3.23.91 \(\int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{2+3 x} \, dx\) [2291]

3.23.91.1 Optimal result

Integrand size = 26, antiderivative size = 130 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{2+3 x} \, dx=-\frac {925}{864} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {5}{24} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {1}{9} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {6553 \sqrt {\frac {5}{2}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2592}+\frac {2}{81} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]

output

6553/5184*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+2/81*arctan(1/7*(1- 
2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-5/24*(3+5*x)^(3/2)*(1-2*x)^(1/2) 
+1/9*(3+5*x)^(5/2)*(1-2*x)^(1/2)-925/864*(1-2*x)^(1/2)*(3+5*x)^(1/2)
 

3.23.91.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{2+3 x} \, dx=\frac {6 \sqrt {1-2 x} \left (-1803+2935 x+17100 x^2+12000 x^3\right )-6553 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+128 \sqrt {7} \sqrt {3+5 x} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5184 \sqrt {3+5 x}} \]

input

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

output

(6*Sqrt[1 - 2*x]*(-1803 + 2935*x + 17100*x^2 + 12000*x^3) - 6553*Sqrt[30 + 
 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]] + 128*Sqrt[7]*Sqrt[3 + 5*x]*A 
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(5184*Sqrt[3 + 5*x])
 

3.23.91.3 Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {112, 27, 171, 27, 171, 27, 175, 64, 104, 217, 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.

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

\(\Big \downarrow \) 112

\(\displaystyle \frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {1}{9} \int -\frac {(5 x+3)^{3/2} (45 x+16)}{2 \sqrt {1-2 x} (3 x+2)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{18} \int \frac {(5 x+3)^{3/2} (45 x+16)}{\sqrt {1-2 x} (3 x+2)}dx+\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{18} \left (-\frac {1}{12} \int -\frac {3 \sqrt {5 x+3} (925 x+654)}{2 \sqrt {1-2 x} (3 x+2)}dx-\frac {15}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{18} \left (\frac {1}{8} \int \frac {\sqrt {5 x+3} (925 x+654)}{\sqrt {1-2 x} (3 x+2)}dx-\frac {15}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{18} \left (\frac {1}{8} \left (-\frac {1}{6} \int -\frac {32765 x+21694}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {925}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {15}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \int \frac {32765 x+21694}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx-\frac {925}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {15}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {32765}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {448}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {925}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {15}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {13106}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {448}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )-\frac {925}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {15}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {13106}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {896}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {925}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {15}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {13106}{3} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {128}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {925}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {15}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{18} \left (\frac {1}{8} \left (\frac {1}{12} \left (\frac {6553}{3} \sqrt {10} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {128}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )-\frac {925}{6} \sqrt {1-2 x} \sqrt {5 x+3}\right )-\frac {15}{4} \sqrt {1-2 x} (5 x+3)^{3/2}\right )+\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}\)

input

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

output

(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/9 + ((-15*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/4 
 + ((-925*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6 + ((6553*Sqrt[10]*ArcSin[Sqrt[2/1 
1]*Sqrt[3 + 5*x]])/3 + (128*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 
 5*x])])/3)/12)/8)/18
 

3.23.91.3.1 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 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 112

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

rule 171

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

rule 175

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

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])
 

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]
 

3.23.91.4 Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (28800 x^{2} \sqrt {-10 x^{2}-x +3}+6553 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-128 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+23760 x \sqrt {-10 x^{2}-x +3}-7212 \sqrt {-10 x^{2}-x +3}\right )}{10368 \sqrt {-10 x^{2}-x +3}}\)	\(115\)
risch	\(-\frac {\left (2400 x^{2}+1980 x -601\right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{864 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {6553 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{10368}+\frac {\sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{81}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(131\)

input

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

output

1/10368*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(28800*x^2*(-10*x^2-x+3)^(1/2)+6553*10 
^(1/2)*arcsin(20/11*x+1/11)-128*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10 
*x^2-x+3)^(1/2))+23760*x*(-10*x^2-x+3)^(1/2)-7212*(-10*x^2-x+3)^(1/2))/(-1 
0*x^2-x+3)^(1/2)
 

3.23.91.5 Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{2+3 x} \, dx=\frac {1}{864} \, {\left (2400 \, x^{2} + 1980 \, x - 601\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {6553}{10368} \, \sqrt {5} \sqrt {2} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + \frac {1}{81} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

input

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

output

1/864*(2400*x^2 + 1980*x - 601)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 6553/10368* 
sqrt(5)*sqrt(2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt( 
-2*x + 1)/(10*x^2 + x - 3)) + 1/81*sqrt(7)*arctan(1/14*sqrt(7)*(37*x + 20) 
*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))
 

3.23.91.6 Sympy [F]

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

input

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

output

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

3.23.91.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{2+3 x} \, dx=-\frac {5}{18} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {145}{72} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {6553}{10368} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1}{81} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {119}{864} \, \sqrt {-10 \, x^{2} - x + 3} \]

input

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

output

-5/18*(-10*x^2 - x + 3)^(3/2) + 145/72*sqrt(-10*x^2 - x + 3)*x + 6553/1036 
8*sqrt(10)*arcsin(20/11*x + 1/11) - 1/81*sqrt(7)*arcsin(37/11*x/abs(3*x + 
2) + 20/11/abs(3*x + 2)) + 119/864*sqrt(-10*x^2 - x + 3)
 

3.23.91.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (92) = 184\).

Time = 0.41 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{2+3 x} \, dx=-\frac {1}{810} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{4320} \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 15 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 925 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {6553}{10368} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \]

input

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

output

-1/810*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sq 
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5 
) - sqrt(22)))) + 1/4320*(12*(8*sqrt(5)*(5*x + 3) - 15*sqrt(5))*(5*x + 3) 
- 925*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 6553/10368*sqrt(10)*(pi + 2 
*arctan(-1/4*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))))
 

3.23.91.9 Mupad [F(-1)]

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

input

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

output

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