Integrand size = 22, antiderivative size = 273 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {5 x}{18 a^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{6 a \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} \sqrt [3]{b} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} \sqrt [3]{b} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b} \sqrt {a^2+2 a b x^3+b^2 x^6}} \] Output:
5/18*x/a^2/((b*x^3+a)^2)^(1/2)+1/6*x/a/(b*x^3+a)/((b*x^3+a)^2)^(1/2)-5/27* (b*x^3+a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(8/3 )/b^(1/3)/((b*x^3+a)^2)^(1/2)+5/27*(b*x^3+a)*ln(a^(1/3)+b^(1/3)*x)/a^(8/3) /b^(1/3)/((b*x^3+a)^2)^(1/2)-5/54*(b*x^3+a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b ^(2/3)*x^2)/a^(8/3)/b^(1/3)/((b*x^3+a)^2)^(1/2)
Time = 1.07 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {24 a^{5/3} \sqrt [3]{b} x+15 a^{2/3} b^{4/3} x^4-10 \sqrt {3} \left (a+b x^3\right )^2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+10 \left (a+b x^3\right )^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-5 a^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-10 a b x^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-5 b^2 x^6 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} \sqrt [3]{b} \left (a+b x^3\right ) \sqrt {\left (a+b x^3\right )^2}} \] Input:
Integrate[(a^2 + 2*a*b*x^3 + b^2*x^6)^(-3/2),x]
Output:
(24*a^(5/3)*b^(1/3)*x + 15*a^(2/3)*b^(4/3)*x^4 - 10*Sqrt[3]*(a + b*x^3)^2* ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 10*(a + b*x^3)^2*Log[a^(1/3) + b^(1/3)*x] - 5*a^2*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 10* a*b*x^3*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 5*b^2*x^6*Log[a^( 2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/3)*b^(1/3)*(a + b*x^3)*S qrt[(a + b*x^3)^2])
Time = 0.39 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.76, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {1384, 749, 749, 750, 16, 27, 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.
| \(\displaystyle \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \int \frac {1}{\left (b^2 x^3+a b\right )^3}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \left (\frac {5 \int \frac {1}{\left (b^2 x^3+a b\right )^2}dx}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \int \frac {1}{b^2 x^3+a b}dx}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {\int \frac {\sqrt [3]{b} \left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right )}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{3 a^{2/3} b^{2/3}}+\frac {\int \frac {1}{b^{2/3} x+\sqrt [3]{a} \sqrt [3]{b}}dx}{3 a^{2/3} b^{2/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {\int \frac {\sqrt [3]{b} \left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right )}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{3 a^{2/3} b^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx-\frac {\int -\frac {b \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{2 b}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx+\frac {\int \frac {b \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{2 b}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {b^3 \left (a+b x^3\right ) \left (\frac {5 \left (\frac {2 \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 b}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
Input:
Int[(a^2 + 2*a*b*x^3 + b^2*x^6)^(-3/2),x]
Output:
(b^3*(a + b*x^3)*(x/(6*a*b^3*(a + b*x^3)^2) + (5*(x/(3*a*b^2*(a + b*x^3)) + (2*(Log[a^(1/3) + b^(1/3)*x]/(3*a^(2/3)*b^(4/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b))/(3*a^(2/3)*b^(1/3))))/(3*a*b)))/(6*a*b)))/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]
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_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
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[((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 = 1.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.32
| method | result | size |
| risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {5 b \,x^{4}}{18 a^{2}}+\frac {4 x}{9 a}\right )}{\left (b \,x^{3}+a \right )^{3}}+\frac {5 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{27 \left (b \,x^{3}+a \right ) b \,a^{2}}\) | \(88\) |
| default | \(\frac {\left (-10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b^{2} x^{6}+10 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{2} x^{6}-5 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) b^{2} x^{6}+15 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2} x^{4}-20 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a b \,x^{3}+20 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a b \,x^{3}-10 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a b \,x^{3}+24 \left (\frac {a}{b}\right )^{\frac {2}{3}} a b x -10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2 x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2}+10 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2}-5 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{2}\right ) \left (b \,x^{3}+a \right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} b \,a^{2} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {3}{2}}}\) | \(299\) |
Input:
int(1/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
