Integrand size = 19, antiderivative size = 241 \[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=-\frac {3 \sqrt {b x^{2/3}+a x}}{7 b x^{8/3}}+\frac {13 a \sqrt {b x^{2/3}+a x}}{28 b^2 x^{7/3}}-\frac {143 a^2 \sqrt {b x^{2/3}+a x}}{280 b^3 x^2}+\frac {1287 a^3 \sqrt {b x^{2/3}+a x}}{2240 b^4 x^{5/3}}-\frac {429 a^4 \sqrt {b x^{2/3}+a x}}{640 b^5 x^{4/3}}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{512 b^6 x}-\frac {1287 a^6 \sqrt {b x^{2/3}+a x}}{1024 b^7 x^{2/3}}+\frac {1287 a^7 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{1024 b^{15/2}} \] Output:
-3/7*(b*x^(2/3)+a*x)^(1/2)/b/x^(8/3)+13/28*a*(b*x^(2/3)+a*x)^(1/2)/b^2/x^( 7/3)-143/280*a^2*(b*x^(2/3)+a*x)^(1/2)/b^3/x^2+1287/2240*a^3*(b*x^(2/3)+a* x)^(1/2)/b^4/x^(5/3)-429/640*a^4*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(4/3)+429/512 *a^5*(b*x^(2/3)+a*x)^(1/2)/b^6/x-1287/1024*a^6*(b*x^(2/3)+a*x)^(1/2)/b^7/x ^(2/3)+1287/1024*a^7*arctanh(b^(1/2)*x^(1/3)/(b*x^(2/3)+a*x)^(1/2))/b^(15/ 2)
Time = 0.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.57 \[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=\frac {\sqrt {b x^{2/3}+a x} \left (-15360 b^6+16640 a b^5 \sqrt [3]{x}-18304 a^2 b^4 x^{2/3}+20592 a^3 b^3 x-24024 a^4 b^2 x^{4/3}+30030 a^5 b x^{5/3}-45045 a^6 x^2\right )}{35840 b^7 x^{8/3}}+\frac {1287 a^7 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{1024 b^{15/2}} \] Input:
Integrate[1/(x^3*Sqrt[b*x^(2/3) + a*x]),x]
Output:
(Sqrt[b*x^(2/3) + a*x]*(-15360*b^6 + 16640*a*b^5*x^(1/3) - 18304*a^2*b^4*x ^(2/3) + 20592*a^3*b^3*x - 24024*a^4*b^2*x^(4/3) + 30030*a^5*b*x^(5/3) - 4 5045*a^6*x^2))/(35840*b^7*x^(8/3)) + (1287*a^7*ArcTanh[(Sqrt[b]*x^(1/3))/S qrt[b*x^(2/3) + a*x]])/(1024*b^(15/2))
Time = 0.83 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1931, 1931, 1931, 1931, 1931, 1931, 1931, 1935, 219}
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.
| \(\displaystyle \int \frac {1}{x^3 \sqrt {a x+b x^{2/3}}} \, dx\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {13 a \int \frac {1}{x^{8/3} \sqrt {x^{2/3} b+a x}}dx}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {13 a \left (-\frac {11 a \int \frac {1}{x^{7/3} \sqrt {x^{2/3} b+a x}}dx}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \int \frac {1}{x^2 \sqrt {x^{2/3} b+a x}}dx}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \int \frac {1}{x^{5/3} \sqrt {x^{2/3} b+a x}}dx}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \int \frac {1}{x^{4/3} \sqrt {x^{2/3} b+a x}}dx}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\) |
| \(\Big \downarrow \) 1935 |
| \(\displaystyle -\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\) |
Input:
Int[1/(x^3*Sqrt[b*x^(2/3) + a*x]),x]
Output:
(-3*Sqrt[b*x^(2/3) + a*x])/(7*b*x^(8/3)) - (13*a*(-1/2*Sqrt[b*x^(2/3) + a* x]/(b*x^(7/3)) - (11*a*((-3*Sqrt[b*x^(2/3) + a*x])/(5*b*x^2) - (9*a*((-3*S qrt[b*x^(2/3) + a*x])/(4*b*x^(5/3)) - (7*a*(-(Sqrt[b*x^(2/3) + a*x]/(b*x^( 4/3))) - (5*a*((-3*Sqrt[b*x^(2/3) + a*x])/(2*b*x) - (3*a*((-3*Sqrt[b*x^(2/ 3) + a*x])/(b*x^(2/3)) + (3*a*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a *x]])/b^(3/2)))/(4*b)))/(6*b)))/(8*b)))/(10*b)))/(12*b)))/(14*b)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 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]
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-2/(n - j) Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
Time = 0.34 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.76
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {x^{\frac {1}{3}} a +b}\, \left (45045 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {3}{2}} a^{6} x^{2}-45045 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) a^{7} b \,x^{\frac {7}{3}}-30030 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {5}{2}} a^{5} x^{\frac {5}{3}}+24024 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {7}{2}} a^{4} x^{\frac {4}{3}}-20592 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {9}{2}} a^{3} x +18304 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {11}{2}} a^{2} x^{\frac {2}{3}}-16640 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {13}{2}} a \,x^{\frac {1}{3}}+15360 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {15}{2}}\right )}{35840 x^{2} \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {17}{2}}}\) | \(183\) |
