\(\int \frac {x^{3/2}}{(a x+b x^3)^{9/2}} \, dx\) [95]

Optimal result

Integrand size = 19, antiderivative size = 159 \[ \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {x^{3/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {9 \sqrt {x}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {3}{5 a^3 \sqrt {x} \left (a x+b x^3\right )^{3/2}}+\frac {3}{a^4 x^{3/2} \sqrt {a x+b x^3}}-\frac {9 \sqrt {a x+b x^3}}{2 a^5 x^{5/2}}+\frac {9 b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^3}}\right )}{2 a^{11/2}} \] Output:

1/7*x^(3/2)/a/(b*x^3+a*x)^(7/2)+9/35*x^(1/2)/a^2/(b*x^3+a*x)^(5/2)+3/5/a^3 
/x^(1/2)/(b*x^3+a*x)^(3/2)+3/a^4/x^(3/2)/(b*x^3+a*x)^(1/2)-9/2*(b*x^3+a*x) 
^(1/2)/a^5/x^(5/2)+9/2*b*arctanh(a^(1/2)*x^(1/2)/(b*x^3+a*x)^(1/2))/a^(11/ 
2)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77 \[ \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {\sqrt {x \left (a+b x^2\right )} \left (-\sqrt {a} \left (35 a^4+528 a^3 b x^2+1218 a^2 b^2 x^4+1050 a b^3 x^6+315 b^4 x^8\right )+315 b x^2 \left (a+b x^2\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )\right )}{70 a^{11/2} x^{5/2} \left (a+b x^2\right )^4} \] Input:

Integrate[x^(3/2)/(a*x + b*x^3)^(9/2),x]
 

Output:

(Sqrt[x*(a + b*x^2)]*(-(Sqrt[a]*(35*a^4 + 528*a^3*b*x^2 + 1218*a^2*b^2*x^4 
 + 1050*a*b^3*x^6 + 315*b^4*x^8)) + 315*b*x^2*(a + b*x^2)^(7/2)*ArcTanh[Sq 
rt[a + b*x^2]/Sqrt[a]]))/(70*a^(11/2)*x^(5/2)*(a + b*x^2)^4)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {1929, 1929, 1929, 1929, 1931, 1935, 219}

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.

\(\displaystyle \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{9/2}} \, dx\)

\(\Big \downarrow \) 1929

\(\displaystyle \frac {9 \int \frac {\sqrt {x}}{\left (b x^3+a x\right )^{7/2}}dx}{7 a}+\frac {x^{3/2}}{7 a \left (a x+b x^3\right )^{7/2}}\)

\(\Big \downarrow \) 1929

\(\displaystyle \frac {9 \left (\frac {7 \int \frac {1}{\sqrt {x} \left (b x^3+a x\right )^{5/2}}dx}{5 a}+\frac {\sqrt {x}}{5 a \left (a x+b x^3\right )^{5/2}}\right )}{7 a}+\frac {x^{3/2}}{7 a \left (a x+b x^3\right )^{7/2}}\)

\(\Big \downarrow \) 1929

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \int \frac {1}{x^{3/2} \left (b x^3+a x\right )^{3/2}}dx}{3 a}+\frac {1}{3 a \sqrt {x} \left (a x+b x^3\right )^{3/2}}\right )}{5 a}+\frac {\sqrt {x}}{5 a \left (a x+b x^3\right )^{5/2}}\right )}{7 a}+\frac {x^{3/2}}{7 a \left (a x+b x^3\right )^{7/2}}\)

\(\Big \downarrow \) 1929

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \int \frac {1}{x^{5/2} \sqrt {b x^3+a x}}dx}{a}+\frac {1}{a x^{3/2} \sqrt {a x+b x^3}}\right )}{3 a}+\frac {1}{3 a \sqrt {x} \left (a x+b x^3\right )^{3/2}}\right )}{5 a}+\frac {\sqrt {x}}{5 a \left (a x+b x^3\right )^{5/2}}\right )}{7 a}+\frac {x^{3/2}}{7 a \left (a x+b x^3\right )^{7/2}}\)

