\(\int \frac {1}{(a+b x^{3/2})^3} \, dx\) [63]

Optimal result

Integrand size = 11, antiderivative size = 165 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^3} \, dx=\frac {x}{3 a \left (a+b x^{3/2}\right )^2}+\frac {4 x}{9 a^2 \left (a+b x^{3/2}\right )}-\frac {4 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{2/3}}-\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{27 a^{7/3} b^{2/3}}+\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x\right )}{27 a^{7/3} b^{2/3}} \] Output:

1/3*x/a/(a+b*x^(3/2))^2+4/9*x/a^2/(a+b*x^(3/2))-4/27*arctan(1/3*(a^(1/3)-2 
*b^(1/3)*x^(1/2))*3^(1/2)/a^(1/3))*3^(1/2)/a^(7/3)/b^(2/3)-4/27*ln(a^(1/3) 
+b^(1/3)*x^(1/2))/a^(7/3)/b^(2/3)+2/27*ln(a^(2/3)-a^(1/3)*b^(1/3)*x^(1/2)+ 
b^(2/3)*x)/a^(7/3)/b^(2/3)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^3} \, dx=\frac {\frac {3 \sqrt [3]{a} x \left (7 a+4 b x^{3/2}\right )}{\left (a+b x^{3/2}\right )^2}-\frac {4 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}-\frac {4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{b^{2/3}}+\frac {2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x\right )}{b^{2/3}}}{27 a^{7/3}} \] Input:

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

Output:

((3*a^(1/3)*x*(7*a + 4*b*x^(3/2)))/(a + b*x^(3/2))^2 - (4*Sqrt[3]*ArcTan[( 
1 - (2*b^(1/3)*Sqrt[x])/a^(1/3))/Sqrt[3]])/b^(2/3) - (4*Log[a^(1/3) + b^(1 
/3)*Sqrt[x]])/b^(2/3) + (2*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sqrt[x] + b^(2/3) 
*x])/b^(2/3))/(27*a^(7/3))
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {774, 819, 819, 821, 16, 1142, 25, 27, 1082, 217, 1103}

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.

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

\(\Big \downarrow \) 774

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

\(\Big \downarrow \) 819

\(\displaystyle 2 \left (\frac {2 \int \frac {\sqrt {x}}{\left (b x^{3/2}+a\right )^2}d\sqrt {x}}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )\)

\(\Big \downarrow \) 819

\(\displaystyle 2 \left (\frac {2 \left (\frac {\int \frac {\sqrt {x}}{b x^{3/2}+a}d\sqrt {x}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )\)

\(\Big \downarrow \) 821

\(\displaystyle 2 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle 2 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )\)

\(\Big \downarrow \) 1142

\(\displaystyle 2 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}\right )}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}\right )}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle 2 \left (\frac {2 \left (\frac {\frac {\frac {3 \int \frac {1}{-x-3}d\left (1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle 2 \left (\frac {2 \left (\frac {\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \sqrt {x}}{b^{2/3} x-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+a^{2/3}}d\sqrt {x}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle 2 \left (\frac {2 \left (\frac {\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x}{3 a \left (a+b x^{3/2}\right )}\right )}{3 a}+\frac {x}{6 a \left (a+b x^{3/2}\right )^2}\right )\)

Input:

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

Output:

2*(x/(6*a*(a + b*x^(3/2))^2) + (2*(x/(3*a*(a + b*x^(3/2))) + (-1/3*Log[a^( 
1/3) + b^(1/3)*Sqrt[x]]/(a^(1/3)*b^(2/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^( 
1/3)*Sqrt[x])/a^(1/3))/Sqrt[3]])/b^(1/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)* 
Sqrt[x] + b^(2/3)*x]/(2*b^(1/3)))/(3*a^(1/3)*b^(1/3)))/(3*a)))/(3*a))
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 774

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, 
Simp[k   Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre 
eQ[{a, b, p}, x] && FractionQ[n]
 

rule 819

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( 
c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 
1) + 1)/(a*n*(p + 1))   Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a 
, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 821

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 
1)   Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) 
 Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 
*x^2), x], x] /; FreeQ[{a, b}, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {x}{3 a \left (a +b \,x^{\frac {3}{2}}\right )^{2}}+\frac {\frac {4 x}{9 a \left (a +b \,x^{\frac {3}{2}}\right )}+\frac {4 \left (-\frac {\ln \left (\sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x -\left (\frac {a}{b}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a}}{a}\)	\(137\)
default	\(\text {Expression too large to display}\)	\(727\)

Input:

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

Output:

1/3*x/a/(a+b*x^(3/2))^2+4/3/a*(1/3*x/a/(a+b*x^(3/2))+1/3/a*(-1/3/b/(a/b)^( 
1/3)*ln(x^(1/2)+(a/b)^(1/3))+1/6/b/(a/b)^(1/3)*ln(x-(a/b)^(1/3)*x^(1/2)+(a 
/b)^(2/3))+1/3*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x^( 
1/2)-1))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (114) = 228\).

