Integrand size = 19, antiderivative size = 236 \[ \int \frac {1}{x^2 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {6}{b x^{5/3} \sqrt {b x^{2/3}+a x}}-\frac {13 \sqrt {b x^{2/3}+a x}}{2 b^2 x^{7/3}}+\frac {143 a \sqrt {b x^{2/3}+a x}}{20 b^3 x^2}-\frac {1287 a^2 \sqrt {b x^{2/3}+a x}}{160 b^4 x^{5/3}}+\frac {3003 a^3 \sqrt {b x^{2/3}+a x}}{320 b^5 x^{4/3}}-\frac {3003 a^4 \sqrt {b x^{2/3}+a x}}{256 b^6 x}+\frac {9009 a^5 \sqrt {b x^{2/3}+a x}}{512 b^7 x^{2/3}}-\frac {9009 a^6 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{512 b^{15/2}} \] Output:
6/b/x^(5/3)/(b*x^(2/3)+a*x)^(1/2)-13/2*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(7/3)+1 43/20*a*(b*x^(2/3)+a*x)^(1/2)/b^3/x^2-1287/160*a^2*(b*x^(2/3)+a*x)^(1/2)/b ^4/x^(5/3)+3003/320*a^3*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(4/3)-3003/256*a^4*(b* x^(2/3)+a*x)^(1/2)/b^6/x+9009/512*a^5*(b*x^(2/3)+a*x)^(1/2)/b^7/x^(2/3)-90 09/512*a^6*arctanh(b^(1/2)*x^(1/3)/(b*x^(2/3)+a*x)^(1/2))/b^(15/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.20 \[ \int \frac {1}{x^2 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {6 a^6 \sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},7,\frac {1}{2},1+\frac {a \sqrt [3]{x}}{b}\right )}{b^7 \sqrt {b x^{2/3}+a x}} \] Input:
Integrate[1/(x^2*(b*x^(2/3) + a*x)^(3/2)),x]
Output:
(6*a^6*x^(1/3)*Hypergeometric2F1[-1/2, 7, 1/2, 1 + (a*x^(1/3))/b])/(b^7*Sq rt[b*x^(2/3) + a*x])
Time = 0.82 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {1929, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a x+b x^{2/3}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1929 |
| \(\displaystyle \frac {13 \int \frac {1}{x^{8/3} \sqrt {x^{2/3} b+a x}}dx}{b}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {13 \left (-\frac {11 a \int \frac {1}{x^{7/3} \sqrt {x^{2/3} b+a x}}dx}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{b}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {13 \left (-\frac {11 a \left (-\frac {9 a \int \frac {1}{x^2 \sqrt {x^{2/3} b+a x}}dx}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{b}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {13 \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \int \frac {1}{x^{5/3} \sqrt {x^{2/3} b+a x}}dx}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{b}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {13 \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \int \frac {1}{x^{4/3} \sqrt {x^{2/3} b+a x}}dx}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{b}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {13 \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{b}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {13 \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{b}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1935 |
| \(\displaystyle \frac {13 \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{b}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {13 \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{b}+\frac {6}{b x^{5/3} \sqrt {a x+b x^{2/3}}}\) |
Input:
Int[1/(x^2*(b*x^(2/3) + a*x)^(3/2)),x]
Output:
6/(b*x^(5/3)*Sqrt[b*x^(2/3) + a*x]) + (13*(-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*Sqrt[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)))/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*(n - j )*(p + 1))), x] + Simp[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1))) In t[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, m}, x] & & !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1]
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 = 0.33 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.53
| method | result | size |
| derivativedivides | \(-\frac {\left (x^{\frac {1}{3}} a +b \right ) \left (45045 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) \sqrt {x^{\frac {1}{3}} a +b}\, a^{6} x^{2}+1280 b^{\frac {13}{2}}-1664 b^{\frac {11}{2}} a \,x^{\frac {1}{3}}+2288 b^{\frac {9}{2}} a^{2} x^{\frac {2}{3}}-3432 b^{\frac {7}{2}} a^{3} x +6006 b^{\frac {5}{2}} a^{4} x^{\frac {4}{3}}-15015 b^{\frac {3}{2}} a^{5} x^{\frac {5}{3}}-45045 a^{6} x^{2} \sqrt {b}\right )}{2560 x \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} b^{\frac {15}{2}}}\) | \(126\) |
