Integrand size = 24, antiderivative size = 359 \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {35 x^2}{243 a^4 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {x^2}{12 a \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {5 x^2}{54 a^2 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 x^2}{324 a^3 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {35 \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {35 \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {35 \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{13/3} b^{2/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \] Output:
35/243*x^2/a^4/((b*x^3+a)^2)^(1/2)+1/12*x^2/a/(b*x^3+a)^3/((b*x^3+a)^2)^(1 /2)+5/54*x^2/a^2/(b*x^3+a)^2/((b*x^3+a)^2)^(1/2)+35/324*x^2/a^3/(b*x^3+a)/ ((b*x^3+a)^2)^(1/2)-35/729*(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^(13/3)/b^(2/3)/((b*x^3+a)^2)^(1/2)-35/729*(b*x^3+a) *ln(a^(1/3)+b^(1/3)*x)/a^(13/3)/b^(2/3)/((b*x^3+a)^2)^(1/2)+35/1458*(b*x^3 +a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(13/3)/b^(2/3)/((b*x^3+a)^ 2)^(1/2)
Time = 1.14 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.61 \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (243 a^{10/3} x^2+270 a^{7/3} x^2 \left (a+b x^3\right )+315 a^{4/3} x^2 \left (a+b x^3\right )^2+420 \sqrt [3]{a} x^2 \left (a+b x^3\right )^3+\frac {140 \sqrt {3} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3}}-\frac {140 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {70 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}\right )}{2916 a^{13/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \] Input:
Integrate[x/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]
Output:
((a + b*x^3)*(243*a^(10/3)*x^2 + 270*a^(7/3)*x^2*(a + b*x^3) + 315*a^(4/3) *x^2*(a + b*x^3)^2 + 420*a^(1/3)*x^2*(a + b*x^3)^3 + (140*Sqrt[3]*(a + b*x ^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/b^(2/3) - (140*( a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + (70*(a + b*x^3)^4*Log[a^( 2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3)))/(2916*a^(13/3)*((a + b* x^3)^2)^(5/2))
Time = 0.44 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.71, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {1384, 27, 819, 819, 819, 819, 821, 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.
| \(\displaystyle \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \int \frac {x}{b^5 \left (b x^3+a\right )^5}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \int \frac {x}{\left (b x^3+a\right )^5}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \int \frac {x}{\left (b x^3+a\right )^4}dx}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {7 \int \frac {x}{\left (b x^3+a\right )^3}dx}{9 a}+\frac {x^2}{9 a \left (a+b x^3\right )^3}\right )}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {7 \left (\frac {2 \int \frac {x}{\left (b x^3+a\right )^2}dx}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x^2}{9 a \left (a+b x^3\right )^3}\right )}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {7 \left (\frac {2 \left (\frac {\int \frac {x}{b x^3+a}dx}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x^2}{9 a \left (a+b x^3\right )^3}\right )}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x^2}{9 a \left (a+b x^3\right )^3}\right )}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x^2}{9 a \left (a+b x^3\right )^3}\right )}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x^2}{9 a \left (a+b x^3\right )^3}\right )}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x^2}{9 a \left (a+b x^3\right )^3}\right )}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x^2}{9 a \left (a+b x^3\right )^3}\right )}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x^2}{9 a \left (a+b x^3\right )^3}\right )}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x^2}{9 a \left (a+b x^3\right )^3}\right )}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {5 \left (\frac {7 \left (\frac {2 \left (\frac {\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {x^2}{3 a \left (a+b x^3\right )}\right )}{3 a}+\frac {x^2}{6 a \left (a+b x^3\right )^2}\right )}{9 a}+\frac {x^2}{9 a \left (a+b x^3\right )^3}\right )}{6 a}+\frac {x^2}{12 a \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
Input:
Int[x/(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]
Output:
((a + b*x^3)*(x^2/(12*a*(a + b*x^3)^4) + (5*(x^2/(9*a*(a + b*x^3)^3) + (7* (x^2/(6*a*(a + b*x^3)^2) + (2*(x^2/(3*a*(a + b*x^3)) + (-1/3*Log[a^(1/3) + b^(1/3)*x]/(a^(1/3)*b^(2/3)) + (-((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^(1/3)*b^(1/3)))/(3*a)))/(3*a)))/(9*a)))/(6*a)))/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[((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[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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 = 1.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.31
| method | result | size |
| risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {35 b^{3} x^{11}}{243 a^{4}}+\frac {175 b^{2} x^{8}}{324 a^{3}}+\frac {20 b \,x^{5}}{27 a^{2}}+\frac {104 x^{2}}{243 a}\right )}{\left (b \,x^{3}+a \right )^{5}}+\frac {35 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{729 \left (b \,x^{3}+a \right ) a^{4} b}\) | \(112\) |
| default | \(\frac {\left (-140 \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}-140 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{12}+70 \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}+420 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4} x^{11}-560 \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}-560 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{9}+280 \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}+1575 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{3} x^{8}-840 \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}-840 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{6}+420 \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}+2160 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b^{2} x^{5}-560 \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}-560 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{3}+280 \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}+1248 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3} b \,x^{2}-140 \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}-140 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4}+70 \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 {1}{3}} b \,a^{4} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}\) | \(521\) |
Input:
int(x/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
