\(\int \frac {(5-x) (3+2 x)^4}{(2+3 x^2)^{3/2}} \, dx\) [244]

Optimal result

Integrand size = 24, antiderivative size = 89 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {7 (2-7 x) (3+2 x)^3}{6 \sqrt {2+3 x^2}}-\frac {151}{27} (3+2 x)^2 \sqrt {2+3 x^2}-\frac {10}{81} (185+207 x) \sqrt {2+3 x^2}+\frac {880 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{3 \sqrt {3}} \] Output:

-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)+880/9*arcsinh(1/2*x*6^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {33914+14715 x-15024 x^2+432 x^3+288 x^4}{162 \sqrt {2+3 x^2}}-\frac {880 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{3 \sqrt {3}} \] Input:

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

Output:

-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])
 

Rubi [A] (verified)

Time = 0.22 (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

\(\displaystyle \frac {1}{3} \left (\frac {10}{9} \left (264 \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-69 \sqrt {3 x^2+2} x-\frac {185}{3} \sqrt {3 x^2+2}\right )-\frac {151}{9} (2 x+3)^2 \sqrt {3 x^2+2}\right )-\frac {7 (2-7 x) (2 x+3)^3}{6 \sqrt {3 x^2+2}}\)

Input:

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

Output:

(-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
 

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])
 

rule 687

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])
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.51

method	result	size
risch	\(-\frac {288 x^{4}+432 x^{3}-15024 x^{2}+14715 x +33914}{162 \sqrt {3 x^{2}+2}}+\frac {880 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{9}\)	\(45\)
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	\(-\frac {545 x}{6 \sqrt {3 x^{2}+2}}+\frac {880 \,\operatorname {arcsinh}\left (\frac {\sqrt {6}\, x}{2}\right ) \sqrt {3}}{9}+\frac {2504 x^{2}}{27 \sqrt {3 x^{2}+2}}-\frac {16957}{81 \sqrt {3 x^{2}+2}}-\frac {8 x^{3}}{3 \sqrt {3 x^{2}+2}}-\frac {16 x^{4}}{9 \sqrt {3 x^{2}+2}}\)	\(79\)
meijerg	\(\frac {405 \sqrt {2}\, x}{4 \sqrt {\frac {3 x^{2}}{2}+1}}-\frac {32 \sqrt {3}\, \left (\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}\, \left (\frac {15 x^{2}}{2}+15\right )}{20 \sqrt {\frac {3 x^{2}}{2}+1}}-\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{2}\right )}{27 \sqrt {\pi }}+\frac {88 \sqrt {2}\, \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (6 x^{2}+8\right )}{4 \sqrt {\frac {3 x^{2}}{2}+1}}\right )}{3 \sqrt {\pi }}+\frac {96 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {2}\, \sqrt {3}}{2 \sqrt {\frac {3 x^{2}}{2}+1}}+\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )\right )}{\sqrt {\pi }}+\frac {333 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {\frac {3 x^{2}}{2}+1}}\right )}{2 \sqrt {\pi }}-\frac {32 \sqrt {2}\, \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-\frac {9}{2} x^{4}+12 x^{2}+16\right )}{6 \sqrt {\frac {3 x^{2}}{2}+1}}\right )}{27 \sqrt {\pi }}\)	\(214\)

Input:

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

Output:

-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*6^(1/2)*x)*3^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {7920 \, \sqrt {3} {\left (3 \, x^{2} + 2\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) - {\left (288 \, x^{4} + 432 \, x^{3} - 15024 \, x^{2} + 14715 \, x + 33914\right )} \sqrt {3 \, x^{2} + 2}}{162 \, {\left (3 \, x^{2} + 2\right )}} \] Input:

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

Output:

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)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

-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)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {16 \, x^{4}}{9 \, \sqrt {3 \, x^{2} + 2}} - \frac {8 \, x^{3}}{3 \, \sqrt {3 \, x^{2} + 2}} + \frac {2504 \, x^{2}}{27 \, \sqrt {3 \, x^{2} + 2}} + \frac {880}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {545 \, x}{6 \, \sqrt {3 \, x^{2} + 2}} - \frac {16957}{81 \, \sqrt {3 \, x^{2} + 2}} \] Input:

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

Output:

-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)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.61 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=-\frac {880}{9} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) - \frac {3 \, {\left (16 \, {\left (3 \, {\left (2 \, x + 3\right )} x - 313\right )} x + 4905\right )} x + 33914}{162 \, \sqrt {3 \, x^{2} + 2}} \] Input:

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

Output:

-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)
 

Mupad [B] (verification not implemented)

Time = 5.86 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {880\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{9}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {16\,x^2}{9}+\frac {8\,x}{3}-\frac {2536}{27}\right )}{3}+\frac {\sqrt {3}\,\sqrt {6}\,\left (-44058+\sqrt {6}\,4809{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}+\frac {\sqrt {3}\,\sqrt {6}\,\left (44058+\sqrt {6}\,4809{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{1944\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \] Input:

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

Output:

(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))
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.47 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+3 x^2\right )^{3/2}} \, dx=\frac {-288 \sqrt {3 x^{2}+2}\, x^{4}-432 \sqrt {3 x^{2}+2}\, x^{3}+15024 \sqrt {3 x^{2}+2}\, x^{2}-14715 \sqrt {3 x^{2}+2}\, x -33914 \sqrt {3 x^{2}+2}+47520 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right ) x^{2}+31680 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {3 x^{2}+2}+\sqrt {3}\, x}{\sqrt {2}}\right )-14499 \sqrt {3}\, x^{2}-9666 \sqrt {3}}{486 x^{2}+324} \] Input:

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

Output:

( - 288*sqrt(3*x**2 + 2)*x**4 - 432*sqrt(3*x**2 + 2)*x**3 + 15024*sqrt(3*x 
**2 + 2)*x**2 - 14715*sqrt(3*x**2 + 2)*x - 33914*sqrt(3*x**2 + 2) + 47520* 
sqrt(3)*log((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2))*x**2 + 31680*sqrt(3)*l 
og((sqrt(3*x**2 + 2) + sqrt(3)*x)/sqrt(2)) - 14499*sqrt(3)*x**2 - 9666*sqr 
t(3))/(162*(3*x**2 + 2))