| default | \(-\frac {\left (x^{\frac {1}{3}} a +b \right ) \left (45045 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) \sqrt {x^{\frac {1}{3}} a +b}\, a^{6} x^{2}+1280 b^{\frac {13}{2}}-1664 b^{\frac {11}{2}} a \,x^{\frac {1}{3}}+2288 b^{\frac {9}{2}} a^{2} x^{\frac {2}{3}}-3432 b^{\frac {7}{2}} a^{3} x +6006 b^{\frac {5}{2}} a^{4} x^{\frac {4}{3}}-15015 b^{\frac {3}{2}} a^{5} x^{\frac {5}{3}}-45045 a^{6} x^{2} \sqrt {b}\right )}{2560 x \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} b^{\frac {15}{2}}}\) | \(126\) |
Input:
int(1/x^2/(b*x^(2/3)+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2560*(x^(1/3)*a+b)*(45045*arctanh((x^(1/3)*a+b)^(1/2)/b^(1/2))*(x^(1/3) *a+b)^(1/2)*a^6*x^2+1280*b^(13/2)-1664*b^(11/2)*a*x^(1/3)+2288*b^(9/2)*a^2 *x^(2/3)-3432*b^(7/2)*a^3*x+6006*b^(5/2)*a^4*x^(4/3)-15015*b^(3/2)*a^5*x^( 5/3)-45045*a^6*x^2*b^(1/2))/x/(b*x^(2/3)+a*x)^(3/2)/b^(15/2)
Timed out. \[ \int \frac {1}{x^2 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(b*x^(2/3)+a*x)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x^2 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x**2/(b*x**(2/3)+a*x)**(3/2),x)
Output:
Integral(1/(x**2*(a*x + b*x**(2/3))**(3/2)), x)
\[ \int \frac {1}{x^2 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b*x^(2/3)+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((a*x + b*x^(2/3))^(3/2)*x^2), x)
Time = 0.33 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^2 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {9009 \, a^{6} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{512 \, \sqrt {-b} b^{7}} + \frac {6 \, a^{6}}{\sqrt {a x^{\frac {1}{3}} + b} b^{7}} + \frac {29685 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{6} - 163095 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{6} b + 364194 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{6} b^{2} - 416094 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{6} b^{3} + 246505 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{6} b^{4} - 62475 \, \sqrt {a x^{\frac {1}{3}} + b} a^{6} b^{5}}{2560 \, a^{6} b^{7} x^{2}} \] Input:
integrate(1/x^2/(b*x^(2/3)+a*x)^(3/2),x, algorithm="giac")
Output:
9009/512*a^6*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + 6*a^6/( sqrt(a*x^(1/3) + b)*b^7) + 1/2560*(29685*(a*x^(1/3) + b)^(11/2)*a^6 - 1630 95*(a*x^(1/3) + b)^(9/2)*a^6*b + 364194*(a*x^(1/3) + b)^(7/2)*a^6*b^2 - 41 6094*(a*x^(1/3) + b)^(5/2)*a^6*b^3 + 246505*(a*x^(1/3) + b)^(3/2)*a^6*b^4 - 62475*sqrt(a*x^(1/3) + b)*a^6*b^5)/(a^6*b^7*x^2)
Timed out. \[ \int \frac {1}{x^2 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2}} \,d x \] Input:
int(1/(x^2*(a*x + b*x^(2/3))^(3/2)),x)
Output:
int(1/(x^2*(a*x + b*x^(2/3))^(3/2)), x)
Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.62 \[ \int \frac {1}{x^2 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {45045 \sqrt {b}\, \sqrt {x^{\frac {1}{3}} a +b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}-\sqrt {b}\right ) a^{6} x^{2}-45045 \sqrt {b}\, \sqrt {x^{\frac {1}{3}} a +b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}+\sqrt {b}\right ) a^{6} x^{2}+30030 x^{\frac {5}{3}} a^{5} b^{2}-4576 x^{\frac {2}{3}} a^{2} b^{5}-12012 x^{\frac {4}{3}} a^{4} b^{3}+3328 x^{\frac {1}{3}} a \,b^{6}+90090 a^{6} b \,x^{2}+6864 a^{3} b^{4} x -2560 b^{7}}{5120 \sqrt {x^{\frac {1}{3}} a +b}\, b^{8} x^{2}} \] Input:
int(1/x^2/(b*x^(2/3)+a*x)^(3/2),x)
Output:
(45045*sqrt(b)*sqrt(x**(1/3)*a + b)*log(sqrt(x**(1/3)*a + b) - sqrt(b))*a* *6*x**2 - 45045*sqrt(b)*sqrt(x**(1/3)*a + b)*log(sqrt(x**(1/3)*a + b) + sq rt(b))*a**6*x**2 + 30030*x**(2/3)*a**5*b**2*x - 4576*x**(2/3)*a**2*b**5 - 12012*x**(1/3)*a**4*b**3*x + 3328*x**(1/3)*a*b**6 + 90090*a**6*b*x**2 + 68 64*a**3*b**4*x - 2560*b**7)/(5120*sqrt(x**(1/3)*a + b)*b**8*x**2)