3.23.89 \(\int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx\) [2289]

3.23.89.1 Optimal result
3.23.89.2 Mathematica [A] (verified)
3.23.89.3 Rubi [A] (verified)
3.23.89.4 Maple [A] (verified)
3.23.89.5 Fricas [A] (verification not implemented)
3.23.89.6 Sympy [F]
3.23.89.7 Maxima [A] (verification not implemented)
3.23.89.8 Giac [B] (verification not implemented)
3.23.89.9 Mupad [F(-1)]

3.23.89.1 Optimal result

Integrand size = 24, antiderivative size = 138 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {334081 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}-\frac {30371 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120}-\frac {2761 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1920}-\frac {251}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {3674891 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}} \]

output
-2761/1920*(1-2*x)^(3/2)*(3+5*x)^(3/2)-251/800*(1-2*x)^(3/2)*(3+5*x)^(5/2) 
-3/50*(1-2*x)^(3/2)*(3+5*x)^(7/2)+3674891/512000*arcsin(1/11*22^(1/2)*(3+5 
*x)^(1/2))*10^(1/2)-30371/5120*(1-2*x)^(3/2)*(3+5*x)^(1/2)+334081/51200*(1 
-2*x)^(1/2)*(3+5*x)^(1/2)
 
3.23.89.2 Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.60 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {10 \sqrt {1-2 x} \left (-3762261-4115415 x+16522420 x^2+37765600 x^3+33936000 x^4+11520000 x^5\right )-11024673 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1536000 \sqrt {3+5 x}} \]

input
Integrate[Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(5/2),x]
 
output
(10*Sqrt[1 - 2*x]*(-3762261 - 4115415*x + 16522420*x^2 + 37765600*x^3 + 33 
936000*x^4 + 11520000*x^5) - 11024673*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5* 
x]/Sqrt[3 + 5*x]])/(1536000*Sqrt[3 + 5*x])
 
3.23.89.3 Rubi [A] (verified)

Time = 0.21 (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.

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {251}{100} \int \sqrt {1-2 x} (5 x+3)^{5/2}dx-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {251}{100} \left (\frac {55}{16} \int \sqrt {1-2 x} (5 x+3)^{3/2}dx-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {251}{100} \left (\frac {55}{16} \left (\frac {11}{4} \int \sqrt {1-2 x} \sqrt {5 x+3}dx-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {251}{100} \left (\frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \int \frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}dx-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {251}{100} \left (\frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {251}{100} \left (\frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11}{25} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {251}{100} \left (\frac {55}{16} \left (\frac {11}{4} \left (\frac {11}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5 \sqrt {10}}+\frac {1}{5} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} (1-2 x)^{3/2} \sqrt {5 x+3}\right )-\frac {1}{6} (1-2 x)^{3/2} (5 x+3)^{3/2}\right )-\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}\right )-\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{7/2}\)

input
Int[Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(5/2),x]
 
output
(-3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/50 + (251*(-1/8*((1 - 2*x)^(3/2)*(3 + 
 5*x)^(5/2)) + (55*(-1/6*((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (11*(-1/4*((1 
 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + (11*((Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5 + (11* 
ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5*Sqrt[10])))/8))/4))/16))/100
 

3.23.89.3.1 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]
 
3.23.89.4 Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78

method result size
risch \(-\frac {\left (2304000 x^{4}+5404800 x^{3}+4310240 x^{2}+718340 x -1254087\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{153600 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {3674891 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{1024000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) \(108\)
default \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (46080000 x^{4} \sqrt {-10 x^{2}-x +3}+108096000 x^{3} \sqrt {-10 x^{2}-x +3}+86204800 x^{2} \sqrt {-10 x^{2}-x +3}+11024673 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+14366800 x \sqrt {-10 x^{2}-x +3}-25081740 \sqrt {-10 x^{2}-x +3}\right )}{3072000 \sqrt {-10 x^{2}-x +3}}\) \(121\)

input
int((2+3*x)*(3+5*x)^(5/2)*(1-2*x)^(1/2),x,method=_RETURNVERBOSE)
 
output
-1/153600*(2304000*x^4+5404800*x^3+4310240*x^2+718340*x-1254087)*(-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)+3674891/1024000*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)
 
3.23.89.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.56 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {1}{153600} \, {\left (2304000 \, x^{4} + 5404800 \, x^{3} + 4310240 \, x^{2} + 718340 \, x - 1254087\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3674891}{1024000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

input
integrate((2+3*x)*(3+5*x)^(5/2)*(1-2*x)^(1/2),x, algorithm="fricas")
 
output
1/153600*(2304000*x^4 + 5404800*x^3 + 4310240*x^2 + 718340*x - 1254087)*sq 
rt(5*x + 3)*sqrt(-2*x + 1) - 3674891/1024000*sqrt(10)*arctan(1/20*sqrt(10) 
*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 
3.23.89.6 Sympy [F]

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

input
integrate((2+3*x)*(3+5*x)**(5/2)*(1-2*x)**(1/2),x)
 
output
Integral(sqrt(1 - 2*x)*(3*x + 2)*(5*x + 3)**(5/2), x)
 
3.23.89.7 Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.63 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=-\frac {3}{2} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {539}{160} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {1121}{384} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {30371}{2560} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3674891}{1024000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {30371}{51200} \, \sqrt {-10 \, x^{2} - x + 3} \]

input
integrate((2+3*x)*(3+5*x)^(5/2)*(1-2*x)^(1/2),x, algorithm="maxima")
 
output
-3/2*(-10*x^2 - x + 3)^(3/2)*x^2 - 539/160*(-10*x^2 - x + 3)^(3/2)*x - 112 
1/384*(-10*x^2 - x + 3)^(3/2) + 30371/2560*sqrt(-10*x^2 - x + 3)*x - 36748 
91/1024000*sqrt(10)*arcsin(-20/11*x - 1/11) + 30371/51200*sqrt(-10*x^2 - x 
 + 3)
 
3.23.89.8 Giac [B] (verification not implemented)

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

Time = 0.35 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.99 \[ \int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \, dx=\frac {1}{2560000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {37}{384000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {57}{8000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {351}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {27}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

input
integrate((2+3*x)*(3+5*x)^(5/2)*(1-2*x)^(1/2),x, algorithm="giac")
 
output
1/2560000*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*sqr 
t(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 37/384000*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))) + 57/8000*sqrt(5) 
*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785* 
sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 351/2000*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))) + 27/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3) 
) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))
 
3.23.89.9 Mupad [F(-1)]

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

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