\(\int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx\) [524]

Optimal result

Integrand size = 26, antiderivative size = 111 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}-\frac {75 \arcsin (1-2 a x)}{128 a^4} \] Output:

-75/64*(a*x)^(1/2)*(-a*x+1)^(1/2)/a^4-25/32*(a*x)^(3/2)*(-a*x+1)^(1/2)/a^4 
-5/8*(a*x)^(5/2)*(-a*x+1)^(1/2)/a^4-1/4*(a*x)^(7/2)*(-a*x+1)^(1/2)/a^4+75/ 
128*arcsin(2*a*x-1)/a^4
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.93 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {\sqrt {a} x \left (-75+25 a x+10 a^2 x^2+24 a^3 x^3+16 a^4 x^4\right )+150 \sqrt {x} \sqrt {1-a x} \arctan \left (\frac {\sqrt {a} \sqrt {x}}{-1+\sqrt {1-a x}}\right )}{64 a^{7/2} \sqrt {-a x (-1+a x)}} \] Input:

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

Output:

(Sqrt[a]*x*(-75 + 25*a*x + 10*a^2*x^2 + 24*a^3*x^3 + 16*a^4*x^4) + 150*Sqr 
t[x]*Sqrt[1 - a*x]*ArcTan[(Sqrt[a]*Sqrt[x])/(-1 + Sqrt[1 - a*x])])/(64*a^( 
7/2)*Sqrt[-(a*x*(-1 + a*x))])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {8, 90, 60, 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.

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

\(\Big \downarrow \) 8

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {\frac {15}{8} \int \frac {(a x)^{5/2}}{\sqrt {1-a x}}dx-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a}}{a^3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {15}{8} \left (\frac {5}{6} \int \frac {(a x)^{3/2}}{\sqrt {1-a x}}dx-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}\right )-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a}}{a^3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {15}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \frac {\sqrt {a x}}{\sqrt {1-a x}}dx-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a}\right )-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}\right )-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a}}{a^3}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\frac {15}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {a x} \sqrt {1-a x}}dx-\frac {\sqrt {a x} \sqrt {1-a x}}{a}\right )-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a}\right )-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}\right )-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a}}{a^3}\)

\(\Big \downarrow \) 62

\(\displaystyle \frac {\frac {15}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {a x-a^2 x^2}}dx-\frac {\sqrt {a x} \sqrt {1-a x}}{a}\right )-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a}\right )-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}\right )-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a}}{a^3}\)

\(\Big \downarrow \) 1090

\(\displaystyle \frac {\frac {15}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\int \frac {1}{\sqrt {1-\frac {\left (a-2 a^2 x\right )^2}{a^2}}}d\left (a-2 a^2 x\right )}{2 a^2}-\frac {\sqrt {a x} \sqrt {1-a x}}{a}\right )-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a}\right )-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}\right )-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a}}{a^3}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\frac {15}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (-\frac {\arcsin \left (\frac {a-2 a^2 x}{a}\right )}{2 a}-\frac {\sqrt {a x} \sqrt {1-a x}}{a}\right )-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a}\right )-\frac {(a x)^{5/2} \sqrt {1-a x}}{3 a}\right )-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a}}{a^3}\)

Input:

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

Output:

(-1/4*((a*x)^(7/2)*Sqrt[1 - a*x])/a + (15*(-1/3*((a*x)^(5/2)*Sqrt[1 - a*x] 
)/a + (5*(-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))/8)/a^3
 

