\(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx\) [1529]

Optimal result

Integrand size = 28, antiderivative size = 218 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx=\frac {229 \sqrt {1-2 x} \sqrt {3+5 x}}{1029 (2+3 x)^{7/2}}-\frac {37117 \sqrt {1-2 x} \sqrt {3+5 x}}{36015 (2+3 x)^{5/2}}-\frac {106772 \sqrt {1-2 x} \sqrt {3+5 x}}{252105 (2+3 x)^{3/2}}-\frac {106558 \sqrt {1-2 x} \sqrt {3+5 x}}{1764735 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{7/2}}+\frac {106558 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{151263 \sqrt {35}}-\frac {213544 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{151263 \sqrt {35}} \] Output:

229/1029*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)-37117/36015*(1-2*x)^(1/ 
2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)-106772/252105*(1-2*x)^(1/2)*(3+5*x)^(1/2)/( 
2+3*x)^(3/2)-106558/1764735*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)+11/7 
*(3+5*x)^(3/2)/(1-2*x)^(1/2)/(2+3*x)^(7/2)+106558/5294205*EllipticE(1/11*5 
5^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-213544/5294205*EllipticF(1 
/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.71 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.48 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx=\frac {2 \left (\frac {3 \sqrt {3+5 x} \left (616327+4889131 x+12020751 x^2+11042235 x^3+2877066 x^4\right )}{\sqrt {1-2 x} (2+3 x)^{7/2}}-i \sqrt {33} \left (53279 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+56735 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{5294205} \] Input:

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

Output:

(2*((3*Sqrt[3 + 5*x]*(616327 + 4889131*x + 12020751*x^2 + 11042235*x^3 + 2 
877066*x^4))/(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) - I*Sqrt[33]*(53279*EllipticE 
[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + 56735*EllipticF[I*ArcSinh[Sqrt[9 + 15 
*x]], -2/33])))/5294205
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {109, 27, 167, 27, 169, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}-\frac {1}{7} \int -\frac {\sqrt {5 x+3} (650 x+357)}{2 \sqrt {1-2 x} (3 x+2)^{9/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \int \frac {\sqrt {5 x+3} (650 x+357)}{\sqrt {1-2 x} (3 x+2)^{9/2}}dx+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{14} \left (\frac {2}{147} \int \frac {147800 x+86161}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{147} \int \frac {147800 x+86161}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{14} \left (\frac {1}{147} \left (\frac {2}{35} \int \frac {3 (185585 x+105928)}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {74234 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{147} \left (\frac {6}{35} \int \frac {185585 x+105928}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {74234 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{14} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {2}{21} \int \frac {533860 x+338147}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {106772 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {74234 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \int \frac {533860 x+338147}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {106772 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {74234 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{14} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {2}{7} \int \frac {5 (89048-53279 x)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {106558 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {106772 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {74234 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \int \frac {89048-53279 x}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {106558 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {106772 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {74234 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{14} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {605077}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {53279}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {106558 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {106772 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {74234 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{14} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {605077}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {53279}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {106558 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {106772 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {74234 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{14} \left (\frac {1}{147} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {53279}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {110014}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )\right )-\frac {106558 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {106772 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {74234 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )+\frac {458 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\)

Input:

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

Output:

(11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) + ((458*Sqrt[1 - 2* 
x]*Sqrt[3 + 5*x])/(147*(2 + 3*x)^(7/2)) + ((-74234*Sqrt[1 - 2*x]*Sqrt[3 + 
5*x])/(35*(2 + 3*x)^(5/2)) + (6*((-106772*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21 
*(2 + 3*x)^(3/2)) + ((-106558*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x 
]) + (10*((53279*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/ 
33])/5 - (110014*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/ 
33])/5))/7)/21))/35)/147)/14
 

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]
 

