\(\int \frac {x^2}{\sqrt {b \sqrt [3]{x}+a x}} \, dx\) [130]

Optimal result

Integrand size = 19, antiderivative size = 216 \[ \int \frac {x^2}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=-\frac {78 b^3 \sqrt {b \sqrt [3]{x}+a x}}{77 a^4}+\frac {234 b^2 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{385 a^3}-\frac {26 b x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{55 a^2}+\frac {2 x^2 \sqrt {b \sqrt [3]{x}+a x}}{5 a}+\frac {39 b^{15/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{77 a^{17/4} \sqrt {b \sqrt [3]{x}+a x}} \] Output:

-78/77*b^3*(b*x^(1/3)+a*x)^(1/2)/a^4+234/385*b^2*x^(2/3)*(b*x^(1/3)+a*x)^( 
1/2)/a^3-26/55*b*x^(4/3)*(b*x^(1/3)+a*x)^(1/2)/a^2+2/5*x^2*(b*x^(1/3)+a*x) 
^(1/2)/a+39/77*b^(15/4)*(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^(17/4)/(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 = 10.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.57 \[ \int \frac {x^2}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {2 \sqrt {b \sqrt [3]{x}+a x} \left (-195 b^4-78 a b^3 x^{2/3}+26 a^2 b^2 x^{4/3}-14 a^3 b x^2+77 a^4 x^{8/3}+195 b^4 \sqrt {1+\frac {a x^{2/3}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {a x^{2/3}}{b}\right )\right )}{385 a^4 \left (b+a x^{2/3}\right )} \] Input:

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

Output:

(2*Sqrt[b*x^(1/3) + a*x]*(-195*b^4 - 78*a*b^3*x^(2/3) + 26*a^2*b^2*x^(4/3) 
 - 14*a^3*b*x^2 + 77*a^4*x^(8/3) + 195*b^4*Sqrt[1 + (a*x^(2/3))/b]*Hyperge 
ometric2F1[1/4, 1/2, 5/4, -((a*x^(2/3))/b)]))/(385*a^4*(b + a*x^(2/3)))
 

Rubi [A] (warning: unable to verify)

Time = 0.73 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.22, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {1924, 1930, 1930, 1930, 1930, 1917, 266, 761}

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.

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

\(\Big \downarrow \) 1924

\(\displaystyle 3 \int \frac {x^{8/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}\)

\(\Big \downarrow \) 1930

\(\displaystyle 3 \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \int \frac {x^2}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{15 a}\right )\)

\(\Big \downarrow \) 1930

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

\(\Big \downarrow \) 1930

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

\(\Big \downarrow \) 1930

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

\(\Big \downarrow \) 1917

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 761

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

Input:

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

Output:

3*((2*x^2*Sqrt[b*x^(1/3) + a*x])/(15*a) - (13*b*((2*x^(4/3)*Sqrt[b*x^(1/3) 
 + a*x])/(11*a) - (9*b*((2*x^(2/3)*Sqrt[b*x^(1/3) + a*x])/(7*a) - (5*b*((2 
*Sqrt[b*x^(1/3) + a*x])/(3*a) - (b^(3/4)*(Sqrt[b] + Sqrt[a]*x^(2/3))*Sqrt[ 
b + a*x^(2/3)]*x^(1/6)*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])/(3*a^(5/4)*Sqrt[b*x^ 
(1/3) + a*x]*Sqrt[b + a*x^(4/3)])))/(7*a)))/(11*a)))/(15*a))
 

Defintions of rubi rules used

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 1917

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + 
b*x^n)^FracPart[p]/(x^(j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])   Int[ 
x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !Integ 
erQ[p] && NeQ[n, j] && PosQ[n - j]
 

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]
 

Maple [A] (verified)

Time = 1.60 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {52 x^{\frac {5}{3}} a^{3} b^{2}-28 x^{\frac {7}{3}} a^{4} b +195 b^{4} \sqrt {-a b}\, \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-156 a^{2} b^{3} x +154 a^{5} x^{3}-390 x^{\frac {1}{3}} a \,b^{4}}{385 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{5}}\)	\(163\)
derivativedivides	\(\frac {2 x^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{5 a}-\frac {26 b \,x^{\frac {4}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{55 a^{2}}+\frac {234 b^{2} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{385 a^{3}}-\frac {78 b^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{77 a^{4}}+\frac {39 b^{4} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{77 a^{5} \sqrt {b \,x^{\frac {1}{3}}+a x}}\)	\(201\)

Input:

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

Output:

1/385*(52*x^(5/3)*a^3*b^2-28*x^(7/3)*a^4*b+195*b^4*(-a*b)^(1/2)*((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))-156*a^2*b^3*x+154*a^5*x^3-390*x^(1/3 
)*a*b^4)/(x^(1/3)*(b+a*x^(2/3)))^(1/2)/a^5
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {234 x^{\frac {5}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a \,b^{2}-182 \sqrt {x}\, \sqrt {x^{\frac {2}{3}} a +b}\, a^{2} b x +154 x^{\frac {13}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a^{3}-390 x^{\frac {1}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, b^{3}+65 \left (\int \frac {\sqrt {x^{\frac {2}{3}} a +b}}{x^{\frac {5}{6}} b +\sqrt {x}\, a x}d x \right ) b^{4}}{385 a^{4}} \] Input:

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

Output:

(234*x**(5/6)*sqrt(x**(2/3)*a + b)*a*b**2 - 182*sqrt(x)*sqrt(x**(2/3)*a + 
b)*a**2*b*x + 154*x**(1/6)*sqrt(x**(2/3)*a + b)*a**3*x**2 - 390*x**(1/6)*s 
qrt(x**(2/3)*a + b)*b**3 + 65*int(sqrt(x**(2/3)*a + b)/(x**(5/6)*b + sqrt( 
x)*a*x),x)*b**4)/(385*a**4)