Integrand size = 26, antiderivative size = 360 \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {5 x}{243 a^3 b \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {x}{12 b \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{108 a b \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x}{81 a^2 b \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {10 \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {10 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {5 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{11/3} b^{4/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
5/243*x/a^3/b/((b*x^3+a)^2)^(1/2)-1/12*x/b/(b*x^3+a)^3/((b*x^3+a)^2)^(1/2) +1/108*x/a/b/(b*x^3+a)^2/((b*x^3+a)^2)^(1/2)+1/81*x/a^2/b/(b*x^3+a)/((b*x^ 3+a)^2)^(1/2)+10/729*(b*x^3+a)*ln(a^(1/3)+b^(1/3)*x)/a^(11/3)/b^(4/3)/((b* x^3+a)^2)^(1/2)-5/729*(b*x^3+a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/ a^(11/3)/b^(4/3)/((b*x^3+a)^2)^(1/2)-10/729*(b*x^3+a)*arctan(1/3*(a^(1/3)- 2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(11/3)/b^(4/3)*3^(1/2)/((b*x^3+a)^2)^(1/2)
Time = 1.09 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.61 \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (-243 a^{11/3} \sqrt [3]{b} x+27 a^{8/3} \sqrt [3]{b} x \left (a+b x^3\right )+36 a^{5/3} \sqrt [3]{b} x \left (a+b x^3\right )^2+60 a^{2/3} \sqrt [3]{b} x \left (a+b x^3\right )^3+40 \sqrt {3} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )+40 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-20 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{2916 a^{11/3} b^{4/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]
((a + b*x^3)*(-243*a^(11/3)*b^(1/3)*x + 27*a^(8/3)*b^(1/3)*x*(a + b*x^3) + 36*a^(5/3)*b^(1/3)*x*(a + b*x^3)^2 + 60*a^(2/3)*b^(1/3)*x*(a + b*x^3)^3 + 40*Sqrt[3]*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3) )] + 40*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] - 20*(a + b*x^3)^4*Log[a^(2 /3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(11/3)*b^(4/3)*((a + b*x^ 3)^2)^(5/2))
Time = 0.43 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.67, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {1384, 27, 817, 749, 749, 749, 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.
| \(\displaystyle \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \int \frac {x^3}{b^5 \left (b x^3+a\right )^5}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \int \frac {x^3}{\left (b x^3+a\right )^5}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\int \frac {1}{\left (b x^3+a\right )^4}dx}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\frac {8 \int \frac {1}{\left (b x^3+a\right )^3}dx}{9 a}+\frac {x}{9 a \left (a+b x^3\right )^3}}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\frac {8 \left (\frac {5 \int \frac {1}{\left (b x^3+a\right )^2}dx}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^3\right )^3}}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\frac {8 \left (\frac {5 \left (\frac {2 \int \frac {1}{b x^3+a}dx}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^3\right )^3}}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^3\right )^3}}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^3\right )^3}}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^3\right )^3}}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^3\right )^3}}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^3\right )^3}}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^3\right )^3}}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^3\right )^3}}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{3 a}+\frac {x}{3 a \left (a+b x^3\right )}\right )}{6 a}+\frac {x}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x}{9 a \left (a+b x^3\right )^3}}{12 b}-\frac {x}{12 b \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
((a + b*x^3)*(-1/12*x/(b*(a + b*x^3)^4) + (x/(9*a*(a + b*x^3)^3) + (8*(x/( 6*a*(a + b*x^3)^2) + (5*(x/(3*a*(a + b*x^3)) + (2*(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))/Sqr t[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))))/(3*a)))/(6*a)))/(9*a))/(12*b)))/Sqrt[a^2 + 2*a*b*x^3 + b ^2*x^6]
3.2.10.3.1 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_)^(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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && 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 = 2.81 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.30
| method | result | size |
| risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {5 b^{2} x^{10}}{243 a^{3}}+\frac {2 b \,x^{7}}{27 a^{2}}+\frac {31 x^{4}}{324 a}-\frac {10 x}{243 b}\right )}{\left (b \,x^{3}+a \right )^{5}}+\frac {10 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{729 \left (b \,x^{3}+a \right ) a^{3} b^{2}}\) | \(107\) |