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.38

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {484 \left (-30 x^{2}-38 x -12\right )}{16807 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {178096 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{7411887 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {106558 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{7411887 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{83349 \left (\frac {2}{3}+x \right )^{4}}+\frac {998 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{972405 \left (\frac {2}{3}+x \right )^{3}}-\frac {30542 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2268945 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {91804}{352947} x^{2}-\frac {45902}{1764735} x +\frac {45902}{588245}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(301\)
default	\(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (49011237 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1438533 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2877066 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1918044 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+426232 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x +5546943\right )}{5294205 \left (2+3 x \right )^{\frac {7}{2}} \left (10 x^{2}+x -3\right )}\)	\(401\)

Input:

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

Output:

(-(3+5*x)*(-1+2*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 
)*(-484/16807*(-30*x^2-38*x-12)/((x-1/2)*(-30*x^2-38*x-12))^(1/2)+178096/7 
411887*(28+42*x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7*x 
+6)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-106558/7411887*(28+4 
2*x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(- 
1/15*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-3/5*EllipticF(1/7*(28+42* 
x)^(1/2),1/2*70^(1/2)))-2/83349*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+998 
/972405*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3-30542/2268945*(-30*x^3-23*x 
^2+7*x+6)^(1/2)/(2/3+x)^2+45902/5294205*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2- 
3*x+9))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.68 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx=-\frac {270 \, {\left (2877066 \, x^{4} + 11042235 \, x^{3} + 12020751 \, x^{2} + 4889131 \, x + 616327\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 9239737 \, \sqrt {-30} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 4795110 \, \sqrt {-30} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{238239225 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \] Input:

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

Output:

-1/238239225*(270*(2877066*x^4 + 11042235*x^3 + 12020751*x^2 + 4889131*x + 
 616327)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 9239737*sqrt(-30)*(1 
62*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*weierstrassPInverse(1159/ 
675, 38998/91125, x + 23/90) + 4795110*sqrt(-30)*(162*x^5 + 351*x^4 + 216* 
x^3 - 24*x^2 - 64*x - 16)*weierstrassZeta(1159/675, 38998/91125, weierstra 
ssPInverse(1159/675, 38998/91125, x + 23/90)))/(162*x^5 + 351*x^4 + 216*x^ 
3 - 24*x^2 - 64*x - 16)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 20*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 70*sqrt(3*x + 2)*s 
qrt(5*x + 3)*sqrt( - 2*x + 1) - 766260*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sq 
rt( - 2*x + 1)*x**2)/(4860*x**8 + 14256*x**7 + 13419*x**6 + 819*x**5 - 609 
0*x**4 - 3080*x**3 + 224*x**2 + 496*x + 96),x)*x**5 - 1660230*int((sqrt(3* 
x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(4860*x**8 + 14256*x**7 + 1341 
9*x**6 + 819*x**5 - 6090*x**4 - 3080*x**3 + 224*x**2 + 496*x + 96),x)*x**4 
 - 1021680*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(4860*x 
**8 + 14256*x**7 + 13419*x**6 + 819*x**5 - 6090*x**4 - 3080*x**3 + 224*x** 
2 + 496*x + 96),x)*x**3 + 113520*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 
2*x + 1)*x**2)/(4860*x**8 + 14256*x**7 + 13419*x**6 + 819*x**5 - 6090*x**4 
 - 3080*x**3 + 224*x**2 + 496*x + 96),x)*x**2 + 302720*int((sqrt(3*x + 2)* 
sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(4860*x**8 + 14256*x**7 + 13419*x**6 
+ 819*x**5 - 6090*x**4 - 3080*x**3 + 224*x**2 + 496*x + 96),x)*x + 75680*i 
nt((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(4860*x**8 + 14256* 
x**7 + 13419*x**6 + 819*x**5 - 6090*x**4 - 3080*x**3 + 224*x**2 + 496*x + 
96),x) + 265518*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(4860*x 
**8 + 14256*x**7 + 13419*x**6 + 819*x**5 - 6090*x**4 - 3080*x**3 + 224*x** 
2 + 496*x + 96),x)*x**5 + 575289*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 
2*x + 1))/(4860*x**8 + 14256*x**7 + 13419*x**6 + 819*x**5 - 6090*x**4 - 30 
80*x**3 + 224*x**2 + 496*x + 96),x)*x**4 + 354024*int((sqrt(3*x + 2)*sq...