\(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^5} \, dx\) [1216]

Optimal result

Integrand size = 26, antiderivative size = 202 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {139745 \sqrt {3+5 x}}{1613472 \sqrt {1-2 x}}+\frac {43 \sqrt {3+5 x}}{588 \sqrt {1-2 x} (2+3 x)^4}-\frac {2717 \sqrt {3+5 x}}{8232 \sqrt {1-2 x} (2+3 x)^3}-\frac {2013 \sqrt {3+5 x}}{10976 \sqrt {1-2 x} (2+3 x)^2}-\frac {14135 \sqrt {3+5 x}}{153664 \sqrt {1-2 x} (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^4}-\frac {547745 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1075648 \sqrt {7}} \] Output:

139745/1613472*(3+5*x)^(1/2)/(1-2*x)^(1/2)+43/588*(3+5*x)^(1/2)/(1-2*x)^(1 
/2)/(2+3*x)^4-2717/8232*(3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^3-2013/10976*( 
3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^2-14135/153664*(3+5*x)^(1/2)/(1-2*x)^(1 
/2)/(2+3*x)+11/21*(3+5*x)^(3/2)/(1-2*x)^(3/2)/(2+3*x)^4-547745/7529536*7^( 
1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 3.47 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.78 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {5 \left (-\frac {7 \sqrt {3+5 x} \left (-2906640-18627988 x-27318504 x^2+25673409 x^3+82071900 x^4+45277380 x^5\right )}{5 (1-2 x)^{3/2} (2+3 x)^4}+328647 \sqrt {7} \arctan \left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+328647 \sqrt {7} \arctan \left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )\right )}{22588608} \] Input:

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

Output:

(5*((-7*Sqrt[3 + 5*x]*(-2906640 - 18627988*x - 27318504*x^2 + 25673409*x^3 
 + 82071900*x^4 + 45277380*x^5))/(5*(1 - 2*x)^(3/2)*(2 + 3*x)^4) + 328647* 
Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(-Sqrt[11] + Sqrt 
[5 - 10*x])] + 328647*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]] 
*(-Sqrt[11] + Sqrt[5 - 10*x]))]))/22588608
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.12, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {109, 27, 166, 27, 168, 27, 168, 27, 168, 27, 169, 27, 104, 217}

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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}-\frac {1}{21} \int -\frac {3 \sqrt {5 x+3} (265 x+148)}{2 (1-2 x)^{3/2} (3 x+2)^5}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \int \frac {\sqrt {5 x+3} (265 x+148)}{(1-2 x)^{3/2} (3 x+2)^5}dx+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{14} \left (\frac {1}{84} \int \frac {11 (3060 x+1793)}{(1-2 x)^{3/2} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {11}{84} \int \frac {3060 x+1793}{(1-2 x)^{3/2} (3 x+2)^4 \sqrt {5 x+3}}dx+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{14} \left (\frac {11}{84} \left (\frac {1}{21} \int \frac {3 (14820 x+8599)}{2 (1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {247 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {11}{84} \left (\frac {1}{14} \int \frac {14820 x+8599}{(1-2 x)^{3/2} (3 x+2)^3 \sqrt {5 x+3}}dx-\frac {247 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{14} \left (\frac {11}{84} \left (\frac {1}{14} \left (\frac {1}{14} \int \frac {35 (4392 x+2671)}{2 (1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {549 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {247 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {11}{84} \left (\frac {1}{14} \left (\frac {5}{4} \int \frac {4392 x+2671}{(1-2 x)^{3/2} (3 x+2)^2 \sqrt {5 x+3}}dx-\frac {549 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {247 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{14} \left (\frac {11}{84} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{7} \int \frac {15420 x+20239}{2 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx-\frac {771 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {549 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {247 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {11}{84} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \int \frac {15420 x+20239}{(1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx-\frac {771 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {549 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {247 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{14} \left (\frac {11}{84} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {111796 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {2}{77} \int -\frac {328647}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {771 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {549 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {247 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{14} \left (\frac {11}{84} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {29877}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {111796 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )-\frac {771 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {549 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {247 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{14} \left (\frac {11}{84} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {59754}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}+\frac {111796 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )-\frac {771 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {549 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {247 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{14} \left (\frac {11}{84} \left (\frac {1}{14} \left (\frac {5}{4} \left (\frac {1}{14} \left (\frac {111796 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {59754 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}\right )-\frac {771 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)}\right )-\frac {549 \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^2}\right )-\frac {247 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^3}\right )+\frac {43 \sqrt {5 x+3}}{42 \sqrt {1-2 x} (3 x+2)^4}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^4}\)

Input:

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

Output:

(11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^4) + ((43*Sqrt[3 + 5*x] 
)/(42*Sqrt[1 - 2*x]*(2 + 3*x)^4) + (11*((-247*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2 
*x]*(2 + 3*x)^3) + ((-549*Sqrt[3 + 5*x])/(2*Sqrt[1 - 2*x]*(2 + 3*x)^2) + ( 
5*((-771*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + ((111796*Sqrt[3 + 5* 
x])/(77*Sqrt[1 - 2*x]) - (59754*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x 
])])/(7*Sqrt[7]))/14))/4)/14))/84)/14
 

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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

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 166

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] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 168

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 169

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(345\) vs. \(2(157)=314\).

