\(\int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\) [1228]

Optimal result

Integrand size = 26, antiderivative size = 116 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {2401}{264 (1-2 x)^{3/2} \sqrt {3+5 x}}-\frac {1500673 \sqrt {1-2 x}}{798600 \sqrt {3+5 x}}-\frac {21952 \sqrt {3+5 x}}{3993 \sqrt {1-2 x}}-\frac {81}{200} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {4887 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}} \] Output:

2401/264/(1-2*x)^(3/2)/(3+5*x)^(1/2)-1500673/798600*(1-2*x)^(1/2)/(3+5*x)^ 
(1/2)-21952/3993*(3+5*x)^(1/2)/(1-2*x)^(1/2)-81/200*(1-2*x)^(1/2)*(3+5*x)^ 
(1/2)+4887/2000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.76 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=-\frac {10 \sqrt {3+5 x} \left (8379147-12657123 x-40488772 x^2+6468660 x^3\right )-19513791 \sqrt {10-20 x} \left (-3+x+10 x^2\right ) \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{7986000 (1-2 x)^{3/2} (3+5 x)} \] Input:

Integrate[(2 + 3*x)^4/((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)),x]
 

Output:

-1/7986000*(10*Sqrt[3 + 5*x]*(8379147 - 12657123*x - 40488772*x^2 + 646866 
0*x^3) - 19513791*Sqrt[10 - 20*x]*(-3 + x + 10*x^2)*ArcTan[Sqrt[5/2 - 5*x] 
/Sqrt[3 + 5*x]])/((1 - 2*x)^(3/2)*(3 + 5*x))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {109, 27, 167, 27, 160, 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.

\(\displaystyle \int \frac {(3 x+2)^4}{(1-2 x)^{5/2} (5 x+3)^{3/2}} \, dx\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {1}{33} \int \frac {(3 x+2)^2 (507 x+296)}{2 (1-2 x)^{3/2} (5 x+3)^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {1}{66} \int \frac {(3 x+2)^2 (507 x+296)}{(1-2 x)^{3/2} (5 x+3)^{3/2}}dx\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{66} \left (-\frac {1}{11} \int -\frac {(3 x+2) (49701 x+28738)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx-\frac {1099 (3 x+2)^2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{66} \left (\frac {1}{22} \int \frac {(3 x+2) (49701 x+28738)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx-\frac {1099 (3 x+2)^2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 160

\(\displaystyle \frac {1}{66} \left (\frac {1}{22} \left (\frac {1773981}{100} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} (8200665 x+4898747)}{550 \sqrt {5 x+3}}\right )-\frac {1099 (3 x+2)^2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{66} \left (\frac {1}{22} \left (\frac {1773981}{250} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {\sqrt {1-2 x} (8200665 x+4898747)}{550 \sqrt {5 x+3}}\right )-\frac {1099 (3 x+2)^2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{66} \left (\frac {1}{22} \left (\frac {1773981 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{50 \sqrt {10}}-\frac {\sqrt {1-2 x} (8200665 x+4898747)}{550 \sqrt {5 x+3}}\right )-\frac {1099 (3 x+2)^2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^3}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

Input:

Int[(2 + 3*x)^4/((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2)),x]
 

Output:

(7*(2 + 3*x)^3)/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + ((-1099*(2 + 3*x)^2)/ 
(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (-1/550*(Sqrt[1 - 2*x]*(4898747 + 82006 
65*x))/Sqrt[3 + 5*x] + (1773981*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(50*Sqrt 
[10]))/22)/66
 

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 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 109

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

rule 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g 
+ e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Simp[(a*d*f*h*m + b*(d* 
(f*g + e*h) - c*f*h*(m + 2)))/(b^2*d)   Int[(a + b*x)^(m + 1)*(c + d*x)^n, 
x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && 
NeQ[m, -1] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])
 

rule 167

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
 

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 = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.24

method	result	size
default	\(\frac {390275820 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{3}-156110328 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-129373200 x^{3} \sqrt {-10 x^{2}-x +3}-136596537 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-167582940 \sqrt {-10 x^{2}-x +3}}{15972000 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\)	\(144\)

