\(\int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx\) [129]

Optimal result

Integrand size = 19, antiderivative size = 414 \[ \int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=-\frac {418 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 a^{11/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}+\frac {418 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 a^{23/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {209 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{221 a^{23/4} \sqrt {b \sqrt [3]{x}+a x}} \] Output:

-418/221*b^5*(b+a*x^(2/3))*x^(1/3)/a^(11/2)/(b^(1/2)+a^(1/2)*x^(1/3))/(b*x 
^(1/3)+a*x)^(1/2)+418/663*b^4*x^(1/3)*(b*x^(1/3)+a*x)^(1/2)/a^5-2090/4641* 
b^3*x*(b*x^(1/3)+a*x)^(1/2)/a^4+570/1547*b^2*x^(5/3)*(b*x^(1/3)+a*x)^(1/2) 
/a^3-38/119*b*x^(7/3)*(b*x^(1/3)+a*x)^(1/2)/a^2+2/7*x^3*(b*x^(1/3)+a*x)^(1 
/2)/a+418/221*b^(21/4)*(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^(23/4)/(b*x^(1/3)+a*x)^(1/2)-209/221*b^(21/4)*(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^(23/ 
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 = 143, normalized size of antiderivative = 0.35 \[ \int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {2 \sqrt {b \sqrt [3]{x}+a x} \left (1463 b^5 \sqrt [3]{x}+418 a b^4 x-190 a^2 b^3 x^{5/3}+114 a^3 b^2 x^{7/3}-78 a^4 b x^3+663 a^5 x^{11/3}-1463 b^5 \sqrt {1+\frac {a x^{2/3}}{b}} \sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {a x^{2/3}}{b}\right )\right )}{4641 a^5 \left (b+a x^{2/3}\right )} \] Input:

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

Output:

(2*Sqrt[b*x^(1/3) + a*x]*(1463*b^5*x^(1/3) + 418*a*b^4*x - 190*a^2*b^3*x^( 
5/3) + 114*a^3*b^2*x^(7/3) - 78*a^4*b*x^3 + 663*a^5*x^(11/3) - 1463*b^5*Sq 
rt[1 + (a*x^(2/3))/b]*x^(1/3)*Hypergeometric2F1[1/2, 3/4, 7/4, -((a*x^(2/3 
))/b)]))/(4641*a^5*(b + a*x^(2/3)))
 

Rubi [A] (warning: unable to verify)

Time = 1.07 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {1924, 1930, 1930, 1930, 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.

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

\(\Big \downarrow \) 1924

\(\displaystyle 3 \int \frac {x^{11/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}\)

\(\Big \downarrow \) 1930

\(\displaystyle 3 \left (\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{21 a}-\frac {19 b \int \frac {x^3}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{21 a}\right )\)

\(\Big \downarrow \) 1930

\(\displaystyle 3 \left (\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{21 a}-\frac {19 b \left (\frac {2 x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{17 a}-\frac {15 b \int \frac {x^{7/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{17 a}\right )}{21 a}\right )\)

\(\Big \downarrow \) 1930

\(\displaystyle 3 \left (\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{21 a}-\frac {19 b \left (\frac {2 x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{17 a}-\frac {15 b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \int \frac {x^{5/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{13 a}\right )}{17 a}\right )}{21 a}\right )\)

\(\Big \downarrow \) 1930

\(\displaystyle 3 \left (\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{21 a}-\frac {19 b \left (\frac {2 x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{17 a}-\frac {15 b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \int \frac {x}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{9 a}\right )}{13 a}\right )}{17 a}\right )}{21 a}\right )\)

\(\Big \downarrow \) 1930

\(\displaystyle 3 \left (\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{21 a}-\frac {19 b \left (\frac {2 x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{17 a}-\frac {15 b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {3 b \int \frac {\sqrt [3]{x}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{5 a}\right )}{9 a}\right )}{13 a}\right )}{17 a}\right )}{21 a}\right )\)

\(\Big \downarrow \) 1938

\(\displaystyle 3 \left (\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{21 a}-\frac {19 b \left (\frac {2 x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{17 a}-\frac {15 b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {3 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {\sqrt [6]{x}}{\sqrt {x^{2/3} a+b}}d\sqrt [3]{x}}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )}{17 a}\right )}{21 a}\right )\)

\(\Big \downarrow \) 266

\(\displaystyle 3 \left (\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{21 a}-\frac {19 b \left (\frac {2 x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{17 a}-\frac {15 b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {6 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )}{17 a}\right )}{21 a}\right )\)

\(\Big \downarrow \) 834

\(\displaystyle 3 \left (\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{21 a}-\frac {19 b \left (\frac {2 x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{17 a}-\frac {15 b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {6 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}-\frac {\sqrt {b} \int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {b} \sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )}{17 a}\right )}{21 a}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 3 \left (\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{21 a}-\frac {19 b \left (\frac {2 x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{17 a}-\frac {15 b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {6 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )}{17 a}\right )}{21 a}\right )\)

\(\Big \downarrow \) 761

\(\displaystyle 3 \left (\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{21 a}-\frac {19 b \left (\frac {2 x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{17 a}-\frac {15 b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {6 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a x^{4/3}+b}}-\frac {\int \frac {\sqrt {b}-\sqrt {a} x^{2/3}}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{\sqrt {a}}\right )}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )}{17 a}\right )}{21 a}\right )\)

\(\Big \downarrow \) 1510

\(\displaystyle 3 \left (\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{21 a}-\frac {19 b \left (\frac {2 x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{17 a}-\frac {15 b \left (\frac {2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{13 a}-\frac {11 b \left (\frac {2 x \sqrt {a x+b \sqrt [3]{x}}}{9 a}-\frac {7 b \left (\frac {2 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{5 a}-\frac {6 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \left (\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{3/4} \sqrt {a x^{4/3}+b}}-\frac {\frac {\sqrt [4]{b} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a x^{4/3}+b}}-\frac {\sqrt [6]{x} \sqrt {a x^{4/3}+b}}{\sqrt {a} x^{2/3}+\sqrt {b}}}{\sqrt {a}}\right )}{5 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{9 a}\right )}{13 a}\right )}{17 a}\right )}{21 a}\right )\)

Input:

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

Output:

3*((2*x^3*Sqrt[b*x^(1/3) + a*x])/(21*a) - (19*b*((2*x^(7/3)*Sqrt[b*x^(1/3) 
 + a*x])/(17*a) - (15*b*((2*x^(5/3)*Sqrt[b*x^(1/3) + a*x])/(13*a) - (11*b* 
((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)))/(13*a)))/ 
(17*a)))/(21*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 = 2.63 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.63

method	result	size
default	\(-\frac {-228 x^{\frac {8}{3}} a^{4} b^{2}+156 x^{\frac {10}{3}} a^{5} b +380 a^{3} b^{3} x^{2}+8778 b^{6} \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-4389 b^{6} \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-1326 a^{6} x^{4}-2926 x^{\frac {2}{3}} a \,b^{5}-836 x^{\frac {4}{3}} a^{2} b^{4}}{4641 a^{6} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\)	\(261\)
derivativedivides	\(\frac {2 x^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{7 a}-\frac {38 b \,x^{\frac {7}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{119 a^{2}}+\frac {570 b^{2} x^{\frac {5}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1547 a^{3}}-\frac {2090 b^{3} x \sqrt {b \,x^{\frac {1}{3}}+a x}}{4641 a^{4}}+\frac {418 b^{4} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{663 a^{5}}-\frac {209 b^{5} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{221 a^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(276\)

Input:

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

Output:

-1/4641/a^6*(-228*x^(8/3)*a^4*b^2+156*x^(10/3)*a^5*b+380*a^3*b^3*x^2+8778* 
b^6*((x^(1/3)*a+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-2*(x^(1/3)*a-(-a*b)^(1 
/2))/(-a*b)^(1/2))^(1/2)*(-x^(1/3)/(-a*b)^(1/2)*a)^(1/2)*EllipticE(((x^(1/ 
3)*a+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))-4389*b^6*((x^(1/3)*a+( 
-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-2*(x^(1/3)*a-(-a*b)^(1/2))/(-a*b)^(1/2) 
)^(1/2)*(-x^(1/3)/(-a*b)^(1/2)*a)^(1/2)*EllipticF(((x^(1/3)*a+(-a*b)^(1/2) 
)/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))-1326*a^6*x^4-2926*x^(2/3)*a*b^5-836*x^( 
4/3)*a^2*b^4)/(x^(1/3)*(b+a*x^(2/3)))^(1/2)
 

Fricas [F]

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

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

Output:

integral((a^2*x^4 - a*b*x^(10/3) + b^2*x^(8/3))*sqrt(a*x + b*x^(1/3))/(a^3 
*x^2 + b^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {1710 x^{\frac {11}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a^{2} b^{2}-1482 \sqrt {x}\, \sqrt {x^{\frac {2}{3}} a +b}\, a^{3} b \,x^{2}+2926 \sqrt {x}\, \sqrt {x^{\frac {2}{3}} a +b}\, b^{4}+1326 x^{\frac {19}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a^{4}-2090 x^{\frac {7}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a \,b^{3}-1463 \left (\int \frac {x^{\frac {1}{6}} \sqrt {x^{\frac {2}{3}} a +b}}{x^{\frac {2}{3}} b +x^{\frac {4}{3}} a}d x \right ) b^{5}}{4641 a^{5}} \] Input:

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

Output:

(1710*x**(5/6)*sqrt(x**(2/3)*a + b)*a**2*b**2*x - 1482*sqrt(x)*sqrt(x**(2/ 
3)*a + b)*a**3*b*x**2 + 2926*sqrt(x)*sqrt(x**(2/3)*a + b)*b**4 + 1326*x**( 
1/6)*sqrt(x**(2/3)*a + b)*a**4*x**3 - 2090*x**(1/6)*sqrt(x**(2/3)*a + b)*a 
*b**3*x - 1463*int((x**(1/6)*sqrt(x**(2/3)*a + b))/(x**(2/3)*b + x**(1/3)* 
a*x),x)*b**5)/(4641*a**5)