((b*x^3+a)^2)^(1/2)/(b*x^3+a)^3*(5/18/a^2*b*x^4+4/9/a*x)+5/27*((b*x^3+a)^2 )^(1/2)/(b*x^3+a)/b/a^2*sum(1/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Time = 0.11 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.83 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\left [\frac {15 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x + 15 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 10 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{3} + a^{6} b\right )}}, \frac {15 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x + 30 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 10 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{3} + a^{6} b\right )}}\right ] \] Input:
integrate(1/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x, algorithm="fricas")
Output:
[1/54*(15*a^2*b^2*x^4 + 24*a^3*b*x + 15*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^2*x ^3 + a^3*b)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-(a ^2*b)^(1/3)/b))/(b*x^3 + a)) - 5*(b^2*x^6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3) *log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 10*(b^2*x^6 + 2*a*b*x^ 3 + a^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^4*b^3*x^6 + 2*a^5*b^ 2*x^3 + a^6*b), 1/54*(15*a^2*b^2*x^4 + 24*a^3*b*x + 30*sqrt(1/3)*(a*b^3*x^ 6 + 2*a^2*b^2*x^3 + a^3*b)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2* b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 5*(b^2*x^6 + 2* a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3) *a) + 10*(b^2*x^6 + 2*a*b*x^3 + a^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/ 3)))/(a^4*b^3*x^6 + 2*a^5*b^2*x^3 + a^6*b)]
\[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\int \frac {1}{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(b**2*x**6+2*a*b*x**3+a**2)**(3/2),x)
Output:
Integral((a**2 + 2*a*b*x**3 + b**2*x**6)**(-3/2), x)
Time = 0.12 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {5 \, b x^{4} + 8 \, a x}{18 \, {\left (a^{2} b^{2} x^{6} + 2 \, a^{3} b x^{3} + a^{4}\right )}} + \frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {5 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(1/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x, algorithm="maxima")
Output:
1/18*(5*b*x^4 + 8*a*x)/(a^2*b^2*x^6 + 2*a^3*b*x^3 + a^4) + 5/27*sqrt(3)*ar ctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^2*b*(a/b)^(2/3)) - 5/ 54*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^2*b*(a/b)^(2/3)) + 5/27*log(x + (a/b)^(1/3))/(a^2*b*(a/b)^(2/3))
Time = 0.15 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.65 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=-\frac {5 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{3} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{3} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {5 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {5 \, b x^{4} + 8 \, a x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} \mathrm {sgn}\left (b x^{3} + a\right )} \] Input:
integrate(1/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x, algorithm="giac")
Output:
-5/27*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^3*sgn(b*x^3 + a)) + 5/27* sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3 ))/(a^3*b*sgn(b*x^3 + a)) + 5/54*(-a*b^2)^(1/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b*sgn(b*x^3 + a)) + 1/18*(5*b*x^4 + 8*a*x)/((b*x^3 + a )^2*a^2*sgn(b*x^3 + a))
Timed out. \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\int \frac {1}{{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{3/2}} \,d x \] Input:
int(1/(a^2 + b^2*x^6 + 2*a*b*x^3)^(3/2),x)
Output:
int(1/(a^2 + b^2*x^6 + 2*a*b*x^3)^(3/2), x)
Time = 0.18 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx=\frac {-10 a^{\frac {7}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right )-20 a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b \,x^{3}-10 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} x^{6}-5 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )-10 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b \,x^{3}-5 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{2} x^{6}+10 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )+20 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b \,x^{3}+10 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{2} x^{6}+24 b^{\frac {1}{3}} a^{2} x +15 b^{\frac {4}{3}} a \,x^{4}}{54 b^{\frac {1}{3}} a^{3} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )} \] Input:
int(1/(b^2*x^6+2*a*b*x^3+a^2)^(3/2),x)
Output:
( - 10*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3))) *a**2 - 20*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt( 3)))*a*b*x**3 - 10*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/ 3)*sqrt(3)))*b**2*x**6 - 5*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b **(2/3)*x**2)*a**2 - 10*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**( 2/3)*x**2)*a*b*x**3 - 5*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**( 2/3)*x**2)*b**2*x**6 + 10*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a**2 + 20*a* *(1/3)*log(a**(1/3) + b**(1/3)*x)*a*b*x**3 + 10*a**(1/3)*log(a**(1/3) + b* *(1/3)*x)*b**2*x**6 + 24*b**(1/3)*a**2*x + 15*b**(1/3)*a*b*x**4)/(54*b**(1 /3)*a**3*(a**2 + 2*a*b*x**3 + b**2*x**6))