\(\int \frac {1}{x^2 (b \sqrt [3]{x}+a x)^{3/2}} \, dx\) [143]

Optimal result

Integrand size = 19, antiderivative size = 383 \[ \int \frac {1}{x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\frac {3}{b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}+\frac {77 a^{5/2} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 b^4 \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {11 \sqrt {b \sqrt [3]{x}+a x}}{3 b^2 x^{5/3}}+\frac {77 a \sqrt {b \sqrt [3]{x}+a x}}{15 b^3 x}-\frac {77 a^2 \sqrt {b \sqrt [3]{x}+a x}}{5 b^4 \sqrt [3]{x}}-\frac {77 a^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {77 a^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{10 b^{15/4} \sqrt {b \sqrt [3]{x}+a x}} \] Output:

3/b/x^(4/3)/(b*x^(1/3)+a*x)^(1/2)+77/5*a^(5/2)*(b+a*x^(2/3))*x^(1/3)/b^4/( 
b^(1/2)+a^(1/2)*x^(1/3))/(b*x^(1/3)+a*x)^(1/2)-11/3*(b*x^(1/3)+a*x)^(1/2)/ 
b^2/x^(5/3)+77/15*a*(b*x^(1/3)+a*x)^(1/2)/b^3/x-77/5*a^2*(b*x^(1/3)+a*x)^( 
1/2)/b^4/x^(1/3)-77/5*a^(9/4)*(b^(1/2)+a^(1/2)*x^(1/3))*((b+a*x^(2/3))/(b^ 
(1/2)+a^(1/2)*x^(1/3))^2)^(1/2)*x^(1/6)*EllipticE(sin(2*arctan(a^(1/4)*x^( 
1/6)/b^(1/4))),1/2*2^(1/2))/b^(15/4)/(b*x^(1/3)+a*x)^(1/2)+77/10*a^(9/4)*( 
b^(1/2)+a^(1/2)*x^(1/3))*((b+a*x^(2/3))/(b^(1/2)+a^(1/2)*x^(1/3))^2)^(1/2) 
*x^(1/6)*InverseJacobiAM(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)),1/2*2^(1/2))/b^ 
(15/4)/(b*x^(1/3)+a*x)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.17 \[ \int \frac {1}{x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=-\frac {2 \sqrt {1+\frac {a x^{2/3}}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {9}{4},\frac {3}{2},-\frac {5}{4},-\frac {a x^{2/3}}{b}\right )}{3 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}} \] Input:

Integrate[1/(x^2*(b*x^(1/3) + a*x)^(3/2)),x]
 

Output:

(-2*Sqrt[1 + (a*x^(2/3))/b]*Hypergeometric2F1[-9/4, 3/2, -5/4, -((a*x^(2/3 
))/b)])/(3*b*x^(4/3)*Sqrt[b*x^(1/3) + a*x])
 

Rubi [A] (warning: unable to verify)

Time = 0.96 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {1924, 1929, 1931, 1931, 1931, 1938, 266, 834, 27, 761, 1510}

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.

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

\(\Big \downarrow \) 1924

\(\displaystyle 3 \int \frac {1}{x^{4/3} \left (\sqrt [3]{x} b+a x\right )^{3/2}}d\sqrt [3]{x}\)

\(\Big \downarrow \) 1929

\(\displaystyle 3 \left (\frac {11 \int \frac {1}{x^{5/3} \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{2 b}+\frac {1}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\)

\(\Big \downarrow \) 1931

\(\displaystyle 3 \left (\frac {11 \left (-\frac {7 a \int \frac {1}{x \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{2 b}+\frac {1}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\)

\(\Big \downarrow \) 1931

\(\displaystyle 3 \left (\frac {11 \left (-\frac {7 a \left (-\frac {3 a \int \frac {1}{\sqrt [3]{x} \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{2 b}+\frac {1}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\)

\(\Big \downarrow \) 1931

\(\displaystyle 3 \left (\frac {11 \left (-\frac {7 a \left (-\frac {3 a \left (\frac {a \int \frac {\sqrt [3]{x}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{2 b}+\frac {1}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\)

\(\Big \downarrow \) 1938

\(\displaystyle 3 \left (\frac {11 \left (-\frac {7 a \left (-\frac {3 a \left (\frac {a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {\sqrt [6]{x}}{\sqrt {x^{2/3} a+b}}d\sqrt [3]{x}}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{2 b}+\frac {1}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\)

\(\Big \downarrow \) 266

\(\displaystyle 3 \left (\frac {11 \left (-\frac {7 a \left (-\frac {3 a \left (\frac {2 a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{2 b}+\frac {1}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\)

\(\Big \downarrow \) 834

\(\displaystyle 3 \left (\frac {11 \left (-\frac {7 a \left (-\frac {3 a \left (\frac {2 a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}-\frac {\sqrt {b} \int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {b} \sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{2 b}+\frac {1}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \left (\frac {11 \left (-\frac {7 a \left (-\frac {3 a \left (\frac {2 a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{2 b}+\frac {1}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\)

