Integrand size = 19, antiderivative size = 276 \[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^5} \, dx=-\frac {6 \sqrt {b \sqrt [3]{x}+a x}}{23 x^4}-\frac {12 a \sqrt {b \sqrt [3]{x}+a x}}{437 b x^{10/3}}+\frac {68 a^2 \sqrt {b \sqrt [3]{x}+a x}}{2185 b^2 x^{8/3}}-\frac {884 a^3 \sqrt {b \sqrt [3]{x}+a x}}{24035 b^3 x^2}+\frac {7956 a^4 \sqrt {b \sqrt [3]{x}+a x}}{168245 b^4 x^{4/3}}-\frac {2652 a^5 \sqrt {b \sqrt [3]{x}+a x}}{33649 b^5 x^{2/3}}-\frac {1326 a^{23/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{33649 b^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \] Output:
-6/23*(b*x^(1/3)+a*x)^(1/2)/x^4-12/437*a*(b*x^(1/3)+a*x)^(1/2)/b/x^(10/3)+ 68/2185*a^2*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(8/3)-884/24035*a^3*(b*x^(1/3)+a*x )^(1/2)/b^3/x^2+7956/168245*a^4*(b*x^(1/3)+a*x)^(1/2)/b^4/x^(4/3)-2652/336 49*a^5*(b*x^(1/3)+a*x)^(1/2)/b^5/x^(2/3)-1326/33649*a^(23/4)*(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)*Inv erseJacobiAM(2*arctan(a^(1/4)*x^(1/6)/b^(1/4)),1/2*2^(1/2))/b^(21/4)/(b*x^ (1/3)+a*x)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.21 \[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^5} \, dx=-\frac {6 \sqrt {b \sqrt [3]{x}+a x} \operatorname {Hypergeometric2F1}\left (-\frac {23}{4},-\frac {1}{2},-\frac {19}{4},-\frac {a x^{2/3}}{b}\right )}{23 \sqrt {1+\frac {a x^{2/3}}{b}} x^4} \] Input:
Integrate[Sqrt[b*x^(1/3) + a*x]/x^5,x]
Output:
(-6*Sqrt[b*x^(1/3) + a*x]*Hypergeometric2F1[-23/4, -1/2, -19/4, -((a*x^(2/ 3))/b)])/(23*Sqrt[1 + (a*x^(2/3))/b]*x^4)
Time = 0.92 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.21, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1924, 1926, 1931, 1931, 1931, 1931, 1931, 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.
| \(\displaystyle \int \frac {\sqrt {a x+b \sqrt [3]{x}}}{x^5} \, dx\) |
| \(\Big \downarrow \) 1924 |
| \(\displaystyle 3 \int \frac {\sqrt {\sqrt [3]{x} b+a x}}{x^{13/3}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 1926 |
| \(\displaystyle 3 \left (\frac {2}{23} a \int \frac {1}{x^{10/3} \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{23 x^4}\right )\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle 3 \left (\frac {2}{23} a \left (-\frac {17 a \int \frac {1}{x^{8/3} \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{19 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{19 b x^{10/3}}\right )-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{23 x^4}\right )\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle 3 \left (\frac {2}{23} a \left (-\frac {17 a \left (-\frac {13 a \int \frac {1}{x^2 \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{15 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{15 b x^{8/3}}\right )}{19 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{19 b x^{10/3}}\right )-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{23 x^4}\right )\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle 3 \left (\frac {2}{23} a \left (-\frac {17 a \left (-\frac {13 a \left (-\frac {9 a \int \frac {1}{x^{4/3} \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{11 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{11 b x^2}\right )}{15 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{15 b x^{8/3}}\right )}{19 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{19 b x^{10/3}}\right )-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{23 x^4}\right )\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle 3 \left (\frac {2}{23} a \left (-\frac {17 a \left (-\frac {13 a \left (-\frac {9 a \left (-\frac {5 a \int \frac {1}{x^{2/3} \sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{7 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{7 b x^{4/3}}\right )}{11 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{11 b x^2}\right )}{15 