\(\int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{2+3 x} \, dx\) [934]

Optimal result

Integrand size = 26, antiderivative size = 106 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {107}{180} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{6} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {4091 \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{540 \sqrt {10}}+\frac {14}{27} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \] Output:

107/180*(1-2*x)^(1/2)*(3+5*x)^(1/2)+1/6*(1-2*x)^(3/2)*(3+5*x)^(1/2)+4091/5 
400*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+14/27*7^(1/2)*arctan(1/7* 
(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.03 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{2+3 x} \, dx=\frac {30 \sqrt {1-2 x} \left (411+505 x-300 x^2\right )-4091 \sqrt {30+50 x} \arctan \left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+2800 \sqrt {7} \sqrt {3+5 x} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5400 \sqrt {3+5 x}} \] Input:

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

Output:

(30*Sqrt[1 - 2*x]*(411 + 505*x - 300*x^2) - 4091*Sqrt[30 + 50*x]*ArcTan[Sq 
rt[5/2 - 5*x]/Sqrt[3 + 5*x]] + 2800*Sqrt[7]*Sqrt[3 + 5*x]*ArcTan[Sqrt[1 - 
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(5400*Sqrt[3 + 5*x])
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {112, 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.

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

\(\Big \downarrow \) 112

\(\displaystyle \frac {1}{6} (1-2 x)^{3/2} \sqrt {5 x+3}-\frac {1}{6} \int -\frac {\sqrt {1-2 x} (107 x+62)}{2 (3 x+2) \sqrt {5 x+3}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{12} \int \frac {\sqrt {1-2 x} (107 x+62)}{(3 x+2) \sqrt {5 x+3}}dx+\frac {1}{6} \sqrt {5 x+3} (1-2 x)^{3/2}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{12} \left (\frac {1}{15} \int \frac {4091 x+2074}{2 \sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {107}{15} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{6} \sqrt {5 x+3} (1-2 x)^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{12} \left (\frac {1}{30} \int \frac {4091 x+2074}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx+\frac {107}{15} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{6} \sqrt {5 x+3} (1-2 x)^{3/2}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {1}{12} \left (\frac {1}{30} \left (\frac {4091}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {1960}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {107}{15} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{6} \sqrt {5 x+3} (1-2 x)^{3/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{12} \left (\frac {1}{30} \left (\frac {8182}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {1960}{3} \int \frac {1}{\sqrt {1-2 x} (3 x+2) \sqrt {5 x+3}}dx\right )+\frac {107}{15} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{6} \sqrt {5 x+3} (1-2 x)^{3/2}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{12} \left (\frac {1}{30} \left (\frac {8182}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}-\frac {3920}{3} \int \frac {1}{-\frac {1-2 x}{5 x+3}-7}d\frac {\sqrt {1-2 x}}{\sqrt {5 x+3}}\right )+\frac {107}{15} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{6} \sqrt {5 x+3} (1-2 x)^{3/2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{12} \left (\frac {1}{30} \left (\frac {8182}{15} \int \frac {1}{\sqrt {\frac {11}{5}-\frac {2}{5} (5 x+3)}}d\sqrt {5 x+3}+\frac {560}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )+\frac {107}{15} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{6} \sqrt {5 x+3} (1-2 x)^{3/2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{12} \left (\frac {1}{30} \left (\frac {4091}{3} \sqrt {\frac {2}{5}} \arcsin \left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {560}{3} \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )+\frac {107}{15} \sqrt {1-2 x} \sqrt {5 x+3}\right )+\frac {1}{6} \sqrt {5 x+3} (1-2 x)^{3/2}\)

Input:

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

Output:

((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/6 + ((107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/15 
+ ((4091*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/3 + (560*Sqrt[7]*ArcT 
an[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/3)/30)/12
 

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]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (4091 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2800 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-3600 x \sqrt {-10 x^{2}-x +3}+8220 \sqrt {-10 x^{2}-x +3}\right )}{10800 \sqrt {-10 x^{2}-x +3}}\)	\(98\)
risch	\(\frac {\left (-137+60 x \right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{180 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {4091 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{10800}-\frac {7 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{27}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(125\)

Input:

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

Output:

1/10800*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(4091*10^(1/2)*arcsin(20/11*x+1/11)-28 
00*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-3600*x*(-10* 
x^2-x+3)^(1/2)+8220*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{2+3 x} \, dx=-\frac {1}{180} \, {\left (60 \, x - 137\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} + \frac {7}{27} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {4091}{10800} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \] Input:

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

Output:

-1/180*(60*x - 137)*sqrt(5*x + 3)*sqrt(-2*x + 1) + 7/27*sqrt(7)*arctan(1/1 
4*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 409 
1/10800*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 
 1)/(10*x^2 + x - 3))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{2+3 x} \, dx=-\frac {1}{3} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {4091}{10800} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {7}{27} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {137}{180} \, \sqrt {-10 \, x^{2} - x + 3} \] Input:

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

Output:

-1/3*sqrt(-10*x^2 - x + 3)*x + 4091/10800*sqrt(10)*arcsin(20/11*x + 1/11) 
- 7/27*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 137/180 
*sqrt(-10*x^2 - x + 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (76) = 152\).

Time = 0.19 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.63 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{2+3 x} \, dx=-\frac {7}{270} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1}{900} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 173 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {4091}{10800} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \] Input:

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

Output:

-7/270*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)))) - 1/900*(12*sqrt(5)*(5*x + 3) - 173*sqrt(5))*sqrt(5*x + 3) 
*sqrt(-10*x + 5) + 4091/10800*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))))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.08 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{2+3 x} \, dx=-\frac {4091 \sqrt {10}\, \mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{5400}-\frac {14 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}-\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )}{27}+\frac {14 \sqrt {7}\, \mathit {atan} \left (\frac {\sqrt {33}+\sqrt {35}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-2 x +1}\, \sqrt {5}}{\sqrt {11}}\right )}{2}\right )}{\sqrt {2}}\right )}{27}-\frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{3}+\frac {137 \sqrt {5 x +3}\, \sqrt {-2 x +1}}{180} \] Input:

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

Output:

( - 4091*sqrt(10)*asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11)) - 2800*sqrt(7) 
*atan((sqrt(33) - sqrt(35)*tan(asin((sqrt( - 2*x + 1)*sqrt(5))/sqrt(11))/2 
))/sqrt(2)) + 2800*sqrt(7)*atan((sqrt(33) + sqrt(35)*tan(asin((sqrt( - 2*x 
 + 1)*sqrt(5))/sqrt(11))/2))/sqrt(2)) - 1800*sqrt(5*x + 3)*sqrt( - 2*x + 1 
)*x + 4110*sqrt(5*x + 3)*sqrt( - 2*x + 1))/5400