\(\int \frac {x}{\sqrt {b \sqrt [3]{x}+a x}} \, dx\) [131]

Optimal result

Integrand size = 17, antiderivative size = 326 \[ \int \frac {x}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {14 b^2 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 a^{5/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {14 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^2}+\frac {2 x \sqrt {b \sqrt [3]{x}+a x}}{3 a}-\frac {14 b^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{11/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {7 b^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{5 a^{11/4} \sqrt {b \sqrt [3]{x}+a x}} \] Output:

14/5*b^2*(b+a*x^(2/3))*x^(1/3)/a^(5/2)/(b^(1/2)+a^(1/2)*x^(1/3))/(b*x^(1/3 
)+a*x)^(1/2)-14/15*b*x^(1/3)*(b*x^(1/3)+a*x)^(1/2)/a^2+2/3*x*(b*x^(1/3)+a* 
x)^(1/2)/a-14/5*b^(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))/a^(11/4)/(b*x^(1/3)+a*x)^(1/2)+7/5*b^(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))/a^(11/4)/( 
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 = 10.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.33 \[ \int \frac {x}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {2 \sqrt {b \sqrt [3]{x}+a x} \left (-7 b^2 \sqrt [3]{x}-2 a b x+5 a^2 x^{5/3}+7 b^2 \sqrt {1+\frac {a x^{2/3}}{b}} \sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {a x^{2/3}}{b}\right )\right )}{15 a^2 \left (b+a x^{2/3}\right )} \] Input:

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

Output:

(2*Sqrt[b*x^(1/3) + a*x]*(-7*b^2*x^(1/3) - 2*a*b*x + 5*a^2*x^(5/3) + 7*b^2 
*Sqrt[1 + (a*x^(2/3))/b]*x^(1/3)*Hypergeometric2F1[1/2, 3/4, 7/4, -((a*x^( 
2/3))/b)]))/(15*a^2*(b + a*x^(2/3)))
 

Rubi [A] (warning: unable to verify)

Time = 0.72 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {1924, 1930, 1930, 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.

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

\(\Big \downarrow \) 1924

\(\displaystyle 3 \int \frac {x^{5/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}\)

\(\Big \downarrow \) 1930

\(\displaystyle 3 \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \int \frac {x}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{9 a}\right )\)

\(\Big \downarrow \) 1930

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

\(\Big \downarrow \) 1938

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

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

Input:

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

Output:

3*((2*x*Sqrt[b*x^(1/3) + a*x])/(9*a) - (7*b*((2*x^(1/3)*Sqrt[b*x^(1/3) + a 
*x])/(5*a) - (6*b*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)*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)])))/(5*a*Sqrt[b*x^(1/3) + a*x])))/(9*a))
 

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 1930

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p 
+ 1))), x] - Simp[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*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]) && Gt 
Q[m + j*p - n + j + 1, 0] && NeQ[m + n*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 = 1.05 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.64

method	result	size
derivativedivides	\(\frac {2 x \sqrt {b \,x^{\frac {1}{3}}+a x}}{3 a}-\frac {14 b \,x^{\frac {1}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{15 a^{2}}+\frac {7 b^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{5 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(210\)
default	\(-\frac {-42 b^{3} \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+21 b^{3} \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+14 x^{\frac {2}{3}} a \,b^{2}+4 x^{\frac {4}{3}} a^{2} b -10 a^{3} x^{2}}{15 a^{3} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\)	\(228\)

Input:

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

Output:

2/3*x*(b*x^(1/3)+a*x)^(1/2)/a-14/15*b*x^(1/3)*(b*x^(1/3)+a*x)^(1/2)/a^2+7/ 
5*b^2/a^3*(-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)*Ellipti 
cF(((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2),1/2*2^(1/2)))
 

Fricas [F]

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

integrate(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^2 + b^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {-14 \sqrt {x}\, \sqrt {x^{\frac {2}{3}} a +b}\, b +10 x^{\frac {7}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a +7 \left (\int \frac {x^{\frac {1}{6}} \sqrt {x^{\frac {2}{3}} a +b}}{x^{\frac {2}{3}} b +x^{\frac {4}{3}} a}d x \right ) b^{2}}{15 a^{2}} \] Input:

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

Output:

( - 14*sqrt(x)*sqrt(x**(2/3)*a + b)*b + 10*x**(1/6)*sqrt(x**(2/3)*a + b)*a 
*x + 7*int((x**(1/6)*sqrt(x**(2/3)*a + b))/(x**(2/3)*b + x**(1/3)*a*x),x)* 
b**2)/(15*a**2)