\(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\) [1465]

Optimal result

Integrand size = 28, antiderivative size = 187 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {6 \sqrt {1-2 x}}{35 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {436 \sqrt {1-2 x}}{245 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {60684 \sqrt {1-2 x}}{1715 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {1344984 \sqrt {1-2 x} \sqrt {2+3 x}}{3773 \sqrt {3+5 x}}+\frac {1344984 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{539 \sqrt {35}}-\frac {38712 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{539 \sqrt {35}} \] Output:

6/35*(1-2*x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2)+436/245*(1-2*x)^(1/2)/(2+3* 
x)^(3/2)/(3+5*x)^(1/2)+60684/1715*(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 
)-1344984/3773*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)+1344984/18865*Ell 
ipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-38712/18865*E 
llipticF(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 = 6.59 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\frac {2 \left (-\frac {\sqrt {1-2 x} \left (25529443+116993058 x+178568982 x^2+90786420 x^3\right )}{(2+3 x)^{5/2} \sqrt {3+5 x}}-4 i \sqrt {33} \left (168123 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-173180 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{18865} \] Input:

Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)),x]
 

Output:

(2*(-((Sqrt[1 - 2*x]*(25529443 + 116993058*x + 178568982*x^2 + 90786420*x^ 
3))/((2 + 3*x)^(5/2)*Sqrt[3 + 5*x])) - (4*I)*Sqrt[33]*(168123*EllipticE[I* 
ArcSinh[Sqrt[9 + 15*x]], -2/33] - 173180*EllipticF[I*ArcSinh[Sqrt[9 + 15*x 
]], -2/33])))/18865
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {115, 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.

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

\(\Big \downarrow \) 115

\(\displaystyle \frac {2}{35} \int \frac {59-75 x}{\sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}dx+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {2}{35} \left (\frac {2}{21} \int \frac {9 (959-1090 x)}{2 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {218 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{35} \left (\frac {3}{7} \int \frac {959-1090 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2}}dx+\frac {218 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {2}{35} \left (\frac {3}{7} \left (\frac {2}{7} \int \frac {5 (8174-5057 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {10114 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {218 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{35} \left (\frac {3}{7} \left (\frac {10}{7} \int \frac {8174-5057 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {10114 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {218 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {2}{35} \left (\frac {3}{7} \left (\frac {10}{7} \left (-\frac {2}{11} \int \frac {336246 x+212873}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {112082 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {10114 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {218 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{35} \left (\frac {3}{7} \left (\frac {10}{7} \left (-\frac {1}{11} \int \frac {336246 x+212873}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {112082 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {10114 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {218 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {2}{35} \left (\frac {3}{7} \left (\frac {10}{7} \left (\frac {1}{11} \left (-\frac {55627}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {336246}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {112082 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {10114 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {218 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {2}{35} \left (\frac {3}{7} \left (\frac {10}{7} \left (\frac {1}{11} \left (\frac {112082}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {55627}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {112082 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {10114 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {218 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {2}{35} \left (\frac {3}{7} \left (\frac {10}{7} \left (\frac {1}{11} \left (\frac {10114}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {112082}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {112082 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {10114 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}\right )+\frac {218 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} \sqrt {5 x+3}}\right )+\frac {6 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} \sqrt {5 x+3}}\)

Input:

Int[1/(Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)),x]
 

Output:

(6*Sqrt[1 - 2*x])/(35*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (2*((218*Sqrt[1 - 2 
*x])/(7*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (3*((10114*Sqrt[1 - 2*x])/(7*Sqrt 
[2 + 3*x]*Sqrt[3 + 5*x]) + (10*((-112082*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11* 
Sqrt[3 + 5*x]) + ((112082*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x 
]], 35/33])/5 + (10114*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x] 
], 35/33])/5)/11))/7))/7))/35
 

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 115

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 
*n, 2*p]
 

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 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 = 0.90 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.48

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {851492 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{26411 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1344984 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{26411 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{105 \left (\frac {2}{3}+x \right )^{3}}-\frac {646 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{735 \left (\frac {2}{3}+x \right )^{2}}-\frac {83294 \left (-30 x^{2}-3 x +9\right )}{1715 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {250 \left (-30 x^{2}-5 x +10\right )}{11 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(277\)
default	\(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (3003858 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-6052428 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+4005144 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-8069904 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1335048 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2689968 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-181572840 x^{4}-266351544 x^{3}-55417134 x^{2}[phone] x [phone]\right )}{18865 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\)	\(308\)

Input:

int(1/(1-2*x)^(1/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),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 
)*(-851492/26411*(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))-1344984/26 
411*(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)*(-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)))-2/105*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x 
)^3-646/735*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2-83294/1715*(-30*x^2-3*x 
+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)-250/11*(-30*x^2-5*x+10)/((x+3/5)*(-30* 
x^2-5*x+10))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=-\frac {2 \, {\left (15 \, {\left (90786420 \, x^{3} + 178568982 \, x^{2} + 116993058 \, x + 25529443\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 3808304 \, \sqrt {-30} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 10087380 \, \sqrt {-30} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{282975 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \] Input:

integrate(1/(1-2*x)^(1/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="fricas 
")
 

Output:

-2/282975*(15*(90786420*x^3 + 178568982*x^2 + 116993058*x + 25529443)*sqrt 
(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 3808304*sqrt(-30)*(135*x^4 + 351* 
x^3 + 342*x^2 + 148*x + 24)*weierstrassPInverse(1159/675, 38998/91125, x + 
 23/90) + 10087380*sqrt(-30)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*we 
ierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/9 
1125, x + 23/90)))/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(1-2*x)**(1/2)/(2+3*x)**(7/2)/(3+5*x)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}} \,d x } \] Input:

integrate(1/(1-2*x)^(1/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="maxima 
")
 

Output:

integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*sqrt(-2*x + 1)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}} \,d x } \] Input:

integrate(1/(1-2*x)^(1/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="giac")
 

Output:

integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*sqrt(-2*x + 1)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=\int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:

int(1/((1 - 2*x)^(1/2)*(3*x + 2)^(7/2)*(5*x + 3)^(3/2)),x)
 

Output:

int(1/((1 - 2*x)^(1/2)*(3*x + 2)^(7/2)*(5*x + 3)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx=-\left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{4050 x^{7}+13635 x^{6}+17388 x^{5}+9039 x^{4}-376 x^{3}-2536 x^{2}-1056 x -144}d x \right ) \] Input:

int(1/(1-2*x)^(1/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x)
 

Output:

 - int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(4050*x**7 + 13635*x 
**6 + 17388*x**5 + 9039*x**4 - 376*x**3 - 2536*x**2 - 1056*x - 144),x)