Input:

int((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/15972000*(390275820*10^(1/2)*arcsin(20/11*x+1/11)*x^3-156110328*10^(1/2) 
*arcsin(20/11*x+1/11)*x^2-129373200*x^3*(-10*x^2-x+3)^(1/2)-136596537*10^( 
1/2)*arcsin(20/11*x+1/11)*x[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^ 
(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-167582940*(-10* 
x^2-x+3)^(1/2))/(1-2*x)^(3/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=-\frac {19513791 \, \sqrt {10} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (6468660 \, x^{3} - 40488772 \, x^{2} - 12657123 \, x + 8379147\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{15972000 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \] Input:

integrate((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

-1/15972000*(19513791*sqrt(10)*(20*x^3 - 8*x^2 - 7*x + 3)*arctan(1/20*sqrt 
(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(64686 
60*x^3 - 40488772*x^2 - 12657123*x + 8379147)*sqrt(5*x + 3)*sqrt(-2*x + 1) 
)/(20*x^3 - 8*x^2 - 7*x + 3)
 

Sympy [F]

\[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int \frac {\left (3 x + 2\right )^{4}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((2+3*x)**4/(1-2*x)**(5/2)/(3+5*x)**(3/2),x)
 

Output:

Integral((3*x + 2)**4/((1 - 2*x)**(5/2)*(5*x + 3)**(3/2)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.82 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {4887}{4000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {81 \, x^{2}}{20 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {18627221 \, x}{798600 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {3910543}{199650 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2401}{264 \, {\left (2 \, \sqrt {-10 \, x^{2} - x + 3} x - \sqrt {-10 \, x^{2} - x + 3}\right )}} \] Input:

integrate((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="maxima")
 

Output:

4887/4000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 81/20*x^2/sqrt(-10*x^2 
- x + 3) - 18627221/798600*x/sqrt(-10*x^2 - x + 3) - 3910543/199650/sqrt(- 
10*x^2 - x + 3) - 2401/264/(2*sqrt(-10*x^2 - x + 3)*x - sqrt(-10*x^2 - x + 
 3))
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.13 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {4887}{2000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{332750 \, \sqrt {5 \, x + 3}} - \frac {{\left (4 \, {\left (323433 \, \sqrt {5} {\left (5 \, x + 3\right )} - 13033138 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 214579893 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{99825000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{166375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \] Input:

integrate((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="giac")
 

Output:

4887/2000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/332750*sqrt(10) 
*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 1/99825000*(4*(32343 
3*sqrt(5)*(5*x + 3) - 13033138*sqrt(5))*(5*x + 3) + 214579893*sqrt(5))*sqr 
t(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 2/166375*sqrt(10)*sqrt(5*x + 3)/( 
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^4}{{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:

int((3*x + 2)^4/((1 - 2*x)^(5/2)*(5*x + 3)^(3/2)),x)
 

Output:

int((3*x + 2)^4/((1 - 2*x)^(5/2)*(5*x + 3)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^4}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {-39027582 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right ) x [phone] \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )[phone] x^{3}-404887720 x^{2}-126571230 x [phone]}{7986000 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (2 x -1\right )} \] Input:

int((2+3*x)^4/(1-2*x)^(5/2)/(3+5*x)^(3/2),x)
 

Output:

( - 39027582*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(10)*asin((sqrt( - 2*x + 1 
)*sqrt(5))/sqrt(11))*x + 19513791*sqrt(5*x + 3)*sqrt( - 2*x + 1)*sqrt(10)* 
asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) + 64686600*x**3 - 404887720*x**2 
 - 126571230*x + 83791470)/(7986000*sqrt(5*x + 3)*sqrt( - 2*x + 1)*(2*x - 
1))