\(\int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\) [1604]

Optimal result

Integrand size = 28, antiderivative size = 222 \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {3218 \sqrt {1-2 x} (2+3 x)^{5/2}}{19965 (3+5 x)^{3/2}}-\frac {217 (2+3 x)^{7/2}}{121 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^{9/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac {110519 \sqrt {1-2 x} (2+3 x)^{3/2}}{1098075 \sqrt {3+5 x}}-\frac {5199979 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3660250}-\frac {90397364 \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{5490375}+\frac {5199979 \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{10980750} \] Output:

3218/19965*(1-2*x)^(1/2)*(2+3*x)^(5/2)/(3+5*x)^(3/2)-217/121*(2+3*x)^(7/2) 
/(1-2*x)^(1/2)/(3+5*x)^(3/2)+7/33*(2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2 
)+110519/1098075*(1-2*x)^(1/2)*(2+3*x)^(3/2)/(3+5*x)^(1/2)-5199979/3660250 
*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-90397364/27451875*EllipticE(1/1 
1*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+5199979/54903750*Ellipt 
icF(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 = 8.53 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.47 \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {-\frac {5 \sqrt {2+3 x} \left (246962693+89252928 x-1696384053 x^2-1825153850 x^3+177888150 x^4\right )}{(1-2 x)^{3/2} (3+5 x)^{3/2}}+i \sqrt {33} \left (180794728 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-186236855 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{54903750} \] Input:

Integrate[(2 + 3*x)^(11/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]
 

Output:

((-5*Sqrt[2 + 3*x]*(246962693 + 89252928*x - 1696384053*x^2 - 1825153850*x 
^3 + 177888150*x^4))/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + I*Sqrt[33]*(18079 
4728*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 186236855*EllipticF[I*A 
rcSinh[Sqrt[9 + 15*x]], -2/33]))/54903750
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.11, 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, 25, 167, 27, 167, 27, 171, 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.

\(\displaystyle \int \frac {(3 x+2)^{11/2}}{(1-2 x)^{5/2} (5 x+3)^{5/2}} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{33} \int \frac {3 (3 x+2)^{7/2} (204 x+115)}{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {1}{22} \int \frac {(3 x+2)^{7/2} (204 x+115)}{(1-2 x)^{3/2} (5 x+3)^{5/2}}dx\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{22} \left (-\frac {1}{11} \int -\frac {(3 x+2)^{5/2} (13131 x+7235)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \int \frac {(3 x+2)^{5/2} (13131 x+7235)}{\sqrt {1-2 x} (5 x+3)^{5/2}}dx-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {2}{165} \int \frac {(3 x+2)^{3/2} (1357893 x+792632)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \int \frac {(3 x+2)^{3/2} (1357893 x+792632)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {2}{55} \int \frac {9 \sqrt {3 x+2} (5199979 x+3208775)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \int \frac {\sqrt {3 x+2} (5199979 x+3208775)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \left (-\frac {1}{15} \int -\frac {361589456 x+228926353}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5199979}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \left (\frac {1}{30} \int \frac {361589456 x+228926353}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5199979}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \left (\frac {1}{30} \left (\frac {59863397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {361589456}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {5199979}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \left (\frac {1}{30} \left (\frac {59863397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {361589456}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5199979}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{22} \left (\frac {1}{11} \left (\frac {1}{165} \left (\frac {9}{55} \left (\frac {1}{30} \left (-\frac {10884254}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {361589456}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5199979}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {221038 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {6436 \sqrt {1-2 x} (3 x+2)^{5/2}}{165 (5 x+3)^{3/2}}\right )-\frac {434 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} (5 x+3)^{3/2}}\right )+\frac {7 (3 x+2)^{9/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}\)

Input:

Int[(2 + 3*x)^(11/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]
 

Output:

(7*(2 + 3*x)^(9/2))/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + ((-434*(2 + 3*x 
)^(7/2))/(11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + ((6436*Sqrt[1 - 2*x]*(2 + 3* 
x)^(5/2))/(165*(3 + 5*x)^(3/2)) + ((221038*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/ 
(55*Sqrt[3 + 5*x]) + (9*((-5199979*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5* 
x])/15 + ((-361589456*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]] 
, 35/33])/5 - (10884254*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x 
]], 35/33])/5)/30))/55)/165)/11)/22
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && 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 = 0.84 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.14

