3.15.21 \(\int \frac {(5-x) (3+2 x)^3}{(2+3 x^2)^{5/2}} \, dx\) [1421]

3.15.21.1 Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {7 (2-7 x) (3+2 x)^2}{18 \left (2+3 x^2\right )^{3/2}}-\frac {556-1461 x}{54 \sqrt {2+3 x^2}}-\frac {8 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{9 \sqrt {3}} \]

output

-7/18*(2-7*x)*(3+2*x)^2/(3*x^2+2)^(3/2)-8/27*arcsinh(1/2*x*6^(1/2))*3^(1/2 
)+1/54*(-556+1461*x)/(3*x^2+2)^(1/2)
 

3.15.21.2 Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {1490-3741 x+72 x^2-4971 x^3}{54 \left (2+3 x^2\right )^{3/2}}+\frac {8 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{9 \sqrt {3}} \]

input

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

output

-1/54*(1490 - 3741*x + 72*x^2 - 4971*x^3)/(2 + 3*x^2)^(3/2) + (8*Log[-(Sqr 
t[3]*x) + Sqrt[2 + 3*x^2]])/(9*Sqrt[3])
 

3.15.21.3 Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.21, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {684, 27, 675, 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.

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

\(\Big \downarrow \) 684

\(\displaystyle \frac {1}{18} \int \frac {2 (157-12 x) (2 x+3)}{\left (3 x^2+2\right )^{3/2}}dx-\frac {7 (2-7 x) (2 x+3)^2}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int \frac {(157-12 x) (2 x+3)}{\left (3 x^2+2\right )^{3/2}}dx-\frac {7 (2-7 x) (2 x+3)^2}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 675

\(\displaystyle \frac {1}{9} \left (-8 \int \frac {1}{\sqrt {3 x^2+2}}dx+\frac {487 x}{2 \sqrt {3 x^2+2}}-\frac {278}{3 \sqrt {3 x^2+2}}\right )-\frac {7 (2-7 x) (2 x+3)^2}{18 \left (3 x^2+2\right )^{3/2}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{9} \left (-\frac {8 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+\frac {487 x}{2 \sqrt {3 x^2+2}}-\frac {278}{3 \sqrt {3 x^2+2}}\right )-\frac {7 (2-7 x) (2 x+3)^2}{18 \left (3 x^2+2\right )^{3/2}}\)

input

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

output

(-7*(2 - 7*x)*(3 + 2*x)^2)/(18*(2 + 3*x^2)^(3/2)) + (-278/(3*Sqrt[2 + 3*x^ 
2]) + (487*x)/(2*Sqrt[2 + 3*x^2]) - (8*ArcSinh[Sqrt[3/2]*x])/Sqrt[3])/9
 

3.15.21.3.1 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 675

Int[((d_) + (e_.)*(x_))*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[a*(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + (- 
Simp[(c*d*f - a*e*g)*x*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] - Simp[(a* 
e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1))   Int[(a + c*x^2)^(p + 1), x], x]) / 
; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1] &&  !(IntegerQ[p] && NiceSqrtQ 
[(-a)*c])
 

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])
 

3.15.21.4 Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.60

method	result	size
risch	\(\frac {4971 x^{3}-72 x^{2}+3741 x -1490}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {8 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{27}\)	\(40\)
trager	\(\frac {4971 x^{3}-72 x^{2}+3741 x -1490}{54 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{27}\)	\(58\)
default	\(\frac {17 x}{2 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {547 x}{18 \sqrt {3 x^{2}+2}}-\frac {745}{27 \left (3 x^{2}+2\right )^{\frac {3}{2}}}+\frac {8 x^{3}}{9 \left (3 x^{2}+2\right )^{\frac {3}{2}}}-\frac {8 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{27}-\frac {4 x^{2}}{3 \left (3 x^{2}+2\right )^{\frac {3}{2}}}\)	\(77\)
meijerg	\(\frac {45 \sqrt {2}\, x \left (3 x^{2}+3\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {27 \sqrt {2}\, \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{2 \sqrt {\pi }}+\frac {21 \sqrt {2}\, x^{3}}{4 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {4 \sqrt {2}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }\, \left (18 x^{2}+8\right )}{8 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}\right )}{27 \sqrt {\pi }}-\frac {16 \sqrt {3}\, \left (-\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \left (30 x^{2}+15\right )}{20 \left (\frac {3 x^{2}}{2}+1\right )^{\frac {3}{2}}}+\frac {3 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{2}\right )}{81 \sqrt {\pi }}\)	\(154\)

input

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

output

1/54*(4971*x^3-72*x^2+3741*x-1490)/(3*x^2+2)^(3/2)-8/27*arcsinh(1/2*x*6^(1 
/2))*3^(1/2)
 

