Integrand size = 19, antiderivative size = 153 \[ \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx=-\frac {3 \sqrt {b x^{2/3}+a x}}{4 b x^{5/3}}+\frac {7 a \sqrt {b x^{2/3}+a x}}{8 b^2 x^{4/3}}-\frac {35 a^2 \sqrt {b x^{2/3}+a x}}{32 b^3 x}+\frac {105 a^3 \sqrt {b x^{2/3}+a x}}{64 b^4 x^{2/3}}-\frac {105 a^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{64 b^{9/2}} \]
-105/64*a^4*arctanh(x^(1/3)*b^(1/2)/(b*x^(2/3)+a*x)^(1/2))/b^(9/2)-3/4*(b* x^(2/3)+a*x)^(1/2)/b/x^(5/3)+7/8*a*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(4/3)-35/32 *a^2*(b*x^(2/3)+a*x)^(1/2)/b^3/x+105/64*a^3*(b*x^(2/3)+a*x)^(1/2)/b^4/x^(2 /3)
Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.66 \[ \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx=\frac {\sqrt {b x^{2/3}+a x} \left (-48 b^3+56 a b^2 \sqrt [3]{x}-70 a^2 b x^{2/3}+105 a^3 x\right )}{64 b^4 x^{5/3}}-\frac {105 a^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{64 b^{9/2}} \]
(Sqrt[b*x^(2/3) + a*x]*(-48*b^3 + 56*a*b^2*x^(1/3) - 70*a^2*b*x^(2/3) + 10 5*a^3*x))/(64*b^4*x^(5/3)) - (105*a^4*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^( 2/3) + a*x]])/(64*b^(9/2))
Time = 0.34 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {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.
\(\displaystyle \int \frac {1}{x^2 \sqrt {a x+b x^{2/3}}} \, dx\) |
\(\Big \downarrow \) 1931 |
\(\displaystyle -\frac {7 a \int \frac {1}{x^{5/3} \sqrt {x^{2/3} b+a x}}dx}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\) |
\(\Big \downarrow \) 1931 |
\(\displaystyle -\frac {7 a \left (-\frac {5 a \int \frac {1}{x^{4/3} \sqrt {x^{2/3} b+a x}}dx}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\) |
\(\Big \downarrow \) 1931 |
\(\displaystyle -\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\) |
\(\Big \downarrow \) 1931 |
\(\displaystyle -\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle -\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\) |
(-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)
3.2.91.3.1 Defintions of rubi rules used
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 = 2.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {\sqrt {b +a \,x^{\frac {1}{3}}}\, \left (48 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {9}{2}}-56 b^{\frac {7}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a \,x^{\frac {1}{3}}+70 b^{\frac {5}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{2} x^{\frac {2}{3}}-105 b^{\frac {3}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{3} x +105 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) a^{4} b \,x^{\frac {4}{3}}\right )}{64 x \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {11}{2}}}\) | \(123\) |
default | \(-\frac {\sqrt {b +a \,x^{\frac {1}{3}}}\, \left (105 x^{\frac {7}{3}} \operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) a^{4} b +70 x^{\frac {5}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {5}{2}} a^{2}-56 x^{\frac {4}{3}} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {7}{2}} a +48 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {9}{2}} x -105 x^{2} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {3}{2}} a^{3}\right )}{64 x^{2} \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {11}{2}}}\) | \(126\) |
-1/64*(b+a*x^(1/3))^(1/2)*(48*(b+a*x^(1/3))^(1/2)*b^(9/2)-56*b^(7/2)*(b+a* x^(1/3))^(1/2)*a*x^(1/3)+70*b^(5/2)*(b+a*x^(1/3))^(1/2)*a^2*x^(2/3)-105*b^ (3/2)*(b+a*x^(1/3))^(1/2)*a^3*x+105*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*a ^4*b*x^(4/3))/x/(b*x^(2/3)+a*x)^(1/2)/b^(11/2)
Timed out. \[ \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx=\int \frac {1}{x^{2} \sqrt {a x + b x^{\frac {2}{3}}}}\, dx \]
\[ \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b x^{\frac {2}{3}}} x^{2}} \,d x } \]
Time = 0.33 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx=\frac {\frac {105 \, a^{5} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{4}} + \frac {105 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{5} - 385 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{5} b + 511 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{5} b^{2} - 279 \, \sqrt {a x^{\frac {1}{3}} + b} a^{5} b^{3}}{a^{4} b^{4} x^{\frac {4}{3}}}}{64 \, a} \]
1/64*(105*a^5*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^4) + (105*( a*x^(1/3) + b)^(7/2)*a^5 - 385*(a*x^(1/3) + b)^(5/2)*a^5*b + 511*(a*x^(1/3 ) + b)^(3/2)*a^5*b^2 - 279*sqrt(a*x^(1/3) + b)*a^5*b^3)/(a^4*b^4*x^(4/3))) /a
Timed out. \[ \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx=\int \frac {1}{x^2\,\sqrt {a\,x+b\,x^{2/3}}} \,d x \]