| 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^{4} x^{12}+40 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{12}-20 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) b^{4} x^{12}+60 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4} x^{10}-160 \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^{3} x^{9}+160 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{9}-80 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a \,b^{3} x^{9}+216 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{3} x^{7}-240 \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} b^{2} x^{6}+240 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{6}-120 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{2} b^{2} x^{6}+279 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b^{2} x^{4}-160 \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^{3} b \,x^{3}+160 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{3}-80 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{3} b \,x^{3}-120 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b x -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^{4}+40 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4}-20 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{4}\right ) \left (b \,x^{3}+a \right )}{2916 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2} a^{3} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}\) | \(519\) |
((b*x^3+a)^2)^(1/2)/(b*x^3+a)^5*(5/243*b^2/a^3*x^10+2/27*b/a^2*x^7+31/324/ a*x^4-10/243/b*x)+10/729*((b*x^3+a)^2)^(1/2)/(b*x^3+a)/a^3/b^2*sum(1/_R^2* ln(x-_R),_R=RootOf(_Z^3*b+a))
Time = 0.29 (sec) , antiderivative size = 723, normalized size of antiderivative = 2.01 \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\left [\frac {60 \, a^{2} b^{4} x^{10} + 216 \, a^{3} b^{3} x^{7} + 279 \, a^{4} b^{2} x^{4} - 120 \, a^{5} b x + 60 \, \sqrt {\frac {1}{3}} {\left (a b^{5} x^{12} + 4 \, a^{2} b^{4} x^{9} + 6 \, a^{3} b^{3} x^{6} + 4 \, a^{4} b^{2} x^{3} + a^{5} 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 ) - 20 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\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 ) + 40 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{2916 \, {\left (a^{5} b^{6} x^{12} + 4 \, a^{6} b^{5} x^{9} + 6 \, a^{7} b^{4} x^{6} + 4 \, a^{8} b^{3} x^{3} + a^{9} b^{2}\right )}}, \frac {60 \, a^{2} b^{4} x^{10} + 216 \, a^{3} b^{3} x^{7} + 279 \, a^{4} b^{2} x^{4} - 120 \, a^{5} b x + 120 \, \sqrt {\frac {1}{3}} {\left (a b^{5} x^{12} + 4 \, a^{2} b^{4} x^{9} + 6 \, a^{3} b^{3} x^{6} + 4 \, a^{4} b^{2} x^{3} + a^{5} 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 ) - 20 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\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 ) + 40 \, {\left (b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right )}{2916 \, {\left (a^{5} b^{6} x^{12} + 4 \, a^{6} b^{5} x^{9} + 6 \, a^{7} b^{4} x^{6} + 4 \, a^{8} b^{3} x^{3} + a^{9} b^{2}\right )}}\right ] \]
[1/2916*(60*a^2*b^4*x^10 + 216*a^3*b^3*x^7 + 279*a^4*b^2*x^4 - 120*a^5*b*x + 60*sqrt(1/3)*(a*b^5*x^12 + 4*a^2*b^4*x^9 + 6*a^3*b^3*x^6 + 4*a^4*b^2*x^ 3 + a^5*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)) - 20*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^ (1/3)*a) + 40*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4) *(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^5*b^6*x^12 + 4*a^6*b^5*x^9 + 6*a^7*b^4*x^6 + 4*a^8*b^3*x^3 + a^9*b^2), 1/2916*(60*a^2*b^4*x^10 + 216*a ^3*b^3*x^7 + 279*a^4*b^2*x^4 - 120*a^5*b*x + 120*sqrt(1/3)*(a*b^5*x^12 + 4 *a^2*b^4*x^9 + 6*a^3*b^3*x^6 + 4*a^4*b^2*x^3 + a^5*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) - 20*(b^4*x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4 )*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 40*(b^4 *x^12 + 4*a*b^3*x^9 + 6*a^2*b^2*x^6 + 4*a^3*b*x^3 + a^4)*(a^2*b)^(2/3)*log (a*b*x + (a^2*b)^(2/3)))/(a^5*b^6*x^12 + 4*a^6*b^5*x^9 + 6*a^7*b^4*x^6 + 4 *a^8*b^3*x^3 + a^9*b^2)]
\[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x^{3}}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.54 \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {20 \, b^{3} x^{10} + 72 \, a b^{2} x^{7} + 93 \, a^{2} b x^{4} - 40 \, a^{3} x}{972 \, {\left (a^{3} b^{5} x^{12} + 4 \, a^{4} b^{4} x^{9} + 6 \, a^{5} b^{3} x^{6} + 4 \, a^{6} b^{2} x^{3} + a^{7} b\right )}} + \frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {5 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {10 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/972*(20*b^3*x^10 + 72*a*b^2*x^7 + 93*a^2*b*x^4 - 40*a^3*x)/(a^3*b^5*x^12 + 4*a^4*b^4*x^9 + 6*a^5*b^3*x^6 + 4*a^6*b^2*x^3 + a^7*b) + 10/729*sqrt(3) *arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*b^2*(a/b)^(2/3)) - 5/729*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b^2*(a/b)^(2/3)) + 10 /729*log(x + (a/b)^(1/3))/(a^3*b^2*(a/b)^(2/3))
Time = 0.29 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.55 \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {10 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {5 \, \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {10 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{4} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {20 \, b^{3} x^{10} + 72 \, a b^{2} x^{7} + 93 \, a^{2} b x^{4} - 40 \, a^{3} x}{972 \, {\left (b x^{3} + a\right )}^{4} a^{3} b \mathrm {sgn}\left (b x^{3} + a\right )} \]
-10/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a *b^2)^(2/3)*a^3*sgn(b*x^3 + a)) - 5/729*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^ (2/3))/((-a*b^2)^(2/3)*a^3*sgn(b*x^3 + a)) - 10/729*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^4*b*sgn(b*x^3 + a)) + 1/972*(20*b^3*x^10 + 72*a*b^2*x ^7 + 93*a^2*b*x^4 - 40*a^3*x)/((b*x^3 + a)^4*a^3*b*sgn(b*x^3 + a))
Timed out. \[ \int \frac {x^3}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x^3}{{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \]