\(\int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\) [872]

Optimal result

Integrand size = 26, antiderivative size = 93 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx=-\frac {11 \sqrt {1-2 x} \sqrt {3+5 x}}{28 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{2 (2+3 x)^2}-\frac {121 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{28 \sqrt {7}} \] Output:

-11*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(56+84*x)+1/2*(1-2*x)^(1/2)*(3+5*x)^(3/2)/ 
(2+3*x)^2-121/196*7^(1/2)*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx=\frac {1}{196} \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (20+37 x)}{(2+3 x)^2}-121 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right ) \] Input:

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

Output:

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(20 + 37*x))/(2 + 3*x)^2 - 121*Sqrt[7]*Arc 
Tan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/196
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {105, 105, 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.

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

\(\displaystyle \frac {11}{4} \left (\frac {11}{14} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {11}{4} \left (\frac {11}{7} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {11}{4} \left (-\frac {11 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{7 \sqrt {7}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)}\right )+\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^2}\)

Input:

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

Output:

(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2*(2 + 3*x)^2) + (11*(-1/7*(Sqrt[1 - 2*x] 
*Sqrt[3 + 5*x])/(2 + 3*x) - (11*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x 
])])/(7*Sqrt[7])))/4
 

Defintions of rubi rules used

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 105

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

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 [A] (verified)

Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.28

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (37 x +20\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{28 \left (2+3 x \right )^{2} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {121 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{392 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(119\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (1089 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+1452 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +484 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+518 x \sqrt {-10 x^{2}-x +3}+280 \sqrt {-10 x^{2}-x +3}\right )}{392 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{2}}\)	\(154\)

Input:

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

Output:

-1/28*(-1+2*x)*(3+5*x)^(1/2)*(37*x+20)/(2+3*x)^2/(-(-1+2*x)*(3+5*x))^(1/2) 
*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)+121/392*7^(1/2)*arctan(9/14*(20/3+3 
7/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2))*((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 = 86, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx=-\frac {121 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{392 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \] Input:

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

Output:

-1/392*(121*sqrt(7)*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqr 
t(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(37*x + 20)*sqrt(5*x + 3) 
*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx=\frac {121}{392} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {5}{21} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{14 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {37 \, \sqrt {-10 \, x^{2} - x + 3}}{84 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

121/392*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 5/21*s 
qrt(-10*x^2 - x + 3) + 3/14*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 3 
7/84*sqrt(-10*x^2 - x + 3)/(3*x + 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (72) = 144\).

Time = 0.20 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.69 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx=\frac {121}{3920} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121 \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {280 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1120 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{14 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \] Input:

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

Output:

121/3920*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)))) - 121/14*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)))^3 - 
 280*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1120*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)^2
 

Mupad [B] (verification not implemented)

Time = 8.11 (sec) , antiderivative size = 1037, normalized size of antiderivative = 11.15 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((2129*((1 - 2*x)^(1/2) - 1)^5)/(875*(3^(1/2) - (5*x + 3)^(1/2))^5) - (425 
8*((1 - 2*x)^(1/2) - 1)^3)/(4375*(3^(1/2) - (5*x + 3)^(1/2))^3) - (158*((1 
 - 2*x)^(1/2) - 1))/(4375*(3^(1/2) - (5*x + 3)^(1/2))) + (79*((1 - 2*x)^(1 
/2) - 1)^7)/(140*(3^(1/2) - (5*x + 3)^(1/2))^7) + (991*3^(1/2)*((1 - 2*x)^ 
(1/2) - 1)^2)/(4375*(3^(1/2) - (5*x + 3)^(1/2))^2) - (376*3^(1/2)*((1 - 2* 
x)^(1/2) - 1)^4)/(875*(3^(1/2) - (5*x + 3)^(1/2))^4) + (991*3^(1/2)*((1 - 
2*x)^(1/2) - 1)^6)/(700*(3^(1/2) - (5*x + 3)^(1/2))^6))/((544*((1 - 2*x)^( 
1/2) - 1)^2)/(625*(3^(1/2) - (5*x + 3)^(1/2))^2) - (1764*((1 - 2*x)^(1/2) 
- 1)^4)/(625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (136*((1 - 2*x)^(1/2) - 1)^6 
)/(25*(3^(1/2) - (5*x + 3)^(1/2))^6) + ((1 - 2*x)^(1/2) - 1)^8/(3^(1/2) - 
(5*x + 3)^(1/2))^8 - (96*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(625*(3^(1/2) - 
(5*x + 3)^(1/2))^3) + (48*3^(1/2)*((1 - 2*x)^(1/2) - 1)^5)/(125*(3^(1/2) - 
 (5*x + 3)^(1/2))^5) + (12*3^(1/2)*((1 - 2*x)^(1/2) - 1)^7)/(5*(3^(1/2) - 
(5*x + 3)^(1/2))^7) - (96*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(625*(3^(1/2) - ( 
5*x + 3)^(1/2))) + 16/625) - (121*7^(1/2)*atan(((121*7^(1/2)*((726*3^(1/2) 
)/875 + (363*((1 - 2*x)^(1/2) - 1))/(875*(3^(1/2) - (5*x + 3)^(1/2))) - (7 
^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + 
 (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 5 
36/125)*121i)/392 - (363*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(175*(3^(1/2) - 
(5*x + 3)^(1/2))^2)))/392 + (121*7^(1/2)*((726*3^(1/2))/875 + (363*((1 ...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.71 \[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx=\frac {1089 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}+1452 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x +484 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )-1089 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x^{2}-1452 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right ) x -484 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )+259 \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +140 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{1764 x^{2}+2352 x +784} \] Input:

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

Output:

(1089*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5) 
)/sqrt(11))/2))/sqrt(2))*x**2 + 1452*sqrt(7)*atan((sqrt(33) - sqrt(35)*tan 
(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x + 484*sqrt(7)*at 
an((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/ 
sqrt(2)) - 1089*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 
1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x**2 - 1452*sqrt(7)*atan((sqrt(33) + sq 
rt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2))*x - 484* 
sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt 
(11))/2))/sqrt(2)) + 259*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 140*sqrt(5*x + 
 3)*sqrt( - 2*x + 1))/(196*(9*x**2 + 12*x + 4))