Integrand size = 19, antiderivative size = 304 \[ \int \frac {x^4}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {11050 b^6 \sqrt {b \sqrt [3]{x}+a x}}{14421 a^7}-\frac {2210 b^5 x^{2/3} \sqrt {b \sqrt [3]{x}+a x}}{4807 a^6}+\frac {15470 b^4 x^{4/3} \sqrt {b \sqrt [3]{x}+a x}}{43263 a^5}-\frac {1190 b^3 x^2 \sqrt {b \sqrt [3]{x}+a x}}{3933 a^4}+\frac {350 b^2 x^{8/3} \sqrt {b \sqrt [3]{x}+a x}}{1311 a^3}-\frac {50 b x^{10/3} \sqrt {b \sqrt [3]{x}+a x}}{207 a^2}+\frac {2 x^4 \sqrt {b \sqrt [3]{x}+a x}}{9 a}-\frac {5525 b^{27/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{14421 a^{29/4} \sqrt {b \sqrt [3]{x}+a x}} \] Output:
11050/14421*b^6*(b*x^(1/3)+a*x)^(1/2)/a^7-2210/4807*b^5*x^(2/3)*(b*x^(1/3) +a*x)^(1/2)/a^6+15470/43263*b^4*x^(4/3)*(b*x^(1/3)+a*x)^(1/2)/a^5-1190/393 3*b^3*x^2*(b*x^(1/3)+a*x)^(1/2)/a^4+350/1311*b^2*x^(8/3)*(b*x^(1/3)+a*x)^( 1/2)/a^3-50/207*b*x^(10/3)*(b*x^(1/3)+a*x)^(1/2)/a^2+2/9*x^4*(b*x^(1/3)+a* x)^(1/2)/a-5525/14421*b^(27/4)*(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^(29/4)/(b*x^(1/3)+a*x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.53 \[ \int \frac {x^4}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {2 \sqrt {b \sqrt [3]{x}+a x} \left (16575 b^7+6630 a b^6 x^{2/3}-2210 a^2 b^5 x^{4/3}+1190 a^3 b^4 x^2-770 a^4 b^3 x^{8/3}+550 a^5 b^2 x^{10/3}-418 a^6 b x^4+4807 a^7 x^{14/3}-16575 b^7 \sqrt {1+\frac {a x^{2/3}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {a x^{2/3}}{b}\right )\right )}{43263 a^7 \left (b+a x^{2/3}\right )} \] Input:
Integrate[x^4/Sqrt[b*x^(1/3) + a*x],x]
Output:
(2*Sqrt[b*x^(1/3) + a*x]*(16575*b^7 + 6630*a*b^6*x^(2/3) - 2210*a^2*b^5*x^ (4/3) + 1190*a^3*b^4*x^2 - 770*a^4*b^3*x^(8/3) + 550*a^5*b^2*x^(10/3) - 41 8*a^6*b*x^4 + 4807*a^7*x^(14/3) - 16575*b^7*Sqrt[1 + (a*x^(2/3))/b]*Hyperg eometric2F1[1/4, 1/2, 5/4, -((a*x^(2/3))/b)]))/(43263*a^7*(b + a*x^(2/3)))
Time = 1.03 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.22, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {1924, 1930, 1930, 1930, 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.