| default | \(\frac {\sqrt {x^{\frac {1}{3}} a +b}\, \left (45045 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) x^{\frac {13}{3}} a^{7} b -24024 x^{\frac {10}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {7}{2}} a^{4}-45045 x^{4} \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {3}{2}} a^{6}+16640 x^{\frac {7}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {13}{2}} a +30030 x^{\frac {11}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {5}{2}} a^{5}+20592 x^{3} \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {9}{2}} a^{3}-18304 x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {11}{2}} a^{2}-15360 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {15}{2}} x^{2}\right )}{35840 x^{4} \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {17}{2}}}\) | \(188\) |
Input:
int(1/x^3/(b*x^(2/3)+a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/35840/x^2*(x^(1/3)*a+b)^(1/2)*(45045*(x^(1/3)*a+b)^(1/2)*b^(3/2)*a^6*x^ 2-45045*arctanh((x^(1/3)*a+b)^(1/2)/b^(1/2))*a^7*b*x^(7/3)-30030*(x^(1/3)* a+b)^(1/2)*b^(5/2)*a^5*x^(5/3)+24024*(x^(1/3)*a+b)^(1/2)*b^(7/2)*a^4*x^(4/ 3)-20592*(x^(1/3)*a+b)^(1/2)*b^(9/2)*a^3*x+18304*(x^(1/3)*a+b)^(1/2)*b^(11 /2)*a^2*x^(2/3)-16640*(x^(1/3)*a+b)^(1/2)*b^(13/2)*a*x^(1/3)+15360*(x^(1/3 )*a+b)^(1/2)*b^(15/2))/(b*x^(2/3)+a*x)^(1/2)/b^(17/2)
Timed out. \[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=\text {Timed out} \] Input:
integrate(1/x^3/(b*x^(2/3)+a*x)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=\int \frac {1}{x^{3} \sqrt {a x + b x^{\frac {2}{3}}}}\, dx \] Input:
integrate(1/x**3/(b*x**(2/3)+a*x)**(1/2),x)
Output:
Integral(1/(x**3*sqrt(a*x + b*x**(2/3))), x)
\[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b x^{\frac {2}{3}}} x^{3}} \,d x } \] Input:
integrate(1/x^3/(b*x^(2/3)+a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(a*x + b*x^(2/3))*x^3), x)
Time = 0.31 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=-\frac {1}{35840} \, a^{7} {\left (\frac {45045 \, \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{7}} + \frac {45045 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} - 300300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b + 849849 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{2} - 1317888 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{3} + 1200199 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{4} - 631540 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{5} + 169995 \, \sqrt {a x^{\frac {1}{3}} + b} b^{6}}{a^{7} b^{7} x^{\frac {7}{3}}}\right )} \] Input:
integrate(1/x^3/(b*x^(2/3)+a*x)^(1/2),x, algorithm="giac")
Output:
-1/35840*a^7*(45045*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + (45045*(a*x^(1/3) + b)^(13/2) - 300300*(a*x^(1/3) + b)^(11/2)*b + 849849*( a*x^(1/3) + b)^(9/2)*b^2 - 1317888*(a*x^(1/3) + b)^(7/2)*b^3 + 1200199*(a* x^(1/3) + b)^(5/2)*b^4 - 631540*(a*x^(1/3) + b)^(3/2)*b^5 + 169995*sqrt(a* x^(1/3) + b)*b^6)/(a^7*b^7*x^(7/3)))
Timed out. \[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=\int \frac {1}{x^3\,\sqrt {a\,x+b\,x^{2/3}}} \,d x \] Input:
int(1/(x^3*(a*x + b*x^(2/3))^(1/2)),x)
Output:
int(1/(x^3*(a*x + b*x^(2/3))^(1/2)), x)
Time = 0.22 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^3 \sqrt {b x^{2/3}+a x}} \, dx=\frac {60060 x^{\frac {5}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{5} b^{2}-36608 x^{\frac {2}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{2} b^{5}-48048 x^{\frac {4}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{4} b^{3}+33280 x^{\frac {1}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a \,b^{6}-90090 \sqrt {x^{\frac {1}{3}} a +b}\, a^{6} b \,x^{2}+41184 \sqrt {x^{\frac {1}{3}} a +b}\, a^{3} b^{4} x -30720 \sqrt {x^{\frac {1}{3}} a +b}\, b^{7}-45045 x^{\frac {7}{3}} \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}-\sqrt {b}\right ) a^{7}+45045 x^{\frac {7}{3}} \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}+\sqrt {b}\right ) a^{7}}{71680 x^{\frac {7}{3}} b^{8}} \] Input:
int(1/x^3/(b*x^(2/3)+a*x)^(1/2),x)
Output:
(60060*x**(2/3)*sqrt(x**(1/3)*a + b)*a**5*b**2*x - 36608*x**(2/3)*sqrt(x** (1/3)*a + b)*a**2*b**5 - 48048*x**(1/3)*sqrt(x**(1/3)*a + b)*a**4*b**3*x + 33280*x**(1/3)*sqrt(x**(1/3)*a + b)*a*b**6 - 90090*sqrt(x**(1/3)*a + b)*a **6*b*x**2 + 41184*sqrt(x**(1/3)*a + b)*a**3*b**4*x - 30720*sqrt(x**(1/3)* a + b)*b**7 - 45045*x**(1/3)*sqrt(b)*log(sqrt(x**(1/3)*a + b) - sqrt(b))*a **7*x**2 + 45045*x**(1/3)*sqrt(b)*log(sqrt(x**(1/3)*a + b) + sqrt(b))*a**7 *x**2)/(71680*x**(1/3)*b**8*x**2)