((b*x^3+a)^2)^(1/2)/(b*x^3+a)^5*(35/243/a^4*b^3*x^11+175/324/a^3*b^2*x^8+2 0/27/a^2*b*x^5+104/243/a*x^2)+35/729*((b*x^3+a)^2)^(1/2)/(b*x^3+a)/a^4/b*s um(1/_R*ln(x-_R),_R=RootOf(_Z^3*b+a))
Time = 0.08 (sec) , antiderivative size = 734, normalized size of antiderivative = 2.04 \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
integrate(x/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")
Output:
[1/2916*(420*a*b^5*x^11 + 1575*a^2*b^4*x^8 + 2160*a^3*b^3*x^5 + 1248*a^4*b ^2*x^2 + 210*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*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3 )*(a*b*x + 2*(-a*b^2)^(2/3)*x^2 + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*x)/(b*x^3 + a)) + 70*(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*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b* x + (-a*b^2)^(2/3)) - 140*(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*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(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*(420*a*b^5*x ^11 + 1575*a^2*b^4*x^8 + 2160*a^3*b^3*x^5 + 1248*a^4*b^2*x^2 + 420*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)*sq rt(-(-a*b^2)^(1/3)/a)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a* b^2)^(1/3)/a)/b) + 70*(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*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 140*(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*b^ 2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(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)]
\[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x}{\left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)
Output:
Integral(x/((a + b*x**3)**2)**(5/2), x)
Time = 0.13 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.53 \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {140 \, b^{3} x^{11} + 525 \, a b^{2} x^{8} + 720 \, a^{2} b x^{5} + 416 \, a^{3} x^{2}}{972 \, {\left (a^{4} b^{4} x^{12} + 4 \, a^{5} b^{3} x^{9} + 6 \, a^{6} b^{2} x^{6} + 4 \, a^{7} b x^{3} + a^{8}\right )}} + \frac {35 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {35 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {35 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(x/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")
Output:
1/972*(140*b^3*x^11 + 525*a*b^2*x^8 + 720*a^2*b*x^5 + 416*a^3*x^2)/(a^4*b^ 4*x^12 + 4*a^5*b^3*x^9 + 6*a^6*b^2*x^6 + 4*a^7*b*x^3 + a^8) + 35/729*sqrt( 3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^4*b*(a/b)^(1/3)) + 35/1458*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^4*b*(a/b)^(1/3)) - 35 /729*log(x + (a/b)^(1/3))/(a^4*b*(a/b)^(1/3))
Time = 0.15 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.55 \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {35 \, \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {35 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {35 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{5} b^{2} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {140 \, b^{3} x^{11} + 525 \, a b^{2} x^{8} + 720 \, a^{2} b x^{5} + 416 \, a^{3} x^{2}}{972 \, {\left (b x^{3} + a\right )}^{4} a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} \] Input:
integrate(x/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")
Output:
-35/1458*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(1/3)*a^4*sgn( b*x^3 + a)) - 35/729*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a^5*sgn(b*x^ 3 + a)) - 35/729*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^( 1/3))/(-a/b)^(1/3))/(a^5*b^2*sgn(b*x^3 + a)) + 1/972*(140*b^3*x^11 + 525*a *b^2*x^8 + 720*a^2*b*x^5 + 416*a^3*x^2)/((b*x^3 + a)^4*a^4*sgn(b*x^3 + a))
Timed out. \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {x}{{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \] Input:
int(x/(a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2),x)
Output:
int(x/(a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2), x)
Time = 0.18 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.34 \[ \int \frac {x}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {-140 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{4}-560 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{3} b \,x^{3}-840 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2} b^{2} x^{6}-560 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a \,b^{3} x^{9}-140 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{4} x^{12}+1248 b^{\frac {2}{3}} a^{\frac {10}{3}} x^{2}+2160 b^{\frac {5}{3}} a^{\frac {7}{3}} x^{5}+1575 b^{\frac {8}{3}} a^{\frac {4}{3}} x^{8}+420 b^{\frac {11}{3}} a^{\frac {1}{3}} x^{11}+70 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{4}+280 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{3} b \,x^{3}+420 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2} b^{2} x^{6}+280 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a \,b^{3} x^{9}+70 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{4} x^{12}-140 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{4}-560 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{3} b \,x^{3}-840 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2} b^{2} x^{6}-560 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a \,b^{3} x^{9}-140 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{4} x^{12}}{2916 b^{\frac {2}{3}} a^{\frac {13}{3}} \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 )} \] Input:
int(x/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)
Output:
( - 140*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**4 - 560*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b*x**3 - 840*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b** 2*x**6 - 560*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a* b**3*x**9 - 140*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3))) *b**4*x**12 + 1248*b**(2/3)*a**(1/3)*a**3*x**2 + 2160*b**(2/3)*a**(1/3)*a* *2*b*x**5 + 1575*b**(2/3)*a**(1/3)*a*b**2*x**8 + 420*b**(2/3)*a**(1/3)*b** 3*x**11 + 70*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**4 + 28 0*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*b*x**3 + 420*lo g(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b**2*x**6 + 280*log (a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**3*x**9 + 70*log(a**( 2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**4*x**12 - 140*log(a**(1/3) + b**(1/3)*x)*a**4 - 560*log(a**(1/3) + b**(1/3)*x)*a**3*b*x**3 - 840*log( a**(1/3) + b**(1/3)*x)*a**2*b**2*x**6 - 560*log(a**(1/3) + b**(1/3)*x)*a*b **3*x**9 - 140*log(a**(1/3) + b**(1/3)*x)*b**4*x**12)/(2916*b**(2/3)*a**(1 /3)*a**4*(a**4 + 4*a**3*b*x**3 + 6*a**2*b**2*x**6 + 4*a*b**3*x**9 + b**4*x **12))