\(\Big \downarrow \) 1931

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (-\frac {b \int \frac {1}{\sqrt {x} \sqrt {b x^3+a x}}dx}{2 a}-\frac {\sqrt {a x+b x^3}}{2 a x^{5/2}}\right )}{a}+\frac {1}{a x^{3/2} \sqrt {a x+b x^3}}\right )}{3 a}+\frac {1}{3 a \sqrt {x} \left (a x+b x^3\right )^{3/2}}\right )}{5 a}+\frac {\sqrt {x}}{5 a \left (a x+b x^3\right )^{5/2}}\right )}{7 a}+\frac {x^{3/2}}{7 a \left (a x+b x^3\right )^{7/2}}\)

\(\Big \downarrow \) 1935

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {b \int \frac {1}{1-\frac {a x}{b x^3+a x}}d\frac {\sqrt {x}}{\sqrt {b x^3+a x}}}{2 a}-\frac {\sqrt {a x+b x^3}}{2 a x^{5/2}}\right )}{a}+\frac {1}{a x^{3/2} \sqrt {a x+b x^3}}\right )}{3 a}+\frac {1}{3 a \sqrt {x} \left (a x+b x^3\right )^{3/2}}\right )}{5 a}+\frac {\sqrt {x}}{5 a \left (a x+b x^3\right )^{5/2}}\right )}{7 a}+\frac {x^{3/2}}{7 a \left (a x+b x^3\right )^{7/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {9 \left (\frac {7 \left (\frac {5 \left (\frac {3 \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^3}}\right )}{2 a^{3/2}}-\frac {\sqrt {a x+b x^3}}{2 a x^{5/2}}\right )}{a}+\frac {1}{a x^{3/2} \sqrt {a x+b x^3}}\right )}{3 a}+\frac {1}{3 a \sqrt {x} \left (a x+b x^3\right )^{3/2}}\right )}{5 a}+\frac {\sqrt {x}}{5 a \left (a x+b x^3\right )^{5/2}}\right )}{7 a}+\frac {x^{3/2}}{7 a \left (a x+b x^3\right )^{7/2}}\)

Input:

Int[x^(3/2)/(a*x + b*x^3)^(9/2),x]
 

Output:

x^(3/2)/(7*a*(a*x + b*x^3)^(7/2)) + (9*(Sqrt[x]/(5*a*(a*x + b*x^3)^(5/2)) 
+ (7*(1/(3*a*Sqrt[x]*(a*x + b*x^3)^(3/2)) + (5*(1/(a*x^(3/2)*Sqrt[a*x + b* 
x^3]) + (3*(-1/2*Sqrt[a*x + b*x^3]/(a*x^(5/2)) + (b*ArcTanh[(Sqrt[a]*Sqrt[ 
x])/Sqrt[a*x + b*x^3]])/(2*a^(3/2))))/a))/(3*a)))/(5*a)))/(7*a)
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1929

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j 
)*(p + 1))), x] + Simp[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1)))   In 
t[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, m}, x] & 
&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, 
 -1]
 

rule 1931

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p 
+ 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1)))   I 
nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] 
 &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[ 
m + j*p + 1, 0]
 

rule 1935

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp 
[-2/(n - j)   Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], 
 x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
 

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.47

method	result	size
default	\(\frac {\sqrt {x \left (b \,x^{2}+a \right )}\, \left (315 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) b^{4} x^{8} \sqrt {b \,x^{2}+a}-315 \sqrt {a}\, b^{4} x^{8}+945 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) a \,b^{3} x^{6} \sqrt {b \,x^{2}+a}-1050 a^{\frac {3}{2}} b^{3} x^{6}+945 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) a^{2} b^{2} x^{4} \sqrt {b \,x^{2}+a}-1218 a^{\frac {5}{2}} b^{2} x^{4}+315 \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right ) a^{3} b \,x^{2} \sqrt {b \,x^{2}+a}-528 a^{\frac {7}{2}} b \,x^{2}-35 a^{\frac {9}{2}}\right )}{70 a^{\frac {11}{2}} x^{\frac {5}{2}} \left (b \,x^{2}+a \right )^{4}}\)	\(234\)
risch	\(-\frac {b \,x^{2}+a}{2 a^{5} x^{\frac {3}{2}} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {\left (\frac {9 b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {11}{2}}}-\frac {2629 b \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{5} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {2629 b \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{5} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {19 \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{280 a^{4} \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )^{3}}+\frac {389 \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{5} \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {19 \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{280 a^{4} \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )^{3}}+\frac {389 \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{1120 a^{5} \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{112 a^{4} b \left (x -\frac {\sqrt {-a b}}{b}\right )^{4}}-\frac {\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{112 a^{4} b \left (x +\frac {\sqrt {-a b}}{b}\right )^{4}}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x}}{\sqrt {x \left (b \,x^{2}+a \right )}}\)	\(590\)