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (\frac {31513109}{181500000}+\frac {52521907 x}{181500000}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (-\frac {2810713793}{4392300000}-\frac {46518287 x}{54903750}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}-\frac {81 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{500}+\frac {228926353 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{153730500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {90397364 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{38432625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(253\)
default	\(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (622345986360+726070317420 x -11314182033 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right )[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-3111729931800 x^{3}-2385659614380 x^{2}[phone] \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2}-75933785760 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2}[phone] \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x -7593378576 \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x -1075133902500 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{4}-656251957620 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{2}[phone] \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x [phone] \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{5}-1835286570390 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{3}\right )}{2305957500 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \left (6 x^{2}+x -2\right ) \left (-1+2 x \right ) \left (3+5 x \right )^{\frac {3}{2}}}\)	\(515\)

Input:

int((2+3*x)^(11/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),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 
)*((31513109/181500000+52521907/181500000*x)*(-30*x^3-23*x^2+7*x+6)^(1/2)/ 
(-3/10+x^2+1/10*x)^2-2*(-20-30*x)*(-2810713793/4392300000-46518287/5490375 
0*x)/((-3/10+x^2+1/10*x)*(-20-30*x))^(1/2)-81/500*(-30*x^3-23*x^2+7*x+6)^( 
1/2)[phone]/153730500*(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))+90 
397364/38432625*(28+42*x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^3-2 
3*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))))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.60 \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=-\frac {450 \, {\left (177888150 \, x^{4} - 1825153850 \, x^{3} - 1696384053 \, x^{2} + 89252928 \, x + 246962693\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 6143407141 \, \sqrt {-30} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 16271525520 \, \sqrt {-30} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{4941337500 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \] Input:

integrate((2+3*x)^(11/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas" 
)
 

Output:

-1/4941337500*(450*(177888150*x^4 - 1825153850*x^3 - 1696384053*x^2 + 8925 
2928*x + 246962693)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 614340714 
1*sqrt(-30)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*weierstrassPInverse(1159 
/675, 38998/91125, x + 23/90) - 16271525520*sqrt(-30)*(100*x^4 + 20*x^3 - 
59*x^2 - 6*x + 9)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInver 
se(1159/675, 38998/91125, x + 23/90)))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 
9)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((2+3*x)**(11/2)/(1-2*x)**(5/2)/(3+5*x)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {11}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((2+3*x)^(11/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima" 
)
 

Output:

integrate((3*x + 2)^(11/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)
 

Giac [F]

\[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {11}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((2+3*x)^(11/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")
 

Output:

integrate((3*x + 2)^(11/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{11/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:

int((3*x + 2)^(11/2)/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)),x)
                                                                                    
                                                                                    
 

Output:

int((3*x + 2)^(11/2)/((1 - 2*x)^(5/2)*(5*x + 3)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {(2+3 x)^{11/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx =\text {Too large to display} \] Input:

int((2+3*x)^(11/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)
 

Output:

( - 121500*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 1223100*sqr 
t(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 2228310*sqrt(3*x + 2)*sqr 
t(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 717624*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt 
( - 2*x + 1)*x - 521036*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) - 679 
87500*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(3000*x**7 + 
 2900*x**6 - 2010*x**5 - 2277*x**4 + 425*x**3 + 603*x**2 - 27*x - 54),x)*x 
**4 - 13597500*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(30 
00*x**7 + 2900*x**6 - 2010*x**5 - 2277*x**4 + 425*x**3 + 603*x**2 - 27*x - 
 54),x)*x**3 + 40112625*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)* 
x**2)/(3000*x**7 + 2900*x**6 - 2010*x**5 - 2277*x**4 + 425*x**3 + 603*x**2 
 - 27*x - 54),x)*x**2 + 4079250*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2 
*x + 1)*x**2)/(3000*x**7 + 2900*x**6 - 2010*x**5 - 2277*x**4 + 425*x**3 + 
603*x**2 - 27*x - 54),x)*x - 6118875*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt 
( - 2*x + 1)*x**2)/(3000*x**7 + 2900*x**6 - 2010*x**5 - 2277*x**4 + 425*x* 
*3 + 603*x**2 - 27*x - 54),x) - 8202600*int((sqrt(3*x + 2)*sqrt(5*x + 3)*s 
qrt( - 2*x + 1))/(3000*x**7 + 2900*x**6 - 2010*x**5 - 2277*x**4 + 425*x**3 
 + 603*x**2 - 27*x - 54),x)*x**4 - 1640520*int((sqrt(3*x + 2)*sqrt(5*x + 3 
)*sqrt( - 2*x + 1))/(3000*x**7 + 2900*x**6 - 2010*x**5 - 2277*x**4 + 425*x 
**3 + 603*x**2 - 27*x - 54),x)*x**3 + 4839534*int((sqrt(3*x + 2)*sqrt(5*x 
+ 3)*sqrt( - 2*x + 1))/(3000*x**7 + 2900*x**6 - 2010*x**5 - 2277*x**4 +...