3.28.93 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\) [2793]

3.28.93.1 Optimal result

Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {76 \sqrt {1-2 x} \sqrt {3+5 x}}{9 (2+3 x)^{5/2}}+\frac {10124 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {703480 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}-\frac {703480 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}-\frac {21160 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1323} \]

output

-703480/3969*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2 
)-21160/3969*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2 
)+2/3*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+76/9*(1-2*x)^(1/2)*(3+5*x) 
^(1/2)/(2+3*x)^(5/2)+10124/189*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+7 
03480/1323*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
 

3.28.93.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.60 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.53 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (2967269+13103724 x+19312866 x^2+9496980 x^3\right )}{4 (2+3 x)^{7/2}}+5 i \sqrt {33} \left (17587 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-18116 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{3969} \]

input

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(9/2)*Sqrt[3 + 5*x]),x]
 

output

(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2967269 + 13103724*x + 19312866*x^2 + 
9496980*x^3))/(4*(2 + 3*x)^(7/2)) + (5*I)*Sqrt[33]*(17587*EllipticE[I*ArcS 
inh[Sqrt[9 + 15*x]], -2/33] - 18116*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], - 
2/33])))/3969
 

3.28.93.3 Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {109, 27, 167, 27, 169, 27, 169, 27, 176, 123, 129}

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.

\(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^{9/2} \sqrt {5 x+3}} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {2}{21} \int \frac {7 (23-13 x) \sqrt {1-2 x}}{(3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \int \frac {(23-13 x) \sqrt {1-2 x}}{(3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {2}{3} \left (\frac {38 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{5/2}}-\frac {2}{15} \int -\frac {5 (481-544 x)}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx\right )+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \left (\frac {1}{3} \int \frac {481-544 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {38 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {2}{3} \left (\frac {1}{3} \left (\frac {2}{21} \int \frac {5 (4175-2531 x)}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {5062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \left (\frac {1}{3} \left (\frac {10}{21} \int \frac {4175-2531 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {5062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {2}{3} \left (\frac {1}{3} \left (\frac {10}{21} \left (\frac {2}{7} \int \frac {175870 x+111341}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {35174 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {5062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{3} \left (\frac {1}{3} \left (\frac {10}{21} \left (\frac {1}{7} \int \frac {175870 x+111341}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {35174 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {5062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {2}{3} \left (\frac {1}{3} \left (\frac {10}{21} \left (\frac {1}{7} \left (5819 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+35174 \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {35174 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {5062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2}{3} \left (\frac {1}{3} \left (\frac {10}{21} \left (\frac {1}{7} \left (5819 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-35174 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {35174 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {5062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {2}{3} \left (\frac {1}{3} \left (\frac {10}{21} \left (\frac {1}{7} \left (-1058 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-35174 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {35174 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {5062 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {38 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{5/2}}\right )+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)^{7/2}}\)

input

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(9/2)*Sqrt[3 + 5*x]),x]
 

output

(2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(7/2)) + (2*((38*Sqrt[1 - 2 
*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(5/2)) + ((5062*Sqrt[1 - 2*x]*Sqrt[3 + 5*x 
])/(21*(2 + 3*x)^(3/2)) + (10*((35174*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt 
[2 + 3*x]) + (-35174*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 
 35/33] - 1058*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33 
])/7))/21)/3))/3
 

3.28.93.3.1 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 109

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

rule 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] 
/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, 
b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !L 
tQ[-(b*c - a*d)/d, 0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d 
), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])
 

rule 129

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ 
Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - 
a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ 
[(b*e - a*f)/b, 0] && PosQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d 
*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(( 
-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ 
[((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f 
/b]))
 

rule 167

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] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
 

rule 169

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] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 176

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* 
Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f   Int[Sqrt[e + f*x]/(Sqrt[a + b*x 
]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f   Int[1/(Sqrt[a + b*x]*Sqrt[c 
 + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim 
plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
 

3.28.93.4 Maple [A] (verified)

Time = 1.31 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.40

method	result	size
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {10124 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1701 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {7034800}{1323} x^{2}-\frac {703480}{1323} x +\frac {703480}{441}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {890728 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{27783 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1406960 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{27783 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729 \left (\frac {2}{3}+x \right )^{4}}+\frac {8 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27 \left (\frac {2}{3}+x \right )^{3}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(267\)
default	\(-\frac {2 \left (9223632 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9496980 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-18993960 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-12662640 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+2732928 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-2813920 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-284909400 x^{5}-607876920 x^{4}-365577498 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{3969 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\)	\(409\)

input

int((1-2*x)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)
 

output

(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 
)*(10124/1701*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+703480/3969*(-30*x^2- 
3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+890728/27783*(10+15*x)^(1/2)*(21-42 
*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x) 
^(1/2),1/35*70^(1/2))+1406960/27783*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x 
-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-7/6*EllipticE((10+15*x)^(1/2),1/3 
5*70^(1/2))+1/2*EllipticF((10+15*x)^(1/2),1/35*70^(1/2)))+14/729*(-30*x^3- 
23*x^2+7*x+6)^(1/2)/(2/3+x)^4+8/27*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3)
 

3.28.93.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (27 \, {\left (9496980 \, x^{3} + 19312866 \, x^{2} + 13103724 \, x + 2967269\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1195136 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 3165660 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{35721 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

input

integrate((1-2*x)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x, algorithm="fricas")
 

output

2/35721*(27*(9496980*x^3 + 19312866*x^2 + 13103724*x + 2967269)*sqrt(5*x + 
 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1195136*sqrt(-30)*(81*x^4 + 216*x^3 + 2 
16*x^2 + 96*x + 16)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) 
+ 3165660*sqrt(-30)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*weierstrassZe 
ta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 2 
3/90)))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
 

3.28.93.6 Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\text {Timed out} \]

input

integrate((1-2*x)**(5/2)/(2+3*x)**(9/2)/(3+5*x)**(1/2),x)
 

output

Timed out
 

3.28.93.7 Maxima [F]

\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]

input

integrate((1-2*x)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x, algorithm="maxima")
 

output

integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(9/2)), x)
 

3.28.93.8 Giac [F]

\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]

input

integrate((1-2*x)^(5/2)/(2+3*x)^(9/2)/(3+5*x)^(1/2),x, algorithm="giac")
 

output

integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(9/2)), x)
 

3.28.93.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{9/2}\,\sqrt {5\,x+3}} \,d x \]

input

int((1 - 2*x)^(5/2)/((3*x + 2)^(9/2)*(5*x + 3)^(1/2)),x)
 

output

int((1 - 2*x)^(5/2)/((3*x + 2)^(9/2)*(5*x + 3)^(1/2)), x)