\(\int \frac {1}{x \sqrt {b \sqrt [3]{x}+a x}} \, dx\) [133]

Optimal result

Integrand size = 19, antiderivative size = 294 \[ \int \frac {1}{x \sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {6 \sqrt {a} \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{b \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{b \sqrt [3]{x}}-\frac {6 \sqrt [4]{a} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{b^{3/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {3 \sqrt [4]{a} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{b^{3/4} \sqrt {b \sqrt [3]{x}+a x}} \] Output:

6*a^(1/2)*(b+a*x^(2/3))*x^(1/3)/b/(b^(1/2)+a^(1/2)*x^(1/3))/(b*x^(1/3)+a*x 
)^(1/2)-6*(b*x^(1/3)+a*x)^(1/2)/b/x^(1/3)-6*a^(1/4)*(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(si 
n(2*arctan(a^(1/4)*x^(1/6)/b^(1/4))),1/2*2^(1/2))/b^(3/4)/(b*x^(1/3)+a*x)^ 
(1/2)+3*a^(1/4)*(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^(3/4)/(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 = 10.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.18 \[ \int \frac {1}{x \sqrt {b \sqrt [3]{x}+a x}} \, dx=-\frac {6 \sqrt {1+\frac {a x^{2/3}}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\frac {a x^{2/3}}{b}\right )}{\sqrt {b \sqrt [3]{x}+a x}} \] Input:

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

Output:

(-6*Sqrt[1 + (a*x^(2/3))/b]*Hypergeometric2F1[-1/4, 1/2, 3/4, -((a*x^(2/3) 
)/b)])/Sqrt[b*x^(1/3) + a*x]
 

Rubi [A] (warning: unable to verify)

Time = 0.68 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {1924, 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.

\(\displaystyle \int \frac {1}{x \sqrt {a x+b \sqrt [3]{x}}} \, dx\)

\(\Big \downarrow \) 1924

\(\displaystyle 3 \int \frac {1}{\sqrt [3]{x} \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}\)

\(\Big \downarrow \) 1931

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

\(\Big \downarrow \) 1938

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

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

Input:

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

Output:

3*((-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]*EllipticE[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(a^(1/4)*S 
qrt[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]))
 

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 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 = 0.82 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.66

method	result	size
derivativedivides	\(-\frac {6 \left (b +a \,x^{\frac {2}{3}}\right )}{b \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}+\frac {3 \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{b \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(195\)
default	\(-\frac {3 \left (-2 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b +\sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) b +2 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {2}{3}} a +2 \sqrt {b \,x^{\frac {1}{3}}+a x}\, b \right )}{x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right ) b}\)	\(253\)

Input:

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

Output:

-6*(b+a*x^(2/3))/b/(x^(1/3)*(b+a*x^(2/3)))^(1/2)+3/b*(-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]

\[ \int \frac {1}{x \sqrt {b \sqrt [3]{x}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b x^{\frac {1}{3}}} x} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {1}{x \sqrt {b \sqrt [3]{x}+a x}} \, dx=\int \frac {1}{x \sqrt {a x + b \sqrt [3]{x}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{x \sqrt {b \sqrt [3]{x}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b x^{\frac {1}{3}}} x} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{x \sqrt {b \sqrt [3]{x}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b x^{\frac {1}{3}}} x} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x \sqrt {b \sqrt [3]{x}+a x}} \, dx=\int \frac {\sqrt {x^{\frac {1}{3}} b +a x}}{x^{\frac {4}{3}} b +a \,x^{2}}d x \] Input:

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

Output:

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