3.24.36 \(\int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx\) [2336]

3.24.36.1 Optimal result
3.24.36.2 Mathematica [A] (verified)
3.24.36.3 Rubi [A] (verified)
3.24.36.4 Maple [A] (verified)
3.24.36.5 Fricas [A] (verification not implemented)
3.24.36.6 Sympy [F]
3.24.36.7 Maxima [A] (verification not implemented)
3.24.36.8 Giac [B] (verification not implemented)
3.24.36.9 Mupad [F(-1)]

3.24.36.1 Optimal result

Integrand size = 26, antiderivative size = 180 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=-\frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{360 (2+3 x)^4}+\frac {7723 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {270463 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}+\frac {28291441 \sqrt {1-2 x} \sqrt {3+5 x}}{1185408 (2+3 x)}-\frac {11988317 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}} \]

output
-11988317/307328*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1 
/15*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^5+41/360*(1-2*x)^(1/2)*(3+5*x)^(1/ 
2)/(2+3*x)^4+7723/15120*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+270463/84672 
*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+28291441/1185408*(1-2*x)^(1/2)*(3+5 
*x)^(1/2)/(2+3*x)
 
3.24.36.2 Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (269759904+1588955864 x+3511594796 x^2+3451770150 x^3+1273114845 x^4\right )}{(2+3 x)^5}-179824755 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{4609920} \]

input
Integrate[((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(2 + 3*x)^6,x]
 
output
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(269759904 + 1588955864*x + 3511594796*x^2 
 + 3451770150*x^3 + 1273114845*x^4))/(2 + 3*x)^5 - 179824755*Sqrt[7]*ArcTa 
n[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/4609920
 
3.24.36.3 Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {108, 27, 166, 27, 168, 27, 168, 27, 168, 27, 104, 217}

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.

\(\displaystyle \int \frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{(3 x+2)^6} \, dx\)

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{15} \int -\frac {\sqrt {1-2 x} (40 x+13)}{2 (3 x+2)^5 \sqrt {5 x+3}}dx-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{15 (3 x+2)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{30} \int \frac {\sqrt {1-2 x} (40 x+13)}{(3 x+2)^5 \sqrt {5 x+3}}dx-\frac {\sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^5}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{30} \left (\frac {1}{12} \int \frac {1361-1820 x}{2 \sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{15 (3 x+2)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{30} \left (\frac {1}{24} \int \frac {1361-1820 x}{\sqrt {1-2 x} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{15 (3 x+2)^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {1}{21} \int \frac {5 (48965-61784 x)}{2 \sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {7723 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{15 (3 x+2)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{42} \int \frac {48965-61784 x}{\sqrt {1-2 x} (3 x+2)^3 \sqrt {5 x+3}}dx+\frac {7723 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{15 (3 x+2)^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{14} \int \frac {5824307-5409260 x}{2 \sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {270463 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {7723 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{15 (3 x+2)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{28} \int \frac {5824307-5409260 x}{\sqrt {1-2 x} (3 x+2)^2 \sqrt {5 x+3}}dx+\frac {270463 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {7723 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{15 (3 x+2)^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{28} \left (\frac {1}{7} \int \frac {323684559}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {28291441 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {270463 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {7723 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{15 (3 x+2)^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{28} \left (\frac {323684559}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {28291441 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {270463 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {7723 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{15 (3 x+2)^5}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{28} \left (\frac {323684559}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {28291441 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {270463 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {7723 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{15 (3 x+2)^5}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{30} \left (\frac {1}{24} \left (\frac {5}{42} \left (\frac {1}{28} \left (\frac {28291441 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}-\frac {323684559 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )+\frac {270463 \sqrt {1-2 x} \sqrt {5 x+3}}{14 (3 x+2)^2}\right )+\frac {7723 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^3}\right )+\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{12 (3 x+2)^4}\right )-\frac {(1-2 x)^{3/2} \sqrt {5 x+3}}{15 (3 x+2)^5}\)

input
Int[((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(2 + 3*x)^6,x]
 
output
-1/15*((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(2 + 3*x)^5 + ((41*Sqrt[1 - 2*x]*Sqr 
t[3 + 5*x])/(12*(2 + 3*x)^4) + ((7723*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 
+ 3*x)^3) + (5*((270463*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14*(2 + 3*x)^2) + (( 
28291441*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)) - (323684559*ArcTan[Sq 
rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/28))/42)/24)/30
 