Input:

int(x^(3/2)/(b*x^3+a*x)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

1/70*(x*(b*x^2+a))^(1/2)/a^(11/2)*(315*ln(2*(a^(1/2)*(b*x^2+a)^(1/2)+a)/x) 
*b^4*x^8*(b*x^2+a)^(1/2)-315*a^(1/2)*b^4*x^8+945*ln(2*(a^(1/2)*(b*x^2+a)^( 
1/2)+a)/x)*a*b^3*x^6*(b*x^2+a)^(1/2)-1050*a^(3/2)*b^3*x^6+945*ln(2*(a^(1/2 
)*(b*x^2+a)^(1/2)+a)/x)*a^2*b^2*x^4*(b*x^2+a)^(1/2)-1218*a^(5/2)*b^2*x^4+3 
15*ln(2*(a^(1/2)*(b*x^2+a)^(1/2)+a)/x)*a^3*b*x^2*(b*x^2+a)^(1/2)-528*a^(7/ 
2)*b*x^2-35*a^(9/2))/x^(5/2)/(b*x^2+a)^4
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.47 \[ \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\left [\frac {315 \, {\left (b^{5} x^{11} + 4 \, a b^{4} x^{9} + 6 \, a^{2} b^{3} x^{7} + 4 \, a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt {a} \log \left (\frac {b x^{3} + 2 \, a x + 2 \, \sqrt {b x^{3} + a x} \sqrt {a} \sqrt {x}}{x^{3}}\right ) - 2 \, {\left (315 \, a b^{4} x^{8} + 1050 \, a^{2} b^{3} x^{6} + 1218 \, a^{3} b^{2} x^{4} + 528 \, a^{4} b x^{2} + 35 \, a^{5}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{140 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}}, -\frac {315 \, {\left (b^{5} x^{11} + 4 \, a b^{4} x^{9} + 6 \, a^{2} b^{3} x^{7} + 4 \, a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {x}}{\sqrt {b x^{3} + a x}}\right ) + {\left (315 \, a b^{4} x^{8} + 1050 \, a^{2} b^{3} x^{6} + 1218 \, a^{3} b^{2} x^{4} + 528 \, a^{4} b x^{2} + 35 \, a^{5}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{70 \, {\left (a^{6} b^{4} x^{11} + 4 \, a^{7} b^{3} x^{9} + 6 \, a^{8} b^{2} x^{7} + 4 \, a^{9} b x^{5} + a^{10} x^{3}\right )}}\right ] \] Input:

integrate(x^(3/2)/(b*x^3+a*x)^(9/2),x, algorithm="fricas")
 

Output:

[1/140*(315*(b^5*x^11 + 4*a*b^4*x^9 + 6*a^2*b^3*x^7 + 4*a^3*b^2*x^5 + a^4* 
b*x^3)*sqrt(a)*log((b*x^3 + 2*a*x + 2*sqrt(b*x^3 + a*x)*sqrt(a)*sqrt(x))/x 
^3) - 2*(315*a*b^4*x^8 + 1050*a^2*b^3*x^6 + 1218*a^3*b^2*x^4 + 528*a^4*b*x 
^2 + 35*a^5)*sqrt(b*x^3 + a*x)*sqrt(x))/(a^6*b^4*x^11 + 4*a^7*b^3*x^9 + 6* 
a^8*b^2*x^7 + 4*a^9*b*x^5 + a^10*x^3), -1/70*(315*(b^5*x^11 + 4*a*b^4*x^9 
+ 6*a^2*b^3*x^7 + 4*a^3*b^2*x^5 + a^4*b*x^3)*sqrt(-a)*arctan(sqrt(-a)*sqrt 
(x)/sqrt(b*x^3 + a*x)) + (315*a*b^4*x^8 + 1050*a^2*b^3*x^6 + 1218*a^3*b^2* 
x^4 + 528*a^4*b*x^2 + 35*a^5)*sqrt(b*x^3 + a*x)*sqrt(x))/(a^6*b^4*x^11 + 4 
*a^7*b^3*x^9 + 6*a^8*b^2*x^7 + 4*a^9*b*x^5 + a^10*x^3)]
 