Time = 0.10 (sec) , antiderivative size = 664, normalized size of antiderivative = 4.02 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/27*(3*a^2*b^4*x^4 - 21*a^4*b^2*x - 6*sqrt(1/3)*(a*b^5*x^6 - 2*a^3*b^3* 
x^3 + a^5*b)*sqrt((-a*b^2)^(1/3)/a)*log((2*b^3*x^3 - 3*(-a*b^2)^(2/3)*b*x^ 
2 + a^2*b + 3*sqrt(1/3)*(a*b^2*x^2 - 2*(-a*b^2)^(2/3)*a*x - (-a*b^2)^(1/3) 
*a^2 + (2*(-a*b^2)^(2/3)*b*x^2 + (-a*b^2)^(1/3)*a*b*x - a^2*b)*sqrt(x))*sq 
rt((-a*b^2)^(1/3)/a) - 3*(a*b^2*x - (-a*b^2)^(2/3)*a)*sqrt(x))/(b^2*x^3 - 
a^2)) - 2*(b^4*x^6 - 2*a^2*b^2*x^3 + a^4)*(-a*b^2)^(2/3)*log(b^2*x + (-a*b 
^2)^(1/3)*b*sqrt(x) + (-a*b^2)^(2/3)) + 4*(b^4*x^6 - 2*a^2*b^2*x^3 + a^4)* 
(-a*b^2)^(2/3)*log(b*sqrt(x) - (-a*b^2)^(1/3)) - 6*(2*a*b^5*x^5 - 5*a^3*b^ 
3*x^2)*sqrt(x))/(a^3*b^6*x^6 - 2*a^5*b^4*x^3 + a^7*b^2), -1/27*(3*a^2*b^4* 
x^4 - 21*a^4*b^2*x - 12*sqrt(1/3)*(a*b^5*x^6 - 2*a^3*b^3*x^3 + a^5*b)*sqrt 
(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*sqrt(x) + (-a*b^2)^(1/3))*sqrt(- 
(-a*b^2)^(1/3)/a)/b) - 2*(b^4*x^6 - 2*a^2*b^2*x^3 + a^4)*(-a*b^2)^(2/3)*lo 
g(b^2*x + (-a*b^2)^(1/3)*b*sqrt(x) + (-a*b^2)^(2/3)) + 4*(b^4*x^6 - 2*a^2* 
b^2*x^3 + a^4)*(-a*b^2)^(2/3)*log(b*sqrt(x) - (-a*b^2)^(1/3)) - 6*(2*a*b^5 
*x^5 - 5*a^3*b^3*x^2)*sqrt(x))/(a^3*b^6*x^6 - 2*a^5*b^4*x^3 + a^7*b^2)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 896 vs. \(2 (155) = 310\).

