\(\int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx\) [1373]

Optimal result

Integrand size = 28, antiderivative size = 280 \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {51920987843 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{22804031250}+\frac {13066171 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{12993750}-\frac {7119037 (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}}{6756750}-\frac {302777 (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x}}{579150}-\frac {4702 (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}}{19305}-\frac {31}{585} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1764163292393 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{19546312500 \sqrt {35}}+\frac {25385346787 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{9773156250 \sqrt {35}} \] Output:

51920987843/22804031250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)[phone] 
/12993750*(1-2*x)^(3/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-7119037/6756750*(1-2*x 
)^(5/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-302777/579150*(1-2*x)^(5/2)*(2+3*x)^(3 
/2)*(3+5*x)^(1/2)-4702/19305*(1-2*x)^(5/2)*(2+3*x)^(5/2)*(3+5*x)^(1/2)-31/ 
585*(1-2*x)^(5/2)*(2+3*x)^(5/2)*(3+5*x)^(3/2)+2/75*(1-2*x)^(5/2)*(2+3*x)^( 
5/2)*(3+5*x)^(5/2)-1764163292393/684120937500*EllipticE(1/11*55^(1/2)*(1-2 
*x)^(1/2),1/35*1155^(1/2))*35^(1/2)[phone]/342060468750*EllipticF(1/1 
1*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 = 8.91 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.42 \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (12155574323+173484591165 x+48836706750 x^2-528977216250 x^3-336683182500 x^4+621672975000 x^5+547296750000 x^6\right )[phone] i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1817234574505 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{684120937500} \] Input:

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

Output:

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(12155574323 + 173484591165* 
x + 48836706750*x^2 - 528977216250*x^3 - 336683182500*x^4 + 621672975000*x 
^5 + 547296750000*x^6) + (1764163292393*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sq 
rt[9 + 15*x]], -2/33] - (1817234574505*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqr 
t[9 + 15*x]], -2/33])/684120937500
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.14, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {112, 27, 171, 27, 171, 27, 171, 27, 171, 25, 171, 27, 171, 25, 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.

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

\(\Big \downarrow \) 112

\(\displaystyle \frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {2}{75} \int -\frac {5}{2} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2} (31 x+23)dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{15} \int (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{3/2} (31 x+23)dx+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{15} \left (\frac {2}{195} \int \frac {1}{2} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2} (1849 x+2656)dx+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \int \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{3/2} (1849 x+2656)dx+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {2}{165} \int \frac {(284773-427173 x) (3 x+2)^{3/2} (5 x+3)^{3/2}}{2 \sqrt {1-2 x}}dx+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \int \frac {(284773-427173 x) (3 x+2)^{3/2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}dx+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {1}{45} \int -\frac {3 \sqrt {3 x+2} (5 x+3)^{3/2} (10251342 x+7830965)}{2 \sqrt {1-2 x}}dx\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \int \frac {\sqrt {3 x+2} (5 x+3)^{3/2} (10251342 x+7830965)}{\sqrt {1-2 x}}dx+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (-\frac {1}{35} \int -\frac {(5 x+3)^{3/2} (1201550979 x+789074087)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \int \frac {(5 x+3)^{3/2} (1201550979 x+789074087)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (53071282112 x+34486181421)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \int \frac {\sqrt {5 x+3} (53071282112 x+34486181421)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (-\frac {1}{9} \int -\frac {1764163292393 x+1116876385759}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {53071282112}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \int \frac {1764163292393 x+1116876385759}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {53071282112}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {291892051616}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1764163292393}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {53071282112}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {291892051616}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1764163292393}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {53071282112}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{15} \left (\frac {1}{195} \left (\frac {1}{165} \left (\frac {1}{30} \left (\frac {1}{35} \left (\frac {1}{10} \left (\frac {1}{9} \left (-\frac {53071282112}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1764163292393}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {53071282112}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {400516993}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {10251342}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {142391}{15} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {3698}{165} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {62}{195} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\right )+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}\)

Input:

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

Output:

(2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/75 + ((62*(1 - 2*x)^(3 
/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(5/2))/195 + ((3698*Sqrt[1 - 2*x]*(2 + 3*x)^ 
(5/2)*(3 + 5*x)^(5/2))/165 + ((142391*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5 
*x)^(5/2))/15 + ((-10251342*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/3 
5 + ((-400516993*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + ((-53071 
282112*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-1764163292393*Sqr 
t[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (5307128211 
2*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/9)/10)/ 
35)/30)/165)/195)/15
 

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 112

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + 
p + 1))), x] - Simp[1/(f*(m + n + p + 1))   Int[(a + b*x)^(m - 1)*(c + d*x) 
^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a 
*f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && 
GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (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 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 = 0.52 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.60

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-492567075000000 x^{9}-937140435000000 x^{8}-11007171000000 x^{7}[phone] x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1764163292393 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{5}-361521647968500 x^{4}-215604575302050 x^{3}[phone] x^{2}[phone] x [phone]\right )}{684120937500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(168\)
risch	\(-\frac {\left (547296750000 x^{6}[phone] x^{5}-336683182500 x^{4}-528977216250 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{22804031250 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {1116876385759 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{2508443437500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1764163292393 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{2508443437500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(272\)
elliptic	\(\frac {\sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {3855213137 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{506756250}+\frac {12155574323 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{22804031250}+\frac {1116876385759 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{957769312500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1764163292393 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{957769312500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1669631 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{779625}-\frac {2239057 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{96525}-\frac {52782 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{3575}+\frac {1772 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{65}+24 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{6}\right )}{\left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}\)	\(334\)

Input:

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

Output:

-1/684120937500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-49256707500000 
0*x^9-937140435000000*x^8-11007171000000*x^7+937455630300000*x^6+875676154 
848*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+4 
2*x)^(1/2),1/2*70^(1/2))-1764163292393*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2 
)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))[phone] 
0*x^5-361521647968500*x^4-215604575302050*x^3+36835025076780*x^2+337798970 
17530*x[phone])/(30*x^3+23*x^2-7*x-6)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.28 \[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=\frac {1}{22804031250} \, {\left (547296750000 \, x^{6} + 621672975000 \, x^{5} - 336683182500 \, x^{4} - 528977216250 \, x^{3} + 48836706750 \, x^{2} + 173484591165 \, x + 12155574323\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {59943118993271}{61570884375000} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {1764163292393}{684120937500} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:

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

Output:

1/22804031250*(547296750000*x^6 + 621672975000*x^5 - 336683182500*x^4 - 52 
8977216250*x^3 + 48836706750*x^2 + 173484591165*x + 12155574323)*sqrt(5*x 
+ 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 59943118993271/61570884375000*sqrt(-30 
)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 1764163292393/68 
4120937500*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPIn 
verse(1159/675, 38998/91125, x + 23/90))
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx=24 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{6}+\frac {1772 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{5}}{65}-\frac {52782 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{3575}-\frac {2239057 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{96525}+\frac {1669631 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{779625}+\frac {3855213137 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{506756250}-\frac {8925976841 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{7770262500}+\frac {1764163292393 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{23310787500}-\frac {52448935631 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{1726725000} \] Input:

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

Output:

(1118917800000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**6 + 1270975 
860000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**5 - 688330062000*sq 
rt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 1081464531000*sqrt(3*x + 
 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 99843933800*sqrt(3*x + 2)*sqrt(5 
*x + 3)*sqrt( - 2*x + 1)*x**2 + 354679608604*sqrt(3*x + 2)*sqrt(5*x + 3)*s 
qrt( - 2*x + 1)*x - 53555861046*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 
1) + 3528326584786*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2) 
/(30*x**3 + 23*x**2 - 7*x - 6),x) - 1416121262037*int((sqrt(3*x + 2)*sqrt( 
5*x + 3)*sqrt( - 2*x + 1))/(30*x**3 + 23*x**2 - 7*x - 6),x))/46621575000