\(\Big \downarrow \) 761

\(\displaystyle 3 \left (\frac {11 \left (-\frac {7 a \left (-\frac {3 a \left (\frac {2 a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a x^{4/3}+b}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{2 b}+\frac {1}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\)

\(\Big \downarrow \) 1510

\(\displaystyle 3 \left (\frac {11 \left (-\frac {7 a \left (-\frac {3 a \left (\frac {2 a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a x^{4/3}+b}}-\frac {\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a x^{4/3}+b}}-\frac {\sqrt [6]{x} \sqrt {a x^{4/3}+b}}{\sqrt {a} x^{2/3}+\sqrt {b}}}{\sqrt {a}}\right )}{b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{b \sqrt [3]{x}}\right )}{5 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{5 b x}\right )}{9 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{9 b x^{5/3}}\right )}{2 b}+\frac {1}{b x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}\right )\)

Input:

Int[1/(x^2*(b*x^(1/3) + a*x)^(3/2)),x]
 

Output:

3*(1/(b*x^(4/3)*Sqrt[b*x^(1/3) + a*x]) + (11*((-2*Sqrt[b*x^(1/3) + a*x])/( 
9*b*x^(5/3)) - (7*a*((-2*Sqrt[b*x^(1/3) + a*x])/(5*b*x) - (3*a*((-2*Sqrt[b 
*x^(1/3) + a*x])/(b*x^(1/3)) + (2*a*Sqrt[b + a*x^(2/3)]*x^(1/6)*(-((-((x^( 
1/6)*Sqrt[b + a*x^(4/3)])/(Sqrt[b] + Sqrt[a]*x^(2/3))) + (b^(1/4)*(Sqrt[b] 
 + Sqrt[a]*x^(2/3))*Sqrt[(b + a*x^(4/3))/(Sqrt[b] + Sqrt[a]*x^(2/3))^2]*El 
lipticE[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(a^(1/4)*Sqrt[b + a*x^( 
4/3)]))/Sqrt[a]) + (b^(1/4)*(Sqrt[b] + Sqrt[a]*x^(2/3))*Sqrt[(b + a*x^(4/3 
))/(Sqrt[b] + Sqrt[a]*x^(2/3))^2]*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^( 
1/4)], 1/2])/(2*a^(3/4)*Sqrt[b + a*x^(4/3)])))/(b*Sqrt[b*x^(1/3) + a*x]))) 
/(5*b)))/(9*b)))/(2*b))
 

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 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 834

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S 
imp[1/q   Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q   Int[(1 - q*x^2)/Sqrt[a 
 + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

rule 1924

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

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 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

Maple [A] (verified)

Time = 2.53 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(-\frac {2 \sqrt {b \,x^{\frac {1}{3}}+a x}}{3 b^{2} x^{\frac {5}{3}}}+\frac {32 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{15 b^{3} x}-\frac {62 \left (b +a \,x^{\frac {2}{3}}\right ) a^{2}}{5 b^{4} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}-\frac {3 x^{\frac {2}{3}} a^{3}}{b^{4} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}+\frac {77 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{10 b^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(267\)
default	\(-\frac {-462 a^{2} b \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \operatorname {EllipticE}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+231 a^{2} b \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \operatorname {EllipticF}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+462 x^{\frac {10}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}\, a^{3}+372 x^{\frac {8}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}\, a^{2} b -44 x^{2} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a \,b^{2}-64 x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} b +20 x^{\frac {4}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, b^{3}}{30 x^{3} \left (b +a \,x^{\frac {2}{3}}\right ) b^{4}}\)	\(339\)

Input:

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

Output:

-2/3*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(5/3)+32/15*a*(b*x^(1/3)+a*x)^(1/2)/b^3/x 
-62/5*(b+a*x^(2/3))*a^2/b^4/(x^(1/3)*(b+a*x^(2/3)))^(1/2)-3*x^(2/3)*a^3/b^ 
4/((x^(2/3)+b/a)*x^(1/3)*a)^(1/2)+77/10*a^2/b^4*(-a*b)^(1/2)*((x^(1/3)+1/a 
*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2)*(-2*(x^(1/3)-1/a*(-a*b)^(1/2))*a/(-a* 
b)^(1/2))^(1/2)*(-x^(1/3)/(-a*b)^(1/2)*a)^(1/2)/(b*x^(1/3)+a*x)^(1/2)*(-2/ 
a*(-a*b)^(1/2)*EllipticE(((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2) 
,1/2*2^(1/2))+1/a*(-a*b)^(1/2)*EllipticF(((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a 
*b)^(1/2))^(1/2),1/2*2^(1/2)))
 

Fricas [F]

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

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

Output:

integral((a^4*x^3 + 3*a^2*b^2*x^(5/3) - 2*a*b^3*x - (2*a^3*b*x^2 - b^4)*x^ 
(1/3))*sqrt(a*x + b*x^(1/3))/(a^6*x^7 + 2*a^3*b^3*x^5 + b^6*x^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{\frac {7}{3}} \sqrt {x^{\frac {1}{3}} b +a x}\, b +\sqrt {x^{\frac {1}{3}} b +a x}\, a \,x^{3}}d x \] Input:

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

Output:

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