3.24.90 \(\int (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x} \, dx\) [2390]

3.24.90.1 Optimal result

Integrand size = 24, antiderivative size = 138 \[ \int (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x} \, dx=\frac {158389 \sqrt {1-2 x} \sqrt {3+5 x}}{320000}+\frac {14399 (1-2 x)^{3/2} \sqrt {3+5 x}}{96000}+\frac {1309 (1-2 x)^{5/2} \sqrt {3+5 x}}{24000}-\frac {119}{800} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {1742279 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320000 \sqrt {10}} \]

output

-3/50*(1-2*x)^(7/2)*(3+5*x)^(3/2)+1742279/3200000*arcsin(1/11*22^(1/2)*(3+ 
5*x)^(1/2))*10^(1/2)+14399/96000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+1309/24000*(1 
-2*x)^(5/2)*(3+5*x)^(1/2)-119/800*(1-2*x)^(7/2)*(3+5*x)^(1/2)+158389/32000 
0*(1-2*x)^(1/2)*(3+5*x)^(1/2)
 

3.24.90.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.60 \[ \int (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x} \, dx=\frac {10 \sqrt {1-2 x} \left (1067751+5104125 x-8380 x^2-12042400 x^3+2256000 x^4+11520000 x^5\right )-5226837 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{9600000 \sqrt {3+5 x}} \]

input

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

output

(10*Sqrt[1 - 2*x]*(1067751 + 5104125*x - 8380*x^2 - 12042400*x^3 + 2256000 
*x^4 + 11520000*x^5) - 5226837*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt 
[3 + 5*x]])/(9600000*Sqrt[3 + 5*x])
 

3.24.90.3 Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {90, 60, 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.

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {119}{100} \int (1-2 x)^{5/2} \sqrt {5 x+3}dx-\frac {3}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {119}{100} \left (\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {5 x+3}}dx-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {119}{100} \left (\frac {11}{16} \left (\frac {11}{6} \int \frac {(1-2 x)^{3/2}}{\sqrt {5 x+3}}dx+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {119}{100} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {119}{100} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {119}{100} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {119}{100} \left (\frac {11}{16} \left (\frac {11}{6} \left (\frac {33}{20} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{3/2}\right )+\frac {1}{15} \sqrt {5 x+3} (1-2 x)^{5/2}\right )-\frac {1}{8} (1-2 x)^{7/2} \sqrt {5 x+3}\right )-\frac {3}{50} (1-2 x)^{7/2} (5 x+3)^{3/2}\)

input

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

output

(-3*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/50 + (119*(-1/8*((1 - 2*x)^(7/2)*Sqrt 
[3 + 5*x]) + (11*(((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[S 
qrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/20))/6))/16))/100
 

3.24.90.3.1 Defintions of rubi rules used

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 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]
 

3.24.90.4 Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78

method	result	size
risch	\(-\frac {\left (2304000 x^{4}-931200 x^{3}-1849760 x^{2}+1108180 x +355917\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{960000 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {1742279 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6400000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(108\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (46080000 x^{4} \sqrt {-10 x^{2}-x +3}-18624000 x^{3} \sqrt {-10 x^{2}-x +3}-36995200 x^{2} \sqrt {-10 x^{2}-x +3}+5226837 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}+7118340 \sqrt {-10 x^{2}-x +3}\right )}{19200000 \sqrt {-10 x^{2}-x +3}}\)	\(121\)

input

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

output

-1/960000*(2304000*x^4-931200*x^3-1849760*x^2+1108180*x+355917)*(-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)+1742279/6400000*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)
 

3.24.90.5 Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.56 \[ \int (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x} \, dx=\frac {1}{960000} \, {\left (2304000 \, x^{4} - 931200 \, x^{3} - 1849760 \, x^{2} + 1108180 \, x + 355917\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1742279}{6400000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

input

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

output

1/960000*(2304000*x^4 - 931200*x^3 - 1849760*x^2 + 1108180*x + 355917)*sqr 
t(5*x + 3)*sqrt(-2*x + 1) - 1742279/6400000*sqrt(10)*arctan(1/20*sqrt(10)* 
(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 

3.24.90.6 Sympy [F]

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

input

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

output

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

3.24.90.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.63 \[ \int (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x} \, dx=-\frac {6}{25} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {121}{1000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1303}{12000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {14399}{16000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1742279}{6400000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {14399}{320000} \, \sqrt {-10 \, x^{2} - x + 3} \]

input

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

output

-6/25*(-10*x^2 - x + 3)^(3/2)*x^2 + 121/1000*(-10*x^2 - x + 3)^(3/2)*x + 1 
303/12000*(-10*x^2 - x + 3)^(3/2) + 14399/16000*sqrt(-10*x^2 - x + 3)*x - 
1742279/6400000*sqrt(10)*arcsin(-20/11*x - 1/11) + 14399/320000*sqrt(-10*x 
^2 - x + 3)
 

3.24.90.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (99) = 198\).

Time = 0.34 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x} \, dx=\frac {1}{16000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {37}{120000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{400} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {3}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

input

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

output

1/16000000*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))) + 1/600000*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))) - 37/120000*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))) - 1/400*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))) + 3/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) 
+ 2*sqrt(5*x + 3)*sqrt(-10*x + 5))
 

3.24.90.9 Mupad [F(-1)]

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

input

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

output

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