3.15.21.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.21 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {8 \, \sqrt {3} {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + {\left (4971 \, x^{3} - 72 \, x^{2} + 3741 \, x - 1490\right )} \sqrt {3 \, x^{2} + 2}}{54 \, {\left (9 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

input

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

output

1/54*(8*sqrt(3)*(9*x^4 + 12*x^2 + 4)*log(sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 
 - 1) + (4971*x^3 - 72*x^2 + 3741*x - 1490)*sqrt(3*x^2 + 2))/(9*x^4 + 12*x 
^2 + 4)
 

3.15.21.6 Sympy [F]

\[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {243 x}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {126 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \left (- \frac {4 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx - \int \frac {8 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\, dx - \int \left (- \frac {135}{9 x^{4} \sqrt {3 x^{2} + 2} + 12 x^{2} \sqrt {3 x^{2} + 2} + 4 \sqrt {3 x^{2} + 2}}\right )\, dx \]

input

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

output

-Integral(-243*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(-126*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(-4*x**3/(9*x**4*s 
qrt(3*x**2 + 2) + 12*x**2*sqrt(3*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x) - Int 
egral(8*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(-135/(9*x**4*sqrt(3*x**2 + 2) + 12*x**2*sqrt(3 
*x**2 + 2) + 4*sqrt(3*x**2 + 2)), x)
 

3.15.21.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.36 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {8}{27} \, x {\left (\frac {9 \, x^{2}}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {4}{{\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}\right )} - \frac {8}{27} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {1609 \, x}{54 \, \sqrt {3 \, x^{2} + 2}} - \frac {4 \, x^{2}}{3 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} + \frac {17 \, x}{2 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} - \frac {745}{27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]

input

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

output

8/27*x*(9*x^2/(3*x^2 + 2)^(3/2) + 4/(3*x^2 + 2)^(3/2)) - 8/27*sqrt(3)*arcs 
inh(1/2*sqrt(6)*x) + 1609/54*x/sqrt(3*x^2 + 2) - 4/3*x^2/(3*x^2 + 2)^(3/2) 
 + 17/2*x/(3*x^2 + 2)^(3/2) - 745/27/(3*x^2 + 2)^(3/2)
 

3.15.21.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{5/2}} \, dx=\frac {8}{27} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) + \frac {3 \, {\left ({\left (1657 \, x - 24\right )} x + 1247\right )} x - 1490}{54 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}} \]

input

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

output

8/27*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2)) + 1/54*(3*((1657*x - 24)*x 
+ 1247)*x - 1490)/(3*x^2 + 2)^(3/2)
 

3.15.21.9 Mupad [B] (verification not implemented)

Time = 10.42 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.99 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+3 x^2\right )^{5/2}} \, dx=-\frac {8\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {-\frac {427}{48}+\frac {\sqrt {6}\,721{}\mathrm {i}}{48}}{x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}+\frac {\sqrt {6}\,\left (-\frac {427}{72}+\frac {\sqrt {6}\,721{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (\frac {\frac {427}{48}+\frac {\sqrt {6}\,721{}\mathrm {i}}{48}}{x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}}-\frac {\sqrt {6}\,\left (\frac {427}{72}+\frac {\sqrt {6}\,721{}\mathrm {i}}{72}\right )\,1{}\mathrm {i}}{2\,{\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}^2}\right )}{27}-\frac {\sqrt {3}\,\sqrt {6}\,\left (-96+\sqrt {6}\,2067{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x-\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )}-\frac {\sqrt {3}\,\sqrt {6}\,\left (96+\sqrt {6}\,2067{}\mathrm {i}\right )\,\sqrt {x^2+\frac {2}{3}}\,1{}\mathrm {i}}{2592\,\left (x+\frac {\sqrt {6}\,1{}\mathrm {i}}{3}\right )} \]

input

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

output

(3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*721i)/48 + 427/48)/(x - (6^(1/2)*1i) 
/3) - (6^(1/2)*((6^(1/2)*721i)/72 + 427/72)*1i)/(2*(x - (6^(1/2)*1i)/3)^2) 
))/27 - (3^(1/2)*(x^2 + 2/3)^(1/2)*(((6^(1/2)*721i)/48 - 427/48)/(x + (6^( 
1/2)*1i)/3) + (6^(1/2)*((6^(1/2)*721i)/72 - 427/72)*1i)/(2*(x + (6^(1/2)*1 
i)/3)^2)))/27 - (8*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/27 - (3^(1/2)*6^( 
1/2)*(6^(1/2)*2067i - 96)*(x^2 + 2/3)^(1/2)*1i)/(2592*(x - (6^(1/2)*1i)/3) 
) - (3^(1/2)*6^(1/2)*(6^(1/2)*2067i + 96)*(x^2 + 2/3)^(1/2)*1i)/(2592*(x + 
 (6^(1/2)*1i)/3))