\(\int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\) [1083]

Optimal result

Integrand size = 26, antiderivative size = 138 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {9458207 \sqrt {1-2 x} \sqrt {3+5 x}}{102400}-\frac {859837 \sqrt {1-2 x} (3+5 x)^{3/2}}{76800}-\frac {78167 \sqrt {1-2 x} (3+5 x)^{5/2}}{48000}-\frac {1803 \sqrt {1-2 x} (3+5 x)^{7/2}}{4000}+\frac {9}{100} (1-2 x)^{3/2} (3+5 x)^{7/2}+\frac {104040277 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{102400 \sqrt {10}} \] Output:

-9458207/102400*(1-2*x)^(1/2)*(3+5*x)^(1/2)-859837/76800*(1-2*x)^(1/2)*(3+ 
5*x)^(3/2)-78167/48000*(1-2*x)^(1/2)*(3+5*x)^(5/2)-1803/4000*(1-2*x)^(1/2) 
*(3+5*x)^(7/2)+9/100*(1-2*x)^(3/2)*(3+5*x)^(7/2)[phone]/1024000*arcsin( 
1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.60 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=\frac {-10 \sqrt {1-2 x} \left (138561867+376912905 x+378014260 x^2+303416800 x^3+152208000 x^4+34560000 x^5\right )-312120831 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{3072000 \sqrt {3+5 x}} \] Input:

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

Output:

(-10*Sqrt[1 - 2*x]*(138561867 + 376912905*x + 378014260*x^2 + 303416800*x^ 
3 + 152208000*x^4 + 34560000*x^5) - 312120831*Sqrt[30 + 50*x]*ArcTan[Sqrt[ 
5/2 - 5*x]/Sqrt[3 + 5*x]])/(3072000*Sqrt[3 + 5*x])
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.18, 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.

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

\(\Big \downarrow \) 101

\(\displaystyle -\frac {1}{50} \int -\frac {(5 x+3)^{5/2} (963 x+628)}{2 \sqrt {1-2 x}}dx-\frac {3}{50} \sqrt {1-2 x} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{100} \int \frac {(5 x+3)^{5/2} (963 x+628)}{\sqrt {1-2 x}}dx-\frac {3}{50} \sqrt {1-2 x} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {1}{100} \left (\frac {78167}{80} \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x}}dx-\frac {963}{40} \sqrt {1-2 x} (5 x+3)^{7/2}\right )-\frac {3}{50} \sqrt {1-2 x} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{100} \left (\frac {78167}{80} \left (\frac {55}{12} \int \frac {(5 x+3)^{3/2}}{\sqrt {1-2 x}}dx-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\right )-\frac {963}{40} \sqrt {1-2 x} (5 x+3)^{7/2}\right )-\frac {3}{50} \sqrt {1-2 x} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{100} \left (\frac {78167}{80} \left (\frac {55}{12} \left (\frac {33}{8} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}dx-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\right )-\frac {963}{40} \sqrt {1-2 x} (5 x+3)^{7/2}\right )-\frac {3}{50} \sqrt {1-2 x} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{100} \left (\frac {78167}{80} \left (\frac {55}{12} \left (\frac {33}{8} \left (\frac {11}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\right )-\frac {963}{40} \sqrt {1-2 x} (5 x+3)^{7/2}\right )-\frac {3}{50} \sqrt {1-2 x} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{100} \left (\frac {78167}{80} \left (\frac {55}{12} \left (\frac {33}{8} \left (\frac {11}{10} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\right )-\frac {963}{40} \sqrt {1-2 x} (5 x+3)^{7/2}\right )-\frac {3}{50} \sqrt {1-2 x} (3 x+2) (5 x+3)^{7/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{100} \left (\frac {78167}{80} \left (\frac {55}{12} \left (\frac {33}{8} \left (\frac {11 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2 \sqrt {10}}-\frac {1}{2} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )-\frac {1}{6} \sqrt {1-2 x} (5 x+3)^{5/2}\right )-\frac {963}{40} \sqrt {1-2 x} (5 x+3)^{7/2}\right )-\frac {3}{50} \sqrt {1-2 x} (3 x+2) (5 x+3)^{7/2}\)

Input:

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

Output:

(-3*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^(7/2))/50 + ((-963*Sqrt[1 - 2*x]*(3 
+ 5*x)^(7/2))/40 + (78167*(-1/6*(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)) + (55*(-1/ 
4*(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) + (33*(-1/2*(Sqrt[1 - 2*x]*Sqrt[3 + 5*x] 
) + (11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2*Sqrt[10])))/8))/12))/80)/100
 

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]
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {\left (6912000 x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{307200 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {104040277 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2048000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(108\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-138240000 x^{4} \sqrt {-10 x^{2}-x +3}-525888000 x^{3} \sqrt {-10 x^{2}-x +3}-898134400 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-973176400 x \sqrt {-10 x^{2}-x +3}-923745780 \sqrt {-10 x^{2}-x +3}\right )}{6144000 \sqrt {-10 x^{2}-x +3}}\)	\(121\)

Input:

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

Output:

1/307200*(6912000*x^4+26294400*x^3+44906720*x^2+48658820*x[phone])*(-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)[phone]/2048000*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)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.56 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {1}{307200} \, {\left (6912000 \, x^{4} + 26294400 \, x^{3} + 44906720 \, x^{2} + 48658820 \, x + 46187289\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {104040277}{2048000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \] Input:

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

Output:

-1/307200*(6912000*x^4 + 26294400*x^3 + 44906720*x^2 + 48658820*x + 461872 
89)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 104040277/2048000*sqrt(10)*arctan(1/20* 
sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.67 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {45}{2} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {2739}{32} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {280667}{1920} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {2432941}{15360} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {104040277}{2048000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {15395763}{102400} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:

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

Output:

-45/2*sqrt(-10*x^2 - x + 3)*x^4 - 2739/32*sqrt(-10*x^2 - x + 3)*x^3 - 2806 
67/1920*sqrt(-10*x^2 - x + 3)*x^2 - 2432941/15360*sqrt(-10*x^2 - x + 3)*x 
- 104040277/2048000*sqrt(10)*arcsin(-20/11*x - 1/11) - 15395763/102400*sqr 
t(-10*x^2 - x + 3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.52 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {1}{15360000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (36 \, {\left (240 \, x + 481\right )} {\left (5 \, x + 3\right )} + 78167\right )} {\left (5 \, x + 3\right )} + 4299185\right )} {\left (5 \, x + 3\right )} + 141873105\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 1560604155 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \] Input:

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

Output:

-1/15360000*sqrt(5)*(2*(4*(8*(36*(240*x + 481)*(5*x + 3) + 78167)*(5*x + 3 
) + 4299185)*(5*x + 3) + 141873105)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 156060 
4155*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.72 \[ \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx=-\frac {104040277 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{1024000}-\frac {45 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{2}-\frac {2739 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{32}-\frac {280667 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{1920}-\frac {2432941 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{15360}-\frac {15395763 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{102400} \] Input:

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

Output:

( - 312120831*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) - 6912000 
0*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 262944000*sqrt(5*x + 3)*sqrt( - 2* 
x + 1)*x**3 - 449067200*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 - 486588200*sq 
rt(5*x + 3)*sqrt( - 2*x + 1)*x - 461872890*sqrt(5*x + 3)*sqrt( - 2*x + 1)) 
/3072000