\(\int \frac {x^2 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx\) [525]

Optimal result

Integrand size = 26, antiderivative size = 87 \[ \int \frac {x^2 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {11 \sqrt {a x} \sqrt {1-a x}}{8 a^3}-\frac {11 (a x)^{3/2} \sqrt {1-a x}}{12 a^3}-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a^3}-\frac {11 \arcsin (1-2 a x)}{16 a^3} \] Output:

-11/8*(a*x)^(1/2)*(-a*x+1)^(1/2)/a^3-11/12*(a*x)^(3/2)*(-a*x+1)^(1/2)/a^3- 
1/3*(a*x)^(5/2)*(-a*x+1)^(1/2)/a^3+11/16*arcsin(2*a*x-1)/a^3
 

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.09 \[ \int \frac {x^2 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {\sqrt {a} x \left (-33+11 a x+14 a^2 x^2+8 a^3 x^3\right )+66 \sqrt {x} \sqrt {1-a x} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{-1+\sqrt {1-a x}}\right )}{24 a^{5/2} \sqrt {-a x (-1+a x)}} \] Input:

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

Output:

(Sqrt[a]*x*(-33 + 11*a*x + 14*a^2*x^2 + 8*a^3*x^3) + 66*Sqrt[x]*Sqrt[1 - a 
*x]*ArcTan[(Sqrt[a]*Sqrt[x])/(-1 + Sqrt[1 - a*x])])/(24*a^(5/2)*Sqrt[-(a*x 
*(-1 + a*x))])
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.21, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {8, 90, 60, 60, 62, 1090, 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.

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

\(\Big \downarrow \) 8

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {11}{6} \int \frac {(a x)^{3/2}}{\sqrt {1-a x}}dx-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}}{a^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {11}{6} \left (\frac {3}{4} \int \frac {\sqrt {a x}}{\sqrt {1-a x}}dx-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a}\right )-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}}{a^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {11}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {a x} \sqrt {1-a x}}dx-\frac {\sqrt {a x} \sqrt {1-a x}}{a}\right )-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a}\right )-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}}{a^2}\)

\(\Big \downarrow \) 62

\(\displaystyle \frac {\frac {11}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {a x-a^2 x^2}}dx-\frac {\sqrt {a x} \sqrt {1-a x}}{a}\right )-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a}\right )-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}}{a^2}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {\frac {11}{6} \left (\frac {3}{4} \left (-\frac {\int \frac {1}{\sqrt {1-\frac {\left (a-2 a^2 x\right )^2}{a^2}}}d\left (a-2 a^2 x\right )}{2 a^2}-\frac {\sqrt {a x} \sqrt {1-a x}}{a}\right )-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a}\right )-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}}{a^2}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\frac {11}{6} \left (\frac {3}{4} \left (-\frac {\arcsin \left (\frac {a-2 a^2 x}{a}\right )}{2 a}-\frac {\sqrt {a x} \sqrt {1-a x}}{a}\right )-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a}\right )-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}}{a^2}\)

Input:

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

Output:

(-1/3*((a*x)^(5/2)*Sqrt[1 - a*x])/a + (11*(-1/2*((a*x)^(3/2)*Sqrt[1 - a*x] 
)/a + (3*(-((Sqrt[a*x]*Sqrt[1 - a*x])/a) - ArcSin[(a - 2*a^2*x)/a]/(2*a))) 
/4))/6)/a^2
 

Defintions of rubi rules used

rule 8

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_))^(p_), x_Symbol] :> Simp[1/a^m   Int[u*(a* 
x)^(m + p), x], x] /; FreeQ[{a, m, p}, x] && IntegerQ[m]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 62

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Int[ 
1/Sqrt[a*c - b*(a - c)*x - b^2*x^2], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b 
+ d, 0] && GtQ[a + c, 0]
 

rule 90

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

rule 1090

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* 
(c/(b^2 - 4*a*c)))^p)   Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, 
b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.28