| \(\displaystyle \int \frac {x^4}{\sqrt {a x+b \sqrt [3]{x}}} \, dx\) |
| \(\Big \downarrow \) 1924 |
| \(\displaystyle 3 \int \frac {x^{14/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2 x^4 \sqrt {a x+b \sqrt [3]{x}}}{27 a}-\frac {25 b \int \frac {x^4}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{27 a}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2 x^4 \sqrt {a x+b \sqrt [3]{x}}}{27 a}-\frac {25 b \left (\frac {2 x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}{23 a}-\frac {21 b \int \frac {x^{10/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{23 a}\right )}{27 a}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2 x^4 \sqrt {a x+b \sqrt [3]{x}}}{27 a}-\frac {25 b \left (\frac {2 x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}{23 a}-\frac {21 b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \int \frac {x^{8/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{19 a}\right )}{23 a}\right )}{27 a}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2 x^4 \sqrt {a x+b \sqrt [3]{x}}}{27 a}-\frac {25 b \left (\frac {2 x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}{23 a}-\frac {21 b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \int \frac {x^2}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{15 a}\right )}{19 a}\right )}{23 a}\right )}{27 a}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2 x^4 \sqrt {a x+b \sqrt [3]{x}}}{27 a}-\frac {25 b \left (\frac {2 x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}{23 a}-\frac {21 b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \int \frac {x^{4/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{11 a}\right )}{15 a}\right )}{19 a}\right )}{23 a}\right )}{27 a}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2 x^4 \sqrt {a x+b \sqrt [3]{x}}}{27 a}-\frac {25 b \left (\frac {2 x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}{23 a}-\frac {21 b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \int \frac {x^{2/3}}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{7 a}\right )}{11 a}\right )}{15 a}\right )}{19 a}\right )}{23 a}\right )}{27 a}\right )\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle 3 \left (\frac {2 x^4 \sqrt {a x+b \sqrt [3]{x}}}{27 a}-\frac {25 b \left (\frac {2 x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}{23 a}-\frac {21 b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b \int \frac {1}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{3 a}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{19 a}\right )}{23 a}\right )}{27 a}\right )\) |
| \(\Big \downarrow \) 1917 |
| \(\displaystyle 3 \left (\frac {2 x^4 \sqrt {a x+b \sqrt [3]{x}}}{27 a}-\frac {25 b \left (\frac {2 x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}{23 a}-\frac {21 b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {x^{2/3} a+b} \sqrt [6]{x}}d\sqrt [3]{x}}{3 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{19 a}\right )}{23 a}\right )}{27 a}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle 3 \left (\frac {2 x^4 \sqrt {a x+b \sqrt [3]{x}}}{27 a}-\frac {25 b \left (\frac {2 x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}{23 a}-\frac {21 b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {2 b \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{3 a \sqrt {a x+b \sqrt [3]{x}}}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{19 a}\right )}{23 a}\right )}{27 a}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle 3 \left (\frac {2 x^4 \sqrt {a x+b \sqrt [3]{x}}}{27 a}-\frac {25 b \left (\frac {2 x^{10/3} \sqrt {a x+b \sqrt [3]{x}}}{23 a}-\frac {21 b \left (\frac {2 x^{8/3} \sqrt {a x+b \sqrt [3]{x}}}{19 a}-\frac {17 b \left (\frac {2 x^2 \sqrt {a x+b \sqrt [3]{x}}}{15 a}-\frac {13 b \left (\frac {2 x^{4/3} \sqrt {a x+b \sqrt [3]{x}}}{11 a}-\frac {9 b \left (\frac {2 x^{2/3} \sqrt {a x+b \sqrt [3]{x}}}{7 a}-\frac {5 b \left (\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 a}-\frac {b^{3/4} \sqrt [6]{x} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {a x^{2/3}+b} \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 a^{5/4} \sqrt {a x+b \sqrt [3]{x}} \sqrt {a x^{4/3}+b}}\right )}{7 a}\right )}{11 a}\right )}{15 a}\right )}{19 a}\right )}{23 a}\right )}{27 a}\right )\) |
Input:
Int[x^4/Sqrt[b*x^(1/3) + a*x],x]
Output:
3*((2*x^4*Sqrt[b*x^(1/3) + a*x])/(27*a) - (25*b*((2*x^(10/3)*Sqrt[b*x^(1/3 ) + a*x])/(23*a) - (21*b*((2*x^(8/3)*Sqrt[b*x^(1/3) + a*x])/(19*a) - (17*b *((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)))/(19*a)))/(23* a)))/(27*a))