3.24.36.3.1 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 104
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 108
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 
*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 166
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 168
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 217
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 
3.24.36.4 Maple [A] (verified)

Time = 3.73 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.74

method result size
risch \(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (1273114845 x^{4}+3451770150 x^{3}+3511594796 x^{2}+1588955864 x +269759904\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{658560 \left (2+3 x \right )^{5} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {11988317 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{614656 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) \(134\)
default \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (43697415465 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}+145658051550 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}+194210735400 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}+17823607830 x^{4} \sqrt {-10 x^{2}-x +3}+129473823600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+48324782100 x^{3} \sqrt {-10 x^{2}-x +3}+43157941200 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +49162327144 x^{2} \sqrt {-10 x^{2}-x +3}+5754392160 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+22245382096 x \sqrt {-10 x^{2}-x +3}+3776638656 \sqrt {-10 x^{2}-x +3}\right )}{9219840 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{5}}\) \(298\)

input
int((1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^6,x,method=_RETURNVERBOSE)
 
output
-1/658560*(-1+2*x)*(3+5*x)^(1/2)*(1273114845*x^4+3451770150*x^3+3511594796 
*x^2+1588955864*x+269759904)/(2+3*x)^5/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)* 
(3+5*x))^(1/2)/(1-2*x)^(1/2)+11988317/614656*7^(1/2)*arctan(9/14*(20/3+37/ 
3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2))*((1-2*x)*(3+5*x))^(1/2)/(1-2* 
x)^(1/2)/(3+5*x)^(1/2)
 
3.24.36.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.73 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=-\frac {179824755 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1273114845 \, x^{4} + 3451770150 \, x^{3} + 3511594796 \, x^{2} + 1588955864 \, x + 269759904\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{9219840 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]

input
integrate((1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^6,x, algorithm="fricas")
 
output
-1/9219840*(179824755*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 24 
0*x + 32)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10 
*x^2 + x - 3)) - 14*(1273114845*x^4 + 3451770150*x^3 + 3511594796*x^2 + 15 
88955864*x + 269759904)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 
 1080*x^3 + 720*x^2 + 240*x + 32)
 
3.24.36.6 Sympy [F]

\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{6}}\, dx \]

input
integrate((1-2*x)**(3/2)*(3+5*x)**(1/2)/(2+3*x)**6,x)
 
output
Integral((1 - 2*x)**(3/2)*sqrt(5*x + 3)/(3*x + 2)**6, x)
 
3.24.36.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.10 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=\frac {11988317}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {495385}{32928} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{5 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {239 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{280 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {8395 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2352 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {297231 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {3665849 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \]

input
integrate((1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^6,x, algorithm="maxima")
 
output
11988317/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) 
+ 495385/32928*sqrt(-10*x^2 - x + 3) + 1/5*(-10*x^2 - x + 3)^(3/2)/(243*x^ 
5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 239/280*(-10*x^2 - x + 3) 
^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 8395/2352*(-10*x^2 - x + 
 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 297231/21952*(-10*x^2 - x + 3)^(3 
/2)/(9*x^2 + 12*x + 4) - 3665849/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2)
 
3.24.36.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (141) = 282\).

Time = 0.59 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.37 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=\frac {11988317}{6146560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (27021 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 52500560 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 18029240320 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 2768103296000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {166086197760000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {664344791040000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{65856 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]

input
integrate((1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^6,x, algorithm="giac")
 
output
11988317/6146560*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x 
 + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt 
(-10*x + 5) - sqrt(22)))) - 1331/65856*sqrt(10)*(27021*((sqrt(2)*sqrt(-10* 
x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5 
) - sqrt(22)))^9 - 52500560*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x 
 + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 - 18029240 
320*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/ 
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 2768103296000*((sqrt(2)*sqrt(-10 
*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 
5) - sqrt(22)))^3 - 166086197760000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s 
qrt(5*x + 3) + 664344791040000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sq 
rt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x 
 + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5
 
3.24.36.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^6} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^6} \,d x \]

input
int(((1 - 2*x)^(3/2)*(5*x + 3)^(1/2))/(3*x + 2)^6,x)
 
output
int(((1 - 2*x)^(3/2)*(5*x + 3)^(1/2))/(3*x + 2)^6, x)