\(\int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx\) [1224]

Optimal result

Integrand size = 26, antiderivative size = 108 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {190 \sqrt {3+5 x}}{1617 (1-2 x)^{3/2}}-\frac {4390 \sqrt {3+5 x}}{124509 \sqrt {1-2 x}}+\frac {3 \sqrt {3+5 x}}{7 (1-2 x)^{3/2} (2+3 x)}-\frac {405 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{343 \sqrt {7}} \] Output:

-190/1617*(3+5*x)^(1/2)/(1-2*x)^(3/2)-4390/124509*(3+5*x)^(1/2)/(1-2*x)^(1 
/2)+3/7*(3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)-405/2401*7^(1/2)*arctan(1/7*(1 
-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {-7 \sqrt {3+5 x} \left (15321-39500 x+26340 x^2\right )-147015 \sqrt {7-14 x} \left (-2+x+6 x^2\right ) \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{871563 (1-2 x)^{3/2} (2+3 x)} \] Input:

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

Output:

-1/871563*(-7*Sqrt[3 + 5*x]*(15321 - 39500*x + 26340*x^2) - 147015*Sqrt[7 
- 14*x]*(-2 + x + 6*x^2)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(( 
1 - 2*x)^(3/2)*(2 + 3*x))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {114, 27, 169, 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.

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

\(\Big \downarrow \) 114

\(\displaystyle \frac {1}{7} \int -\frac {5 (24 x+7)}{2 (1-2 x)^{5/2} (3 x+2) \sqrt {5 x+3}}dx+\frac {3 \sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{14} \int \frac {24 x+7}{(1-2 x)^{5/2} (3 x+2) \sqrt {5 x+3}}dx\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {3 \sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{14} \left (\frac {76 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}-\frac {2}{231} \int \frac {131-1140 x}{2 (1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{14} \left (\frac {76 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}-\frac {1}{231} \int \frac {131-1140 x}{(1-2 x)^{3/2} (3 x+2) \sqrt {5 x+3}}dx\right )\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {3 \sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{14} \left (\frac {1}{231} \left (\frac {2}{77} \int -\frac {29403}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {1756 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {76 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{14} \left (\frac {1}{231} \left (\frac {1756 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {2673}{7} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {76 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}\right )\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {3 \sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{14} \left (\frac {1}{231} \left (\frac {1756 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}-\frac {5346}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {76 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 \sqrt {5 x+3}}{7 (1-2 x)^{3/2} (3 x+2)}-\frac {5}{14} \left (\frac {1}{231} \left (\frac {5346 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}+\frac {1756 \sqrt {5 x+3}}{77 \sqrt {1-2 x}}\right )+\frac {76 \sqrt {5 x+3}}{231 (1-2 x)^{3/2}}\right )\)

Input:

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

Output:

(3*Sqrt[3 + 5*x])/(7*(1 - 2*x)^(3/2)*(2 + 3*x)) - (5*((76*Sqrt[3 + 5*x])/( 
231*(1 - 2*x)^(3/2)) + ((1756*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) + (5346*Ar 
cTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(7*Sqrt[7]))/231))/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 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(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] + Simp[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*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

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. \(201\) vs. \(2(81)=162\).

Time = 0.26 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.87

method	result	size
default	\(\frac {\left (1764180 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}-588060 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}-735075 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +368760 x^{2} \sqrt {-10 x^{2}-x +3}+294030 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-553000 x \sqrt {-10 x^{2}-x +3}+214494 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}}{1743126 \left (2+3 x \right ) \left (1-2 x \right )^{\frac {3}{2}} \sqrt {-10 x^{2}-x +3}}\)	\(202\)

Input:

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

Output:

1/1743126*(1764180*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/ 
2))*x^3-588060*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))* 
x^2-735075*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+36 
8760*x^2*(-10*x^2-x+3)^(1/2)+294030*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/ 
(-10*x^2-x+3)^(1/2))-553000*x*(-10*x^2-x+3)^(1/2)+214494*(-10*x^2-x+3)^(1/ 
2))*(3+5*x)^(1/2)/(2+3*x)/(1-2*x)^(3/2)/(-10*x^2-x+3)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=-\frac {147015 \, \sqrt {7} {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (26340 \, x^{2} - 39500 \, x + 15321\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1743126 \, {\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )}} \] Input:

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

Output:

-1/1743126*(147015*sqrt(7)*(12*x^3 - 4*x^2 - 5*x + 2)*arctan(1/14*sqrt(7)* 
(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(26340*x^2 
 - 39500*x + 15321)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(12*x^3 - 4*x^2 - 5*x + 
2)
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (81) = 162\).

Time = 0.21 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.15 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {81}{9604} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {594 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{343 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} - \frac {8 \, {\left (536 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3333 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{3112725 \, {\left (2 \, x - 1\right )}^{2}} \] Input:

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

Output:

81/9604*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((s 
qrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 
5) - sqrt(22)))) + 594/343*sqrt(10)*((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)))/(((s 
qrt(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) - 8/3112725*(536*sqrt(5)*(5*x + 3) 
 - 3333*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.72 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}} \, dx=\frac {882090 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}+147015 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x -294030 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-882090 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}-147015 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x +294030 \sqrt {-2 x +1}\, \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-184380 \sqrt {5 x +3}\, x^{2}+276500 \sqrt {5 x +3}\, x -107247 \sqrt {5 x +3}}{871563 \sqrt {-2 x +1}\, \left (6 x^{2}+x -2\right )} \] Input:

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

Output:

(882090*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 + 147015*sqrt( - 2*x + 1) 
*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqr 
t(11))/2))/sqrt(2))*x - 294030*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) - s 
qrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 88209 
0*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 - 147015*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 + 294030*sqrt( - 2*x + 1)*sqrt(7)*atan((sqrt(33) + sqrt(35 
)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 184380*sqrt 
(5*x + 3)*x**2 + 276500*sqrt(5*x + 3)*x - 107247*sqrt(5*x + 3))/(871563*sq 
rt( - 2*x + 1)*(6*x**2 + x - 2))