2] + ((-2027*Sqrt[2 + 3*x^2])/3 - 16*Sqrt[3]*ArcSi 
nh[Sqrt[3/2]*x])/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 455

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]
 

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 = 0.77 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.52

method	result	size
risch	\(-\frac {864 x^{4}-57285 x^{3}+16560 x^{2}-33381 x +25342}{162 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {16 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{27}\)	\(45\)
trager	\(-\frac {864 x^{4}-57285 x^{3}+16560 x^{2}-33381 x +25342}{162 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{27}\)	\(62\)
default	\(-\frac {57 x}{2 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {2111 x}{18 \sqrt {3 x^{2}+2}}-\frac {12671}{81 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {920 x^{2}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {16 x^{3}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {16 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{27}-\frac {16 x^{4}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\)	\(91\)
meijerg	\(\frac {135 \sqrt {2}\, x \left (3 x^{2}+3\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}-\frac {32 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (30 x^{2}+15\right )}{20 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{2}\right )}{81 \sqrt {\pi }}+\frac {88 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (18 x^{2}+8\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{9 \sqrt {\pi }}+\frac {36 \sqrt {2}\, x^{3}}{\left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {111 \sqrt {2}\, \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{2 \sqrt {\pi }}-\frac {32 \sqrt {2}\, \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (\frac {27}{2} x^{4}+36 x^{2}+16\right )}{4 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{81 \sqrt {\pi }}\)	\(194\)

Input:

int((5-x)*(2*x+3)^4/(3*x^2+2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/162*(864*x^4-57285*x^3+16560*x^2-33381*x+25342)/(3*x^2+2)^(3/2)-16/27*a 
rcsinh(1/2*6^(1/2)*x)*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {48 \, \sqrt {3} {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - {\left (864 \, x^{4} - 57285 \, x^{3} + 16560 \, x^{2} - 33381 \, x + 25342\right )} \sqrt {3 \, x^{2} + 2}}{162 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \] Input:

integrate((5-x)*(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="fricas")
 

Output:

1/162*(48*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x 
^2 - 1) - (864*x^4 - 57285*x^3 + 16560*x^2 - 33381*x + 25342)*sqrt(3*x^2 + 
 2))/(9*x^4 + 12*x^2 + 4)
 

Sympy [F]

\[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {999 x}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {864 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {264 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {16 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \frac {16 x^{5}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {405}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx \] Input:

integrate((5-x)*(3+2*x)**4/(3*x**2+2)**(5/2),x)
 

Output:

-Integral(-999*x/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*s 
qrt(3*x**2 + 2)), x) - Integral(-864*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(-264*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) - I 
ntegral(16*x**4/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sq 
rt(3*x**2 + 2)), x) - Integral(16*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(-405/(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 = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.21 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {16 \, x^{4}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {16}{27} \, x {\left (\frac {9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {4}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}\right )} - \frac {16}{27} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {6269 \, x}{54 \, \sqrt {3 \, x^{2} + 2}} - \frac {920 \, x^{2}}{9 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {57 \, x}{2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {12671}{81 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

integrate((5-x)*(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="maxima")
 

Output:

-16/3*x^4/(3*x^2 + 2)^(3/2) + 16/27*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 
+ 2)^(3/2)) - 16/27*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 6269/54*x/sqrt(3*x^2 
+ 2) - 920/9*x^2/(3*x^2 + 2)^(3/2) - 57/2*x/(3*x^2 + 2)^(3/2) - 12671/81/( 
3*x^2 + 2)^(3/2)
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.60 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {16}{27} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {9 \, {\left ({\left ({\left (96 \, x - 6365\right )} x + 1840\right )} x - 3709\right )} x + 25342}{162 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \] Input:

integrate((5-x)*(3+2*x)^4/(3*x^2+2)^(5/2),x, algorithm="giac")
 

Output:

16/27*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) - 1/162*(9*(((96*x - 6365) 
*x + 1840)*x - 3709)*x + 25342)/(3*x^2 + 2)^(3/2)
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.44 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {16\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{27}-\frac {16\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {1603}{48}+\frac {\sqrt {6}\,7343{}\mathrm {i}}{144}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (-\frac {1603}{72}+\frac {\sqrt {6}\,7343{}\mathrm {i}}{216}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {1603}{48}+\frac {\sqrt {6}\,7343{}\mathrm {i}}{144}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (\frac {1603}{72}+\frac {\sqrt {6}\,7343{}\mathrm {i}}{216}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-20544+\sqrt {6}\,27063{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{7776\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (20544+\sqrt {6}\,27063{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{7776\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:

int(-((2*x + 3)^4*(x - 5))/(3*x^2 + 2)^(5/2),x)
 

Output:

(3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*7343i)/144 - 1603/48)/(x - (6^(1/2)* 
1i)/3) - (6^(1/2)*((6^(1/2)*7343i)/216 - 1603/72)*1i)/(2*(x - (6^(1/2)*1i) 
/3)^2)))/27 - (16*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/27 - (16*3^(1/2)*( 
x^2 + 2/3)^(1/2))/27 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*7343i)/144 + 
1603/48)/(x + (6^(1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*7343i)/216 + 1603/72)*1 
i)/(2*(x + (6^(1/2)*1i)/3)^2)))/27 - (3^(1/2)*6^(1/2)*(6^(1/2)*27063i - 20 
544)*(x^2 + 2/3)^(1/2)*1i)/(7776*(x - (6^(1/2)*1i)/3)) - (3^(1/2)*6^(1/2)* 
(6^(1/2)*27063i + 20544)*(x^2 + 2/3)^(1/2)*1i)/(7776*(x + (6^(1/2)*1i)/3))
 

Reduce [B] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.94 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {-864 \sqrt {3 x^{2}+2}\, x^{4}+57285 \sqrt {3 x^{2}+2}\, x^{3}-16560 \sqrt {3 x^{2}+2}\, x^{2}+33381 \sqrt {3 x^{2}+2}\, x -25342 \sqrt {3 x^{2}+2}-864 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right ) x^{4}-1152 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right ) x^{2}-384 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )-9477 \sqrt {3}\, x^{4}-12636 \sqrt {3}\, x^{2}-4212 \sqrt {3}}{1458 x^{4}+1944 x^{2}+648} \] Input:

int((5-x)*(3+2*x)^4/(3*x^2+2)^(5/2),x)
 

Output:

( - 864*sqrt(3*x**2 + 2)*x**4 + 57285*sqrt(3*x**2 + 2)*x**3 - 16560*sqrt(3 
*x**2 + 2)*x**2 + 33381*sqrt(3*x**2 + 2)*x - 25342*sqrt(3*x**2 + 2) - 864* 
sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2))*x**4 - 1152*sqrt(3)*lo 
g((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2))*x**2 - 384*sqrt(3)*log((sqrt(3*x 
**2 + 2) + sqrt(3)*x)/sqrt(2)) - 9477*sqrt(3)*x**4 - 12636*sqrt(3)*x**2 - 
4212*sqrt(3))/(162*(9*x**4 + 12*x**2 + 4))