method	result	size
default	\(-\frac {\sqrt {-x a +1}\, x \left (16 \,\operatorname {csgn}\left (a \right ) x^{2} a^{2} \sqrt {-x \left (x a -1\right ) a}+44 \,\operatorname {csgn}\left (a \right ) \sqrt {-x \left (x a -1\right ) a}\, a x +66 \,\operatorname {csgn}\left (a \right ) \sqrt {-x \left (x a -1\right ) a}-33 \arctan \left (\frac {\operatorname {csgn}\left (a \right ) \left (2 x a -1\right )}{2 \sqrt {-x \left (x a -1\right ) a}}\right )\right ) \operatorname {csgn}\left (a \right )}{48 a^{2} \sqrt {x a}\, \sqrt {-x \left (x a -1\right ) a}}\)	\(111\)
risch	\(\frac {\left (8 a^{2} x^{2}+22 x a +33\right ) x \left (x a -1\right ) \sqrt {x a \left (-x a +1\right )}}{24 a^{2} \sqrt {-x \left (x a -1\right ) a}\, \sqrt {x a}\, \sqrt {-x a +1}}+\frac {11 \arctan \left (\frac {\sqrt {a^{2}}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a^{2} x^{2}+x a}}\right ) \sqrt {x a \left (-x a +1\right )}}{16 a^{2} \sqrt {a^{2}}\, \sqrt {x a}\, \sqrt {-x a +1}}\)	\(124\)
meijerg	\(-\frac {\sqrt {x}\, \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (-a \right )^{\frac {7}{2}} \left (56 a^{2} x^{2}+70 x a +105\right ) \sqrt {-x a +1}}{168 a^{3}}+\frac {5 \sqrt {\pi }\, \left (-a \right )^{\frac {7}{2}} \arcsin \left (\sqrt {x}\, \sqrt {a}\right )}{8 a^{\frac {7}{2}}}\right )}{\left (-a \right )^{\frac {5}{2}} \sqrt {x a}\, \sqrt {\pi }}-\frac {\sqrt {x}\, \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (-a \right )^{\frac {5}{2}} \left (10 x a +15\right ) \sqrt {-x a +1}}{20 a^{2}}+\frac {3 \sqrt {\pi }\, \left (-a \right )^{\frac {5}{2}} \arcsin \left (\sqrt {x}\, \sqrt {a}\right )}{4 a^{\frac {5}{2}}}\right )}{\left (-a \right )^{\frac {3}{2}} \sqrt {x a}\, \sqrt {\pi }\, a}\)	\(153\)

Input:

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

Output:

-1/48*(-a*x+1)^(1/2)*x*(16*csgn(a)*x^2*a^2*(-x*(a*x-1)*a)^(1/2)+44*csgn(a) 
*(-x*(a*x-1)*a)^(1/2)*a*x+66*csgn(a)*(-x*(a*x-1)*a)^(1/2)-33*arctan(1/2*cs 
gn(a)*(2*a*x-1)/(-x*(a*x-1)*a)^(1/2)))*csgn(a)/a^2/(x*a)^(1/2)/(-x*(a*x-1) 
*a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {x^2 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {{\left (8 \, a^{2} x^{2} + 22 \, a x + 33\right )} \sqrt {a x} \sqrt {-a x + 1} + 33 \, \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x - 1}\right )}{24 \, a^{3}} \] Input:

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

Output:

-1/24*((8*a^2*x^2 + 22*a*x + 33)*sqrt(a*x)*sqrt(-a*x + 1) + 33*arctan(sqrt 
(a*x)*sqrt(-a*x + 1)/(a*x - 1)))/a^3
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 10.35 (sec) , antiderivative size = 393, normalized size of antiderivative = 4.52 \[ \int \frac {x^2 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=a \left (\begin {cases} - \frac {5 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} - \frac {i x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {a x - 1}} - \frac {5 i x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {a x - 1}} + \frac {5 i \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {5 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} + \frac {x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {- a x + 1}} + \frac {5 x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {- a x + 1}} - \frac {5 \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {3 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{4 a^{3}} - \frac {i x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {3}{2}}}{4 a^{\frac {3}{2}} \sqrt {a x - 1}} + \frac {3 i \sqrt {x}}{4 a^{\frac {5}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {3 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{4 a^{3}} + \frac {x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {3}{2}}}{4 a^{\frac {3}{2}} \sqrt {- a x + 1}} - \frac {3 \sqrt {x}}{4 a^{\frac {5}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases} \] Input:

integrate(x**2*(a*x+1)/(a*x)**(1/2)/(-a*x+1)**(1/2),x)
 

Output:

a*Piecewise((-5*I*acosh(sqrt(a)*sqrt(x))/(8*a**4) - I*x**(7/2)/(3*sqrt(a)* 
sqrt(a*x - 1)) - I*x**(5/2)/(12*a**(3/2)*sqrt(a*x - 1)) - 5*I*x**(3/2)/(24 
*a**(5/2)*sqrt(a*x - 1)) + 5*I*sqrt(x)/(8*a**(7/2)*sqrt(a*x - 1)), Abs(a*x 
) > 1), (5*asin(sqrt(a)*sqrt(x))/(8*a**4) + x**(7/2)/(3*sqrt(a)*sqrt(-a*x 
+ 1)) + x**(5/2)/(12*a**(3/2)*sqrt(-a*x + 1)) + 5*x**(3/2)/(24*a**(5/2)*sq 
rt(-a*x + 1)) - 5*sqrt(x)/(8*a**(7/2)*sqrt(-a*x + 1)), True)) + Piecewise( 
(-3*I*acosh(sqrt(a)*sqrt(x))/(4*a**3) - I*x**(5/2)/(2*sqrt(a)*sqrt(a*x - 1 
)) - I*x**(3/2)/(4*a**(3/2)*sqrt(a*x - 1)) + 3*I*sqrt(x)/(4*a**(5/2)*sqrt( 
a*x - 1)), Abs(a*x) > 1), (3*asin(sqrt(a)*sqrt(x))/(4*a**3) + x**(5/2)/(2* 
sqrt(a)*sqrt(-a*x + 1)) + x**(3/2)/(4*a**(3/2)*sqrt(-a*x + 1)) - 3*sqrt(x) 
/(4*a**(5/2)*sqrt(-a*x + 1)), True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.95 \[ \int \frac {x^2 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + a x} x^{2}}{3 \, a} - \frac {11 \, \sqrt {-a^{2} x^{2} + a x} x}{12 \, a^{2}} - \frac {11 \, \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right )}{16 \, a^{3}} - \frac {11 \, \sqrt {-a^{2} x^{2} + a x}}{8 \, a^{3}} \] Input:

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

Output:

-1/3*sqrt(-a^2*x^2 + a*x)*x^2/a - 11/12*sqrt(-a^2*x^2 + a*x)*x/a^2 - 11/16 
*arcsin(-(2*a^2*x - a)/a)/a^3 - 11/8*sqrt(-a^2*x^2 + a*x)/a^3
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.46 \[ \int \frac {x^2 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {{\left (2 \, {\left (4 \, a x + 11\right )} a x + 33\right )} \sqrt {a x} \sqrt {-a x + 1} - 33 \, \arcsin \left (\sqrt {a x}\right )}{24 \, a^{3}} \] Input:

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

Output:

-1/24*((2*(4*a*x + 11)*a*x + 33)*sqrt(a*x)*sqrt(-a*x + 1) - 33*arcsin(sqrt 
(a*x)))/a^3
 

Mupad [B] (verification not implemented)

Time = 25.82 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.09 \[ \int \frac {x^2 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {11\,\mathrm {atan}\left (\frac {\sqrt {a\,x}}{\sqrt {1-a\,x}-1}\right )}{4\,a^3}-\frac {\frac {5\,\sqrt {a\,x}}{4\,\left (\sqrt {1-a\,x}-1\right )}+\frac {85\,{\left (a\,x\right )}^{3/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^3}+\frac {33\,{\left (a\,x\right )}^{5/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^5}-\frac {33\,{\left (a\,x\right )}^{7/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^7}-\frac {85\,{\left (a\,x\right )}^{9/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^9}-\frac {5\,{\left (a\,x\right )}^{11/2}}{4\,{\left (\sqrt {1-a\,x}-1\right )}^{11}}}{a^3\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^6}-\frac {\frac {3\,\sqrt {a\,x}}{2\,\left (\sqrt {1-a\,x}-1\right )}+\frac {11\,{\left (a\,x\right )}^{3/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^3}-\frac {11\,{\left (a\,x\right )}^{5/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^5}-\frac {3\,{\left (a\,x\right )}^{7/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^7}}{a^3\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^4} \] Input:

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

Output:

(11*atan((a*x)^(1/2)/((1 - a*x)^(1/2) - 1)))/(4*a^3) - ((5*(a*x)^(1/2))/(4 
*((1 - a*x)^(1/2) - 1)) + (85*(a*x)^(3/2))/(12*((1 - a*x)^(1/2) - 1)^3) + 
(33*(a*x)^(5/2))/(2*((1 - a*x)^(1/2) - 1)^5) - (33*(a*x)^(7/2))/(2*((1 - a 
*x)^(1/2) - 1)^7) - (85*(a*x)^(9/2))/(12*((1 - a*x)^(1/2) - 1)^9) - (5*(a* 
x)^(11/2))/(4*((1 - a*x)^(1/2) - 1)^11))/(a^3*((a*x)/((1 - a*x)^(1/2) - 1) 
^2 + 1)^6) - ((3*(a*x)^(1/2))/(2*((1 - a*x)^(1/2) - 1)) + (11*(a*x)^(3/2)) 
/(2*((1 - a*x)^(1/2) - 1)^3) - (11*(a*x)^(5/2))/(2*((1 - a*x)^(1/2) - 1)^5 
) - (3*(a*x)^(7/2))/(2*((1 - a*x)^(1/2) - 1)^7))/(a^3*((a*x)/((1 - a*x)^(1 
/2) - 1)^2 + 1)^4)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.82 \[ \int \frac {x^2 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {-8 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a^{2} x^{2}-22 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a x -33 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}-33 \,\mathrm {log}\left (\sqrt {-a x +1}+\sqrt {x}\, \sqrt {a}\, i \right ) i}{24 a^{3}} \] Input:

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

Output:

( - 8*sqrt(x)*sqrt(a)*sqrt( - a*x + 1)*a**2*x**2 - 22*sqrt(x)*sqrt(a)*sqrt 
( - a*x + 1)*a*x - 33*sqrt(x)*sqrt(a)*sqrt( - a*x + 1) - 33*log(sqrt( - a* 
x + 1) + sqrt(x)*sqrt(a)*i)*i)/(24*a**3)