Sympy [F]

\[ \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {x^{\frac {3}{2}}}{\left (x \left (a + b x^{2}\right )\right )^{\frac {9}{2}}}\, dx \] Input:

integrate(x**(3/2)/(b*x**3+a*x)**(9/2),x)
 

Output:

Integral(x**(3/2)/(x*(a + b*x**2))**(9/2), x)
 

Maxima [F]

\[ \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int { \frac {x^{\frac {3}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}} \,d x } \] Input:

integrate(x^(3/2)/(b*x^3+a*x)^(9/2),x, algorithm="maxima")
 

Output:

integrate(x^(3/2)/(b*x^3 + a*x)^(9/2), x)
 

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.65 \[ \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=-\frac {9 \, b \arctan \left (\frac {\sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a} a^{5}} - \frac {\sqrt {b x^{2} + a}}{2 \, a^{5} x^{2}} - \frac {140 \, {\left (b x^{2} + a\right )}^{3} b + 35 \, {\left (b x^{2} + a\right )}^{2} a b + 14 \, {\left (b x^{2} + a\right )} a^{2} b + 5 \, a^{3} b}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{5}} \] Input:

integrate(x^(3/2)/(b*x^3+a*x)^(9/2),x, algorithm="giac")
 

Output:

-9/2*b*arctan(sqrt(b*x^2 + a)/sqrt(-a))/(sqrt(-a)*a^5) - 1/2*sqrt(b*x^2 + 
a)/(a^5*x^2) - 1/35*(140*(b*x^2 + a)^3*b + 35*(b*x^2 + a)^2*a*b + 14*(b*x^ 
2 + a)*a^2*b + 5*a^3*b)/((b*x^2 + a)^(7/2)*a^5)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {x^{3/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \] Input:

int(x^(3/2)/(a*x + b*x^3)^(9/2),x)
 

Output:

int(x^(3/2)/(a*x + b*x^3)^(9/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 472, normalized size of antiderivative = 2.97 \[ \int \frac {x^{3/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {-35 \sqrt {b \,x^{2}+a}\, a^{5}-528 \sqrt {b \,x^{2}+a}\, a^{4} b \,x^{2}-1218 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} x^{4}-1050 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{6}-315 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{8}-315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b \,x^{2}-1260 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} x^{4}-1890 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{3} x^{6}-1260 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} x^{8}-315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}-\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{10}+315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b \,x^{2}+1260 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b^{2} x^{4}+1890 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{3} x^{6}+1260 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{4} x^{8}+315 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{5} x^{10}}{70 a^{6} x^{2} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:

int(x^(3/2)/(b*x^3+a*x)^(9/2),x)
 

Output:

( - 35*sqrt(a + b*x**2)*a**5 - 528*sqrt(a + b*x**2)*a**4*b*x**2 - 1218*sqr 
t(a + b*x**2)*a**3*b**2*x**4 - 1050*sqrt(a + b*x**2)*a**2*b**3*x**6 - 315* 
sqrt(a + b*x**2)*a*b**4*x**8 - 315*sqrt(a)*log((sqrt(a + b*x**2) - sqrt(a) 
 + sqrt(b)*x)/sqrt(a))*a**4*b*x**2 - 1260*sqrt(a)*log((sqrt(a + b*x**2) - 
sqrt(a) + sqrt(b)*x)/sqrt(a))*a**3*b**2*x**4 - 1890*sqrt(a)*log((sqrt(a + 
b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a**2*b**3*x**6 - 1260*sqrt(a)*log( 
(sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*a*b**4*x**8 - 315*sqrt(a 
)*log((sqrt(a + b*x**2) - sqrt(a) + sqrt(b)*x)/sqrt(a))*b**5*x**10 + 315*s 
qrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a**4*b*x**2 + 
 1260*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt(a))*a**3*b 
**2*x**4 + 1890*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b)*x)/sqrt( 
a))*a**2*b**3*x**6 + 1260*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + sqrt(b 
)*x)/sqrt(a))*a*b**4*x**8 + 315*sqrt(a)*log((sqrt(a + b*x**2) + sqrt(a) + 
sqrt(b)*x)/sqrt(a))*b**5*x**10)/(70*a**6*x**2*(a**4 + 4*a**3*b*x**2 + 6*a* 
*2*b**2*x**4 + 4*a*b**3*x**6 + b**4*x**8))