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 = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.19

method	result	size
default	\(-\frac {\sqrt {-x a +1}\, x \left (32 \,\operatorname {csgn}\left (a \right ) a^{3} x^{3} \sqrt {-x \left (x a -1\right ) a}+80 \,\operatorname {csgn}\left (a \right ) x^{2} a^{2} \sqrt {-x \left (x a -1\right ) a}+100 \,\operatorname {csgn}\left (a \right ) \sqrt {-x \left (x a -1\right ) a}\, a x +150 \,\operatorname {csgn}\left (a \right ) \sqrt {-x \left (x a -1\right ) a}-75 \arctan \left (\frac {\operatorname {csgn}\left (a \right ) \left (2 x a -1\right )}{2 \sqrt {-x \left (x a -1\right ) a}}\right )\right ) \operatorname {csgn}\left (a \right )}{128 a^{3} \sqrt {x a}\, \sqrt {-x \left (x a -1\right ) a}}\)	\(132\)
risch	\(\frac {\left (16 a^{3} x^{3}+40 a^{2} x^{2}+50 x a +75\right ) x \left (x a -1\right ) \sqrt {x a \left (-x a +1\right )}}{64 a^{3} \sqrt {-x \left (x a -1\right ) a}\, \sqrt {x a}\, \sqrt {-x a +1}}+\frac {75 \arctan \left (\frac {\sqrt {a^{2}}\, \left (x -\frac {1}{2 a}\right )}{\sqrt {-a^{2} x^{2}+x a}}\right ) \sqrt {x a \left (-x a +1\right )}}{128 a^{3} \sqrt {a^{2}}\, \sqrt {x a}\, \sqrt {-x a +1}}\)	\(132\)
meijerg	\(-\frac {\sqrt {x}\, \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (-a \right )^{\frac {9}{2}} \left (144 a^{3} x^{3}+168 a^{2} x^{2}+210 x a +315\right ) \sqrt {-x a +1}}{576 a^{4}}+\frac {35 \sqrt {\pi }\, \left (-a \right )^{\frac {9}{2}} \arcsin \left (\sqrt {x}\, \sqrt {a}\right )}{64 a^{\frac {9}{2}}}\right )}{\left (-a \right )^{\frac {7}{2}} \sqrt {x a}\, \sqrt {\pi }}-\frac {\sqrt {x}\, \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \left (-a \right )^{\frac {7}{2}} \left (56 a^{2} x^{2}+70 x a +105\right ) \sqrt {-x a +1}}{168 a^{3}}+\frac {5 \sqrt {\pi }\, \left (-a \right )^{\frac {7}{2}} \arcsin \left (\sqrt {x}\, \sqrt {a}\right )}{8 a^{\frac {7}{2}}}\right )}{\left (-a \right )^{\frac {5}{2}} \sqrt {x a}\, \sqrt {\pi }\, a}\)	\(169\)

Input:

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

Output:

-1/128*(-a*x+1)^(1/2)*x*(32*csgn(a)*a^3*x^3*(-x*(a*x-1)*a)^(1/2)+80*csgn(a 
)*x^2*a^2*(-x*(a*x-1)*a)^(1/2)+100*csgn(a)*(-x*(a*x-1)*a)^(1/2)*a*x+150*cs 
gn(a)*(-x*(a*x-1)*a)^(1/2)-75*arctan(1/2*csgn(a)*(2*a*x-1)/(-x*(a*x-1)*a)^ 
(1/2)))*csgn(a)/a^3/(x*a)^(1/2)/(-x*(a*x-1)*a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.59 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {{\left (16 \, a^{3} x^{3} + 40 \, a^{2} x^{2} + 50 \, a x + 75\right )} \sqrt {a x} \sqrt {-a x + 1} + 75 \, \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x - 1}\right )}{64 \, a^{4}} \] Input:

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

Output:

-1/64*((16*a^3*x^3 + 40*a^2*x^2 + 50*a*x + 75)*sqrt(a*x)*sqrt(-a*x + 1) + 
75*arctan(sqrt(a*x)*sqrt(-a*x + 1)/(a*x - 1)))/a^4
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 43.55 (sec) , antiderivative size = 484, normalized size of antiderivative = 4.36 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=a \left (\begin {cases} - \frac {35 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{64 a^{5}} - \frac {i x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {7}{2}}}{24 a^{\frac {3}{2}} \sqrt {a x - 1}} - \frac {7 i x^{\frac {5}{2}}}{96 a^{\frac {5}{2}} \sqrt {a x - 1}} - \frac {35 i x^{\frac {3}{2}}}{192 a^{\frac {7}{2}} \sqrt {a x - 1}} + \frac {35 i \sqrt {x}}{64 a^{\frac {9}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {35 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{64 a^{5}} + \frac {x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {7}{2}}}{24 a^{\frac {3}{2}} \sqrt {- a x + 1}} + \frac {7 x^{\frac {5}{2}}}{96 a^{\frac {5}{2}} \sqrt {- a x + 1}} + \frac {35 x^{\frac {3}{2}}}{192 a^{\frac {7}{2}} \sqrt {- a x + 1}} - \frac {35 \sqrt {x}}{64 a^{\frac {9}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {5 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} - \frac {i x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {a x - 1}} - \frac {5 i x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {a x - 1}} + \frac {5 i \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {5 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} + \frac {x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {- a x + 1}} + \frac {5 x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {- a x + 1}} - \frac {5 \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases} \] Input:

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

Output:

a*Piecewise((-35*I*acosh(sqrt(a)*sqrt(x))/(64*a**5) - I*x**(9/2)/(4*sqrt(a 
)*sqrt(a*x - 1)) - I*x**(7/2)/(24*a**(3/2)*sqrt(a*x - 1)) - 7*I*x**(5/2)/( 
96*a**(5/2)*sqrt(a*x - 1)) - 35*I*x**(3/2)/(192*a**(7/2)*sqrt(a*x - 1)) + 
35*I*sqrt(x)/(64*a**(9/2)*sqrt(a*x - 1)), Abs(a*x) > 1), (35*asin(sqrt(a)* 
sqrt(x))/(64*a**5) + x**(9/2)/(4*sqrt(a)*sqrt(-a*x + 1)) + x**(7/2)/(24*a* 
*(3/2)*sqrt(-a*x + 1)) + 7*x**(5/2)/(96*a**(5/2)*sqrt(-a*x + 1)) + 35*x**( 
3/2)/(192*a**(7/2)*sqrt(-a*x + 1)) - 35*sqrt(x)/(64*a**(9/2)*sqrt(-a*x + 1 
)), True)) + 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)*sqrt(-a*x + 1)) - 5*sqrt(x)/(8*a**(7/2)*sqrt(-a*x + 1)), True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {\sqrt {-a^{2} x^{2} + a x} x^{3}}{4 \, a} - \frac {5 \, \sqrt {-a^{2} x^{2} + a x} x^{2}}{8 \, a^{2}} - \frac {25 \, \sqrt {-a^{2} x^{2} + a x} x}{32 \, a^{3}} - \frac {75 \, \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right )}{128 \, a^{4}} - \frac {75 \, \sqrt {-a^{2} x^{2} + a x}}{64 \, a^{4}} \] Input:

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

Output:

-1/4*sqrt(-a^2*x^2 + a*x)*x^3/a - 5/8*sqrt(-a^2*x^2 + a*x)*x^2/a^2 - 25/32 
*sqrt(-a^2*x^2 + a*x)*x/a^3 - 75/128*arcsin(-(2*a^2*x - a)/a)/a^4 - 75/64* 
sqrt(-a^2*x^2 + a*x)/a^4
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.41 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=-\frac {{\left (2 \, {\left (4 \, {\left (2 \, a x + 5\right )} a x + 25\right )} a x + 75\right )} \sqrt {a x} \sqrt {-a x + 1} - 75 \, \arcsin \left (\sqrt {a x}\right )}{64 \, a^{4}} \] Input:

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

Output:

-1/64*((2*(4*(2*a*x + 5)*a*x + 25)*a*x + 75)*sqrt(a*x)*sqrt(-a*x + 1) - 75 
*arcsin(sqrt(a*x)))/a^4
 