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{15 b x^{8/3}}\right )}{19 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{19 b x^{10/3}}\right )-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{23 x^4}\right )\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle 3 \left (\frac {2}{23} a \left (-\frac {17 a \left (-\frac {13 a \left (-\frac {9 a \left (-\frac {5 a \left (-\frac {a \int \frac {1}{\sqrt {\sqrt [3]{x} b+a x}}d\sqrt [3]{x}}{3 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 b x^{2/3}}\right )}{7 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{7 b x^{4/3}}\right )}{11 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{11 b x^2}\right )}{15 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{15 b x^{8/3}}\right )}{19 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{19 b x^{10/3}}\right )-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{23 x^4}\right )\) |
| \(\Big \downarrow \) 1917 |
| \(\displaystyle 3 \left (\frac {2}{23} a \left (-\frac {17 a \left (-\frac {13 a \left (-\frac {9 a \left (-\frac {5 a \left (-\frac {a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {x^{2/3} a+b} \sqrt [6]{x}}d\sqrt [3]{x}}{3 b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 b x^{2/3}}\right )}{7 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{7 b x^{4/3}}\right )}{11 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{11 b x^2}\right )}{15 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{15 b x^{8/3}}\right )}{19 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{19 b x^{10/3}}\right )-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{23 x^4}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle 3 \left (\frac {2}{23} a \left (-\frac {17 a \left (-\frac {13 a \left (-\frac {9 a \left (-\frac {5 a \left (-\frac {2 a \sqrt [6]{x} \sqrt {a x^{2/3}+b} \int \frac {1}{\sqrt {a x^{4/3}+b}}d\sqrt [6]{x}}{3 b \sqrt {a x+b \sqrt [3]{x}}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 b x^{2/3}}\right )}{7 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{7 b x^{4/3}}\right )}{11 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{11 b x^2}\right )}{15 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{15 b x^{8/3}}\right )}{19 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{19 b x^{10/3}}\right )-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{23 x^4}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle 3 \left (\frac {2}{23} a \left (-\frac {17 a \left (-\frac {13 a \left (-\frac {9 a \left (-\frac {5 a \left (-\frac {a^{3/4} \sqrt [6]{x} \left (\sqrt {a} x^{2/3}+\sqrt {b}\right ) \sqrt {a x^{2/3}+b} \sqrt {\frac {a x^{4/3}+b}{\left (\sqrt {a} x^{2/3}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{3 b^{5/4} \sqrt {a x+b \sqrt [3]{x}} \sqrt {a x^{4/3}+b}}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{3 b x^{2/3}}\right )}{7 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{7 b x^{4/3}}\right )}{11 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{11 b x^2}\right )}{15 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{15 b x^{8/3}}\right )}{19 b}-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{19 b x^{10/3}}\right )-\frac {2 \sqrt {a x+b \sqrt [3]{x}}}{23 x^4}\right )\) |
Input:
Int[Sqrt[b*x^(1/3) + a*x]/x^5,x]
Output:
3*((-2*Sqrt[b*x^(1/3) + a*x])/(23*x^4) + (2*a*((-2*Sqrt[b*x^(1/3) + a*x])/ (19*b*x^(10/3)) - (17*a*((-2*Sqrt[b*x^(1/3) + a*x])/(15*b*x^(8/3)) - (13*a *((-2*Sqrt[b*x^(1/3) + a*x])/(11*b*x^2) - (9*a*((-2*Sqrt[b*x^(1/3) + a*x]) /(7*b*x^(4/3)) - (5*a*((-2*Sqrt[b*x^(1/3) + a*x])/(3*b*x^(2/3)) - (a^(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*b^(5/4)*Sqrt[b*x^(1/3) + a*x]*Sqrt[b + a*x^(4/3)])))/(7*b )))/(11*b)))/(15*b)))/(19*b)))/23)
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*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + j*p + 1))), x] - Simp[b*p *((n - j)/(c^n*(m + j*p + 1))) Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Integer sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 0]
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]
Time = 1.54 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.89
| method | result | size |
| derivativedivides | \(-\frac {6 \sqrt {b \,x^{\frac {1}{3}}+a x}}{23 x^{4}}-\frac {12 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{437 b \,x^{\frac {10}{3}}}+\frac {68 a^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{2185 b^{2} x^{\frac {8}{3}}}-\frac {884 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{24035 b^{3} x^{2}}+\frac {7956 a^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}{168245 b^{4} x^{\frac {4}{3}}}-\frac {2652 a^{5} \sqrt {b \,x^{\frac {1}{3}}+a x}}{33649 b^{5} x^{\frac {2}{3}}}-\frac {1326 a^{5} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{33649 b^{5} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(245\) |
| default | \(-\frac {6 \sqrt {b \,x^{\frac {1}{3}}+a x}}{23 x^{4}}-\frac {12 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{437 b \,x^{\frac {10}{3}}}+\frac {68 a^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{2185 b^{2} x^{\frac {8}{3}}}-\frac {884 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{24035 b^{3} x^{2}}+\frac {7956 a^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}{168245 b^{4} x^{\frac {4}{3}}}-\frac {2652 a^{5} \sqrt {b \,x^{\frac {1}{3}}+a x}}{33649 b^{5} x^{\frac {2}{3}}}-\frac {1326 a^{5} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{33649 b^{5} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(245\) |
Input:
int((b*x^(1/3)+a*x)^(1/2)/x^5,x,method=_RETURNVERBOSE)
Output:
-6/23*(b*x^(1/3)+a*x)^(1/2)/x^4-12/437*a*(b*x^(1/3)+a*x)^(1/2)/b/x^(10/3)+ 68/2185*a^2*(b*x^(1/3)+a*x)^(1/2)/b^2/x^(8/3)-884/24035*a^3*(b*x^(1/3)+a*x )^(1/2)/b^3/x^2+7956/168245*a^4*(b*x^(1/3)+a*x)^(1/2)/b^4/x^(4/3)-2652/336 49*a^5*(b*x^(1/3)+a*x)^(1/2)/b^5/x^(2/3)-1326/33649*a^5/b^5*(-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)*EllipticF(((x^(1/3)+1/a*(-a*b)^(1/2))*a/(-a*b)^(1/2))^(1/2),1/2*2^ (1/2))
\[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^5} \, dx=\int { \frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x^{5}} \,d x } \] Input:
integrate((b*x^(1/3)+a*x)^(1/2)/x^5,x, algorithm="fricas")
Output:
integral(sqrt(a*x + b*x^(1/3))/x^5, x)
\[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^5} \, dx=\int \frac {\sqrt {a x + b \sqrt [3]{x}}}{x^{5}}\, dx \] Input:
integrate((b*x**(1/3)+a*x)**(1/2)/x**5,x)
Output:
Integral(sqrt(a*x + b*x**(1/3))/x**5, x)
\[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^5} \, dx=\int { \frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x^{5}} \,d x } \] Input:
integrate((b*x^(1/3)+a*x)^(1/2)/x^5,x, algorithm="maxima")
Output:
integrate(sqrt(a*x + b*x^(1/3))/x^5, x)
\[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^5} \, dx=\int { \frac {\sqrt {a x + b x^{\frac {1}{3}}}}{x^{5}} \,d x } \] Input:
integrate((b*x^(1/3)+a*x)^(1/2)/x^5,x, algorithm="giac")
Output:
integrate(sqrt(a*x + b*x^(1/3))/x^5, x)
Timed out. \[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^5} \, dx=\int \frac {\sqrt {a\,x+b\,x^{1/3}}}{x^5} \,d x \] Input:
int((a*x + b*x^(1/3))^(1/2)/x^5,x)
Output:
int((a*x + b*x^(1/3))^(1/2)/x^5, x)
\[ \int \frac {\sqrt {b \sqrt [3]{x}+a x}}{x^5} \, dx=\frac {-\frac {2 \sqrt {x^{\frac {2}{3}} a +b}}{7}-\frac {2 x^{\frac {23}{6}} \left (\int \frac {\sqrt {x^{\frac {2}{3}} a +b}}{x^{\frac {29}{6}} b +\sqrt {x}\, a \,x^{5}}d x \right ) b}{21}}{x^{\frac {23}{6}}} \] Input:
int((b*x^(1/3)+a*x)^(1/2)/x^5,x)
Output:
(2*( - 3*sqrt(x**(2/3)*a + b) - x**(5/6)*int(sqrt(x**(2/3)*a + b)/(x**(5/6 )*b*x**4 + sqrt(x)*a*x**5),x)*b*x**3))/(21*x**(5/6)*x**3)