Integrand size = 26, antiderivative size = 316 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {4}{9 a^2 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{6 a x^2 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {10 \left (a+b x^3\right )}{9 a^3 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {20 b^{2/3} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {20 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {10 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
4/9/a^2/x^2/((b*x^3+a)^2)^(1/2)+1/6/a/x^2/(b*x^3+a)/((b*x^3+a)^2)^(1/2)-10 /9*(b*x^3+a)/a^3/x^2/((b*x^3+a)^2)^(1/2)-20/27*b^(2/3)*(b*x^3+a)*ln(a^(1/3 )+b^(1/3)*x)/a^(11/3)/((b*x^3+a)^2)^(1/2)+10/27*b^(2/3)*(b*x^3+a)*ln(a^(2/ 3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(11/3)/((b*x^3+a)^2)^(1/2)+20/27*b^(2/ 3)*(b*x^3+a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(11/3)*3^ (1/2)/((b*x^3+a)^2)^(1/2)
Time = 1.07 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {-27 a^{8/3}-96 a^{5/3} b x^3-60 a^{2/3} b^2 x^6+40 \sqrt {3} b^{2/3} x^2 \left (a+b x^3\right )^2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-40 b^{2/3} x^2 \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+20 a^2 b^{2/3} x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+40 a b^{5/3} x^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+20 b^{8/3} x^8 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} x^2 \left (a+b x^3\right ) \sqrt {\left (a+b x^3\right )^2}} \]
(-27*a^(8/3) - 96*a^(5/3)*b*x^3 - 60*a^(2/3)*b^2*x^6 + 40*Sqrt[3]*b^(2/3)* x^2*(a + b*x^3)^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 40*b^(2/3) *x^2*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] + 20*a^2*b^(2/3)*x^2*Log[a^(2/ 3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] + 40*a*b^(5/3)*x^5*Log[a^(2/3) - a^( 1/3)*b^(1/3)*x + b^(2/3)*x^2] + 20*b^(8/3)*x^8*Log[a^(2/3) - a^(1/3)*b^(1/ 3)*x + b^(2/3)*x^2])/(54*a^(11/3)*x^2*(a + b*x^3)*Sqrt[(a + b*x^3)^2])
Time = 0.40 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.67, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1384, 27, 819, 819, 847, 750, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \int \frac {1}{b^3 x^3 \left (b x^3+a\right )^3}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \int \frac {1}{x^3 \left (b x^3+a\right )^3}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {4 \int \frac {1}{x^3 \left (b x^3+a\right )^2}dx}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {4 \left (\frac {5 \int \frac {1}{x^3 \left (b x^3+a\right )}dx}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {4 \left (\frac {5 \left (-\frac {b \int \frac {1}{b x^3+a}dx}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
((a + b*x^3)*(1/(6*a*x^2*(a + b*x^3)^2) + (4*(1/(3*a*x^2*(a + b*x^3)) + (5 *(-1/2*1/(a*x^2) - (b*(Log[a^(1/3) + b^(1/3)*x]/(3*a^(2/3)*b^(1/3)) + (-(( Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log[a^(2/3 ) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3)))/(3*a^(2/3))))/a))/(3*a)) )/(3*a)))/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]
3.2.5.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.80 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.37
| method | result | size |
| risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {10 b^{2} x^{6}}{9 a^{3}}-\frac {16 b \,x^{3}}{9 a^{2}}-\frac {1}{2 a}\right )}{\left (b \,x^{3}+a \right )^{3} x^{2}}+\frac {20 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{11}-3 b^{2}\right ) x -a^{4} b \textit {\_R} \right )\right )}{27 \left (b \,x^{3}+a \right )}\) | \(116\) |
| default | \(-\frac {\left (-40 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b^{2} x^{8}+40 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{2} x^{8}-20 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) b^{2} x^{8}+60 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2} x^{6}-80 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a b \,x^{5}+80 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a b \,x^{5}-40 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a b \,x^{5}+96 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b \,x^{3}-40 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2} x^{2}+40 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} x^{2}-20 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{2} x^{2}+27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}\right ) \left (b \,x^{3}+a \right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} x^{2} a^{3} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}\) | \(322\) |
((b*x^3+a)^2)^(1/2)/(b*x^3+a)^3*(-10/9*b^2/a^3*x^6-16/9*b/a^2*x^3-1/2/a)/x ^2+20/27*((b*x^3+a)^2)^(1/2)/(b*x^3+a)*sum(_R*ln((-4*_R^3*a^11-3*b^2)*x-a^ 4*b*_R),_R=RootOf(_Z^3*a^11+b^2))
Time = 0.27 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.77 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=-\frac {60 \, b^{2} x^{6} + 96 \, a b x^{3} - 40 \, \sqrt {3} {\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 20 \, {\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - 40 \, {\left (b^{2} x^{8} + 2 \, a b x^{5} + a^{2} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 27 \, a^{2}}{54 \, {\left (a^{3} b^{2} x^{8} + 2 \, a^{4} b x^{5} + a^{5} x^{2}\right )}} \]
-1/54*(60*b^2*x^6 + 96*a*b*x^3 - 40*sqrt(3)*(b^2*x^8 + 2*a*b*x^5 + a^2*x^2 )*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b^2/a^2)^(2/3) - sqrt(3)*b) /b) + 20*(b^2*x^8 + 2*a*b*x^5 + a^2*x^2)*(-b^2/a^2)^(1/3)*log(b^2*x^2 + a* b*x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 40*(b^2*x^8 + 2*a*b*x^5 + a ^2*x^2)*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/3)) + 27*a^2)/(a^3*b^2* x^8 + 2*a^4*b*x^5 + a^5*x^2)
\[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.47 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=-\frac {20 \, b^{2} x^{6} + 32 \, a b x^{3} + 9 \, a^{2}}{18 \, {\left (a^{3} b^{2} x^{8} + 2 \, a^{4} b x^{5} + a^{5} x^{2}\right )}} - \frac {20 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {10 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {20 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
-1/18*(20*b^2*x^6 + 32*a*b*x^3 + 9*a^2)/(a^3*b^2*x^8 + 2*a^4*b*x^5 + a^5*x ^2) - 20/27*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a ^3*(a/b)^(2/3)) + 10/27*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*(a/b)^ (2/3)) - 20/27*log(x + (a/b)^(1/3))/(a^3*(a/b)^(2/3))
Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.58 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {20 \, b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {20 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {10 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {20 \, b^{2} x^{6} + 32 \, a b x^{3} + 9 \, a^{2}}{18 \, {\left (b x^{4} + a x\right )}^{2} a^{3} \mathrm {sgn}\left (b x^{3} + a\right )} \]
20/27*b*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^4*sgn(b*x^3 + a)) - 20/ 27*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^( 1/3))/(a^4*sgn(b*x^3 + a)) - 10/27*(-a*b^2)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*sgn(b*x^3 + a)) - 1/18*(20*b^2*x^6 + 32*a*b*x^3 + 9* a^2)/((b*x^4 + a*x)^2*a^3*sgn(b*x^3 + a))
Timed out. \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}} \,d x \]