Mupad [B] (verification not implemented)

Time = 28.26 (sec) , antiderivative size = 345, normalized size of antiderivative = 3.11 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {75\,\mathrm {atan}\left (\frac {\sqrt {a\,x}}{\sqrt {1-a\,x}-1}\right )}{32\,a^4}-\frac {\frac {5\,\sqrt {a\,x}}{4\,\left (\sqrt {1-a\,x}-1\right )}+\frac {85\,{\left (a\,x\right )}^{3/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^3}+\frac {33\,{\left (a\,x\right )}^{5/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^5}-\frac {33\,{\left (a\,x\right )}^{7/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^7}-\frac {85\,{\left (a\,x\right )}^{9/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^9}-\frac {5\,{\left (a\,x\right )}^{11/2}}{4\,{\left (\sqrt {1-a\,x}-1\right )}^{11}}}{a^4\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^6}-\frac {\frac {35\,\sqrt {a\,x}}{32\,\left (\sqrt {1-a\,x}-1\right )}+\frac {805\,{\left (a\,x\right )}^{3/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^3}+\frac {2681\,{\left (a\,x\right )}^{5/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^5}+\frac {5053\,{\left (a\,x\right )}^{7/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^7}-\frac {5053\,{\left (a\,x\right )}^{9/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^9}-\frac {2681\,{\left (a\,x\right )}^{11/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^{11}}-\frac {805\,{\left (a\,x\right )}^{13/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^{13}}-\frac {35\,{\left (a\,x\right )}^{15/2}}{32\,{\left (\sqrt {1-a\,x}-1\right )}^{15}}}{a^4\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^8} \] Input:

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

Output:

(75*atan((a*x)^(1/2)/((1 - a*x)^(1/2) - 1)))/(32*a^4) - ((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^4*((a*x)/((1 - a*x)^(1/2) - 1 
)^2 + 1)^6) - ((35*(a*x)^(1/2))/(32*((1 - a*x)^(1/2) - 1)) + (805*(a*x)^(3 
/2))/(96*((1 - a*x)^(1/2) - 1)^3) + (2681*(a*x)^(5/2))/(96*((1 - a*x)^(1/2 
) - 1)^5) + (5053*(a*x)^(7/2))/(96*((1 - a*x)^(1/2) - 1)^7) - (5053*(a*x)^ 
(9/2))/(96*((1 - a*x)^(1/2) - 1)^9) - (2681*(a*x)^(11/2))/(96*((1 - a*x)^( 
1/2) - 1)^11) - (805*(a*x)^(13/2))/(96*((1 - a*x)^(1/2) - 1)^13) - (35*(a* 
x)^(15/2))/(32*((1 - a*x)^(1/2) - 1)^15))/(a^4*((a*x)/((1 - a*x)^(1/2) - 1 
)^2 + 1)^8)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.81 \[ \int \frac {x^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx=\frac {-16 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a^{3} x^{3}-40 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a^{2} x^{2}-50 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}\, a x -75 \sqrt {x}\, \sqrt {a}\, \sqrt {-a x +1}-75 \,\mathrm {log}\left (\sqrt {-a x +1}+\sqrt {x}\, \sqrt {a}\, i \right ) i}{64 a^{4}} \] Input:

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

Output:

( - 16*sqrt(x)*sqrt(a)*sqrt( - a*x + 1)*a**3*x**3 - 40*sqrt(x)*sqrt(a)*sqr 
t( - a*x + 1)*a**2*x**2 - 50*sqrt(x)*sqrt(a)*sqrt( - a*x + 1)*a*x - 75*sqr 
t(x)*sqrt(a)*sqrt( - a*x + 1) - 75*log(sqrt( - a*x + 1) + sqrt(x)*sqrt(a)* 
i)*i)/(64*a**4)