Time = 0.28 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.71

method	result	size
default	\(\frac {\left (532408140 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}-633883320 x^{5} \sqrt {-10 x^{2}-x +3}-433814040 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-1149006600 x^{4} \sqrt {-10 x^{2}-x +3}-170896440 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-359427726 x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}}{45177216 \left (2+3 x \right )^{4} \left (1-2 x \right )^{\frac {3}{2}} \sqrt {-10 x^{2}-x +3}}\)	\(346\)

Input:

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

Output:

1/45177216*(532408140*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^ 
(1/2))*x^6+887346900*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^( 
1/2))*x^5+133102035*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1 
/2))*x^4-633883320*x^5*(-10*x^2-x+3)^(1/2)-433814040*7^(1/2)*arctan(1/14*( 
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-1149006600*x^4*(-10*x^2-x+3)^(1/ 
2)-170896440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^ 
2-359427726*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20) 
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone] 
*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(- 
10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)/(2+3*x)^4/(1 
-2*x)^(3/2)/(-10*x^2-x+3)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.72 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=-\frac {1643235 \, \sqrt {7} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (45277380 \, x^{5} + 82071900 \, x^{4} + 25673409 \, x^{3} - 27318504 \, x^{2} - 18627988 \, x - 2906640\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{45177216 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \] Input:

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

Output:

-1/45177216*(1643235*sqrt(7)*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x 
^2 + 32*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 
1)/(10*x^2 + x - 3)) + 14*(45277380*x^5 + 82071900*x^4 + 25673409*x^3 - 27 
318504*x^2 - 18627988*x - 2906640)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(324*x^6 
+ 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (157) = 314\).

Time = 0.12 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.61 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {547745}{15059072} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {698725 \, x}{1613472 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343745}{3226944 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {633875 \, x}{691488 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1}{2268 \, {\left (81 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 96 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 16 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {331}{31752 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {9313}{98784 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {659891}{1778112 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {296615}{12446784 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \] Input:

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

Output:

547745/15059072*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) 
+ 698725/1613472*x/sqrt(-10*x^2 - x + 3) + 343745/3226944/sqrt(-10*x^2 - x 
 + 3) + 633875/691488*x/(-10*x^2 - x + 3)^(3/2) - 1/2268/(81*(-10*x^2 - x 
+ 3)^(3/2)*x^4 + 216*(-10*x^2 - x + 3)^(3/2)*x^3 + 216*(-10*x^2 - x + 3)^( 
3/2)*x^2 + 96*(-10*x^2 - x + 3)^(3/2)*x + 16*(-10*x^2 - x + 3)^(3/2)) + 33 
1/31752/(27*(-10*x^2 - x + 3)^(3/2)*x^3 + 54*(-10*x^2 - x + 3)^(3/2)*x^2 + 
 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) - 9313/98784/(9 
*(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - 
 x + 3)^(3/2)) + 659891/1778112/(3*(-10*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 
- x + 3)^(3/2)) + 296615/12446784/(-10*x^2 - x + 3)^(3/2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (157) = 314\).

Time = 0.63 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.01 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^5} \, dx=\frac {109549}{30118144} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {88 \, {\left (100 \, \sqrt {5} {\left (5 \, x + 3\right )} - 627 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1764735 \, {\left (2 \, x - 1\right )}^{2}} - \frac {55 \, \sqrt {10} {\left (79441 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 82486488 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 31196222400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {1487445568000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {5949782272000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{3764768 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \] Input:

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

Output:

109549/30118144*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x 
+ 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt( 
-10*x + 5) - sqrt(22)))) - 88/1764735*(100*sqrt(5)*(5*x + 3) - 627*sqrt(5) 
)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 55/3764768*sqrt(10)*(79441*( 
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt 
(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 82486488*((sqrt(2)*sqrt(-10*x + 5) - 
sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt( 
22)))^5 + 31196222400*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) 
- 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 1487445568000* 
(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 5949782272000*sqrt(5* 
x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x + 5) - 
sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt( 
22)))^2 + 280)^4
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 599, normalized size of antiderivative = 2.97 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(266204070*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sq 
rt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**5 + 576775485*sqrt( - 2* 
x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5 
))/sqrt(11))/2))/sqrt(2))*x**4 + 354938760*sqrt( - 2*x + 1)*sqrt(7)*atan(( 
sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt 
(2))*x**3 - 39437640*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*ta 
n(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 105167040* 
sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 
 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 26291760*sqrt( - 2*x + 1)*sqrt(7)* 
atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2) 
)/sqrt(2)) - 266204070*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)* 
tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**5 - 57677548 
5*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x 
 + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**4 - 354938760*sqrt( - 2*x + 1)*sq 
rt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(1 
1))/2))/sqrt(2))*x**3 + 39437640*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + 
 sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 
+ 105167040*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((s 
qrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 26291760*sqrt( - 2*x + 
 1)*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5...