3.25.26 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx\) [2426]

3.25.26.1 Optimal result

Integrand size = 26, antiderivative size = 143 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx=\frac {593747 \sqrt {1-2 x} \sqrt {3+5 x}}{1600000}+\frac {53977 (1-2 x)^{3/2} \sqrt {3+5 x}}{480000}+\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {6531217 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600000 \sqrt {10}} \]

output

6531217/16000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+53977/480000 
*(1-2*x)^(3/2)*(3+5*x)^(1/2)+4907/120000*(1-2*x)^(5/2)*(3+5*x)^(1/2)-369/4 
000*(1-2*x)^(7/2)*(3+5*x)^(1/2)-3/50*(1-2*x)^(7/2)*(2+3*x)*(3+5*x)^(1/2)+5 
93747/1600000*(1-2*x)^(1/2)*(3+5*x)^(1/2)
 

3.25.26.2 Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.58 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx=\frac {10 \sqrt {1-2 x} \left (4495473+17644875 x-1848740 x^2-37935200 x^3+9648000 x^4+34560000 x^5\right )-19593651 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{48000000 \sqrt {3+5 x}} \]

input

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

output

(10*Sqrt[1 - 2*x]*(4495473 + 17644875*x - 1848740*x^2 - 37935200*x^3 + 964 
8000*x^4 + 34560000*x^5) - 19593651*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x] 
/Sqrt[3 + 5*x]])/(48000000*Sqrt[3 + 5*x])
 

3.25.26.3 Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.14, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {101, 27, 90, 60, 60, 60, 64, 223}

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.

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

\(\Big \downarrow \) 101

\(\displaystyle -\frac {1}{50} \int -\frac {(1-2 x)^{5/2} (369 x+232)}{2 \sqrt {5 x+3}}dx-\frac {3}{50} (3 x+2) \sqrt {5 x+3} (1-2 x)^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{100} \int \frac {(1-2 x)^{5/2} (369 x+232)}{\sqrt {5 x+3}}dx-\frac {3}{50} (1-2 x)^{7/2} (3 x+2) \sqrt {5 x+3}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {1}{100} \left (\frac {4907}{80} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {369}{40} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (3 x+2) \sqrt {5 x+3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{100} \left (\frac {4907}{80} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {369}{40} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (3 x+2) \sqrt {5 x+3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{100} \left (\frac {4907}{80} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {369}{40} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (3 x+2) \sqrt {5 x+3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{100} \left (\frac {4907}{80} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {369}{40} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (3 x+2) \sqrt {5 x+3}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{100} \left (\frac {4907}{80} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {369}{40} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (3 x+2) \sqrt {5 x+3}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{100} \left (\frac {4907}{80} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {369}{40} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (3 x+2) \sqrt {5 x+3}\)

input

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

output

(-3*(1 - 2*x)^(7/2)*(2 + 3*x)*Sqrt[3 + 5*x])/50 + ((-369*(1 - 2*x)^(7/2)*S 
qrt[3 + 5*x])/40 + (4907*(((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/15 + (11*(((1 - 
2*x)^(3/2)*Sqrt[3 + 5*x])/10 + (33*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11* 
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20))/6))/80)/100
 

3.25.26.3.1 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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 64

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp 
[2/b   Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] 
 /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] 
 || PosQ[b])
 

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

3.25.26.4 Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.76

method	result	size
risch	\(-\frac {\left (6912000 x^{4}-2217600 x^{3}-6256480 x^{2}+3384140 x +1498491\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4800000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {6531217 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{32000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(108\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (138240000 x^{4} \sqrt {-10 x^{2}-x +3}-44352000 x^{3} \sqrt {-10 x^{2}-x +3}-125129600 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{96000000 \sqrt {-10 x^{2}-x +3}}\)	\(121\)

input

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

output

-1/4800000*(6912000*x^4-2217600*x^3-6256480*x^2+3384140*x+1498491)*(-1+2*x 
)*(3+5*x)^(1/2)/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^ 
(1/2)+6531217/32000000*10^(1/2)*arcsin(20/11*x+1/11)*((1-2*x)*(3+5*x))^(1/ 
2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)
 

3.25.26.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.54 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx=\frac {1}{4800000} \, {\left (6912000 \, x^{4} - 2217600 \, x^{3} - 6256480 \, x^{2} + 3384140 \, x + 1498491\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {6531217}{32000000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

input

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

output

1/4800000*(6912000*x^4 - 2217600*x^3 - 6256480*x^2 + 3384140*x + 1498491)* 
sqrt(5*x + 3)*sqrt(-2*x + 1) - 6531217/32000000*sqrt(10)*arctan(1/20*sqrt( 
10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 

3.25.26.6 Sympy [F]

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

input

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

output

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

3.25.26.7 Maxima [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.64 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx=\frac {36}{25} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {231}{500} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {39103}{30000} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {169207}{240000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {6531217}{32000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {499497}{1600000} \, \sqrt {-10 \, x^{2} - x + 3} \]

input

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

output

36/25*sqrt(-10*x^2 - x + 3)*x^4 - 231/500*sqrt(-10*x^2 - x + 3)*x^3 - 3910 
3/30000*sqrt(-10*x^2 - x + 3)*x^2 + 169207/240000*sqrt(-10*x^2 - x + 3)*x 
- 6531217/32000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 499497/1600000*sqrt( 
-10*x^2 - x + 3)
 

3.25.26.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (104) = 208\).

Time = 0.32 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.92 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx=\frac {3}{80000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{800000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {23}{120000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {2}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

input

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

output

3/80000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) 
- 506185)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sq 
rt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/800000*sqrt(5)*(2*(4*(8*(60 
*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 
5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 23/120000*sqrt( 
5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 478 
5*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/500*sqrt(5)*(2*(20*x - 
23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt( 
5*x + 3))) + 2/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) 
+ 2*sqrt(5*x + 3)*sqrt(-10*x + 5))
 

3.25.26.9 Mupad [F(-1)]

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

input

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

output

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