Time = 108.28 (sec) , antiderivative size = 896, normalized size of antiderivative = 5.43 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (x/a**3, Eq(b, 0)), (-2/(7* 
b**3*x**(7/2)), Eq(a, 0)), (21*a**2*x/(27*a**5 + 54*a**4*b*x**(3/2) + 27*a 
**3*b**2*x**3) - 4*a**2*(-a/b)**(2/3)*log(sqrt(x) - (-a/b)**(1/3))/(27*a** 
5 + 54*a**4*b*x**(3/2) + 27*a**3*b**2*x**3) + 2*a**2*(-a/b)**(2/3)*log(4*s 
qrt(x)*(-a/b)**(1/3) + 4*x + 4*(-a/b)**(2/3))/(27*a**5 + 54*a**4*b*x**(3/2 
) + 27*a**3*b**2*x**3) - 4*sqrt(3)*a**2*(-a/b)**(2/3)*atan(2*sqrt(3)*sqrt( 
x)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(27*a**5 + 54*a**4*b*x**(3/2) + 27*a**3* 
b**2*x**3) - 4*a**2*(-a/b)**(2/3)*log(2)/(27*a**5 + 54*a**4*b*x**(3/2) + 2 
7*a**3*b**2*x**3) + 12*a*b*x**(5/2)/(27*a**5 + 54*a**4*b*x**(3/2) + 27*a** 
3*b**2*x**3) - 8*a*b*x**(3/2)*(-a/b)**(2/3)*log(sqrt(x) - (-a/b)**(1/3))/( 
27*a**5 + 54*a**4*b*x**(3/2) + 27*a**3*b**2*x**3) + 4*a*b*x**(3/2)*(-a/b)* 
*(2/3)*log(4*sqrt(x)*(-a/b)**(1/3) + 4*x + 4*(-a/b)**(2/3))/(27*a**5 + 54* 
a**4*b*x**(3/2) + 27*a**3*b**2*x**3) - 8*sqrt(3)*a*b*x**(3/2)*(-a/b)**(2/3 
)*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/3)) + sqrt(3)/3)/(27*a**5 + 54*a**4 
*b*x**(3/2) + 27*a**3*b**2*x**3) - 8*a*b*x**(3/2)*(-a/b)**(2/3)*log(2)/(27 
*a**5 + 54*a**4*b*x**(3/2) + 27*a**3*b**2*x**3) - 4*b**2*x**3*(-a/b)**(2/3 
)*log(sqrt(x) - (-a/b)**(1/3))/(27*a**5 + 54*a**4*b*x**(3/2) + 27*a**3*b** 
2*x**3) + 2*b**2*x**3*(-a/b)**(2/3)*log(4*sqrt(x)*(-a/b)**(1/3) + 4*x + 4* 
(-a/b)**(2/3))/(27*a**5 + 54*a**4*b*x**(3/2) + 27*a**3*b**2*x**3) - 4*sqrt 
(3)*b**2*x**3*(-a/b)**(2/3)*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/3)) + ...
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^3} \, dx=\frac {4 \, b x^{\frac {5}{2}} + 7 \, a x}{9 \, {\left (a^{2} b^{2} x^{3} + 2 \, a^{3} b x^{\frac {3}{2}} + a^{4}\right )}} + \frac {4 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {2 \, \log \left (x - \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {4 \, \log \left (\sqrt {x} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:

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

Output:

1/9*(4*b*x^(5/2) + 7*a*x)/(a^2*b^2*x^3 + 2*a^3*b*x^(3/2) + a^4) + 4/27*sqr 
t(3)*arctan(1/3*sqrt(3)*(2*sqrt(x) - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b*(a/b 
)^(1/3)) + 2/27*log(x - sqrt(x)*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a/b)^(1 
/3)) - 4/27*log(sqrt(x) + (a/b)^(1/3))/(a^2*b*(a/b)^(1/3))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^3} \, dx=-\frac {4 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | \sqrt {x} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3}} - \frac {4 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, \sqrt {x} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} b^{2}} + \frac {4 \, b x^{\frac {5}{2}} + 7 \, a x}{9 \, {\left (b x^{\frac {3}{2}} + a\right )}^{2} a^{2}} + \frac {2 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x + \sqrt {x} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{3} b^{2}} \] Input:

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

Output:

-4/27*(-a/b)^(2/3)*log(abs(sqrt(x) - (-a/b)^(1/3)))/a^3 - 4/27*sqrt(3)*(-a 
*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*sqrt(x) + (-a/b)^(1/3))/(-a/b)^(1/3))/(a 
^3*b^2) + 1/9*(4*b*x^(5/2) + 7*a*x)/((b*x^(3/2) + a)^2*a^2) + 2/27*(-a*b^2 
)^(2/3)*log(x + sqrt(x)*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^2)
 

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^3} \, dx=\frac {\frac {7\,x}{9\,a}+\frac {4\,b\,x^{5/2}}{9\,a^2}}{a^2+b^2\,x^3+2\,a\,b\,x^{3/2}}+\frac {4\,\ln \left (\frac {16\,b\,\sqrt {x}}{81\,a^4}-\frac {16\,b^{2/3}}{81\,{\left (-a\right )}^{11/3}}\right )}{27\,{\left (-a\right )}^{7/3}\,b^{2/3}}+\frac {\ln \left (\frac {16\,b\,\sqrt {x}}{81\,a^4}-\frac {b^{2/3}\,{\left (-2+\sqrt {3}\,2{}\mathrm {i}\right )}^2}{81\,{\left (-a\right )}^{11/3}}\right )\,\left (-2+\sqrt {3}\,2{}\mathrm {i}\right )}{27\,{\left (-a\right )}^{7/3}\,b^{2/3}}-\frac {\ln \left (\frac {16\,b\,\sqrt {x}}{81\,a^4}-\frac {b^{2/3}\,{\left (2+\sqrt {3}\,2{}\mathrm {i}\right )}^2}{81\,{\left (-a\right )}^{11/3}}\right )\,\left (2+\sqrt {3}\,2{}\mathrm {i}\right )}{27\,{\left (-a\right )}^{7/3}\,b^{2/3}} \] Input:

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

Output:

((7*x)/(9*a) + (4*b*x^(5/2))/(9*a^2))/(a^2 + b^2*x^3 + 2*a*b*x^(3/2)) + (4 
*log((16*b*x^(1/2))/(81*a^4) - (16*b^(2/3))/(81*(-a)^(11/3))))/(27*(-a)^(7 
/3)*b^(2/3)) + (log((16*b*x^(1/2))/(81*a^4) - (b^(2/3)*(3^(1/2)*2i - 2)^2) 
/(81*(-a)^(11/3)))*(3^(1/2)*2i - 2))/(27*(-a)^(7/3)*b^(2/3)) - (log((16*b* 
x^(1/2))/(81*a^4) - (b^(2/3)*(3^(1/2)*2i + 2)^2)/(81*(-a)^(11/3)))*(3^(1/2 
)*2i + 2))/(27*(-a)^(7/3)*b^(2/3))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.64 \[ \int \frac {1}{\left (a+b x^{3/2}\right )^3} \, dx=\frac {-8 \sqrt {x}\, \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 \sqrt {x}\, b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a b x -4 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 \sqrt {x}\, b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2}-4 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 \sqrt {x}\, b^{\frac {1}{3}}}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} x^{3}+12 \sqrt {x}\, b^{\frac {5}{3}} a^{\frac {1}{3}} x^{2}+21 b^{\frac {2}{3}} a^{\frac {4}{3}} x +4 \sqrt {x}\, \mathrm {log}\left (a^{\frac {2}{3}}-\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}} x \right ) a b x -8 \sqrt {x}\, \mathrm {log}\left (a^{\frac {1}{3}}+\sqrt {x}\, b^{\frac {1}{3}}\right ) a b x +2 \,\mathrm {log}\left (a^{\frac {2}{3}}-\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}} x \right ) a^{2}+2 \,\mathrm {log}\left (a^{\frac {2}{3}}-\sqrt {x}\, b^{\frac {1}{3}} a^{\frac {1}{3}}+b^{\frac {2}{3}} x \right ) b^{2} x^{3}-4 \,\mathrm {log}\left (a^{\frac {1}{3}}+\sqrt {x}\, b^{\frac {1}{3}}\right ) a^{2}-4 \,\mathrm {log}\left (a^{\frac {1}{3}}+\sqrt {x}\, b^{\frac {1}{3}}\right ) b^{2} x^{3}}{27 b^{\frac {2}{3}} a^{\frac {7}{3}} \left (2 \sqrt {x}\, a b x +a^{2}+b^{2} x^{3}\right )} \] Input:

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

Output:

( - 8*sqrt(x)*sqrt(3)*atan((a**(1/3) - 2*sqrt(x)*b**(1/3))/(a**(1/3)*sqrt( 
3)))*a*b*x - 4*sqrt(3)*atan((a**(1/3) - 2*sqrt(x)*b**(1/3))/(a**(1/3)*sqrt 
(3)))*a**2 - 4*sqrt(3)*atan((a**(1/3) - 2*sqrt(x)*b**(1/3))/(a**(1/3)*sqrt 
(3)))*b**2*x**3 + 12*sqrt(x)*b**(2/3)*a**(1/3)*b*x**2 + 21*b**(2/3)*a**(1/ 
3)*a*x + 4*sqrt(x)*log(a**(2/3) - sqrt(x)*b**(1/3)*a**(1/3) + b**(2/3)*x)* 
a*b*x - 8*sqrt(x)*log(a**(1/3) + sqrt(x)*b**(1/3))*a*b*x + 2*log(a**(2/3) 
- sqrt(x)*b**(1/3)*a**(1/3) + b**(2/3)*x)*a**2 + 2*log(a**(2/3) - sqrt(x)* 
b**(1/3)*a**(1/3) + b**(2/3)*x)*b**2*x**3 - 4*log(a**(1/3) + sqrt(x)*b**(1 
/3))*a**2 - 4*log(a**(1/3) + sqrt(x)*b**(1/3))*b**2*x**3)/(27*b**(2/3)*a** 
(1/3)*a**2*(2*sqrt(x)*a*b*x + a**2 + b**2*x**3))