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]
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]
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]
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 ]
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]
Time = 4.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.64
| method | result | size |
| default | \(-\frac {-1100 x^{\frac {11}{3}} a^{6} b^{2}+836 x^{\frac {13}{3}} a^{7} b +1540 a^{5} b^{3} x^{3}+4420 x^{\frac {5}{3}} a^{3} b^{5}-2380 x^{\frac {7}{3}} a^{4} b^{4}-9614 a^{8} x^{5}+16575 b^{7} \sqrt {-a b}\, \sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}} a -\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x^{\frac {1}{3}} a +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-13260 a^{2} b^{6} x -33150 x^{\frac {1}{3}} a \,b^{7}}{43263 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{8}}\) | \(196\) |
| derivativedivides | \(\frac {2 x^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}{9 a}-\frac {50 b \,x^{\frac {10}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{207 a^{2}}+\frac {350 b^{2} x^{\frac {8}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1311 a^{3}}-\frac {1190 b^{3} x^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{3933 a^{4}}+\frac {15470 b^{4} x^{\frac {4}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{43263 a^{5}}-\frac {2210 b^{5} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{4807 a^{6}}+\frac {11050 b^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}{14421 a^{7}}-\frac {5525 b^{7} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{14421 a^{8} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(267\) |
Input:
int(x^4/(b*x^(1/3)+a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/43263*(-1100*x^(11/3)*a^6*b^2+836*x^(13/3)*a^7*b+1540*a^5*b^3*x^3+4420* x^(5/3)*a^3*b^5-2380*x^(7/3)*a^4*b^4-9614*a^8*x^5+16575*b^7*(-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))-13260*a^2*b^6*x-33150*x^(1/ 3)*a*b^7)/(x^(1/3)*(b+a*x^(2/3)))^(1/2)/a^8
\[ \int \frac {x^4}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\int { \frac {x^{4}}{\sqrt {a x + b x^{\frac {1}{3}}}} \,d x } \] Input:
integrate(x^4/(b*x^(1/3)+a*x)^(1/2),x, algorithm="fricas")
Output:
integral((a^2*x^5 - a*b*x^(13/3) + b^2*x^(11/3))*sqrt(a*x + b*x^(1/3))/(a^ 3*x^2 + b^3), x)
\[ \int \frac {x^4}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\int \frac {x^{4}}{\sqrt {a x + b \sqrt [3]{x}}}\, dx \] Input:
integrate(x**4/(b*x**(1/3)+a*x)**(1/2),x)
Output:
Integral(x**4/sqrt(a*x + b*x**(1/3)), x)
\[ \int \frac {x^4}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\int { \frac {x^{4}}{\sqrt {a x + b x^{\frac {1}{3}}}} \,d x } \] Input:
integrate(x^4/(b*x^(1/3)+a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(x^4/sqrt(a*x + b*x^(1/3)), x)
\[ \int \frac {x^4}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\int { \frac {x^{4}}{\sqrt {a x + b x^{\frac {1}{3}}}} \,d x } \] Input:
integrate(x^4/(b*x^(1/3)+a*x)^(1/2),x, algorithm="giac")
Output:
integrate(x^4/sqrt(a*x + b*x^(1/3)), x)
Timed out. \[ \int \frac {x^4}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\int \frac {x^4}{\sqrt {a\,x+b\,x^{1/3}}} \,d x \] Input:
int(x^4/(a*x + b*x^(1/3))^(1/2),x)
Output:
int(x^4/(a*x + b*x^(1/3))^(1/2), x)
\[ \int \frac {x^4}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {11550 x^{\frac {17}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a^{4} b^{2}-19890 x^{\frac {5}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a \,b^{5}-10450 \sqrt {x}\, \sqrt {x^{\frac {2}{3}} a +b}\, a^{5} b \,x^{3}+15470 \sqrt {x}\, \sqrt {x^{\frac {2}{3}} a +b}\, a^{2} b^{4} x +9614 x^{\frac {25}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a^{6}-13090 x^{\frac {13}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, a^{3} b^{3}+33150 x^{\frac {1}{6}} \sqrt {x^{\frac {2}{3}} a +b}\, b^{6}-5525 \left (\int \frac {\sqrt {x^{\frac {2}{3}} a +b}}{x^{\frac {5}{6}} b +\sqrt {x}\, a x}d x \right ) b^{7}}{43263 a^{7}} \] Input:
int(x^4/(b*x^(1/3)+a*x)^(1/2),x)
Output:
(11550*x**(5/6)*sqrt(x**(2/3)*a + b)*a**4*b**2*x**2 - 19890*x**(5/6)*sqrt( x**(2/3)*a + b)*a*b**5 - 10450*sqrt(x)*sqrt(x**(2/3)*a + b)*a**5*b*x**3 + 15470*sqrt(x)*sqrt(x**(2/3)*a + b)*a**2*b**4*x + 9614*x**(1/6)*sqrt(x**(2/ 3)*a + b)*a**6*x**4 - 13090*x**(1/6)*sqrt(x**(2/3)*a + b)*a**3*b**3*x**2 + 33150*x**(1/6)*sqrt(x**(2/3)*a + b)*b**6 - 5525*int(sqrt(x**(2/3)*a + b)/ (x**(5/6)*b + sqrt(x)*a*x),x)*b**7)/(43263*a**7)