Integrand size = 22, antiderivative size = 364 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {x \left (a+b x^3\right )}{12 a \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^2}{108 a^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {11 x \left (a+b x^3\right )^3}{81 a^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {55 x \left (a+b x^3\right )^4}{243 a^4 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {110 \left (a+b x^3\right )^5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}+\frac {110 \left (a+b x^3\right )^5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}}-\frac {55 \left (a+b x^3\right )^5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{14/3} \sqrt [3]{b} \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \]
1/12*x*(b*x^3+a)/a/(b^2*x^6+2*a*b*x^3+a^2)^(5/2)+11/108*x*(b*x^3+a)^2/a^2/ (b^2*x^6+2*a*b*x^3+a^2)^(5/2)+11/81*x*(b*x^3+a)^3/a^3/(b^2*x^6+2*a*b*x^3+a ^2)^(5/2)+55/243*x*(b*x^3+a)^4/a^4/(b^2*x^6+2*a*b*x^3+a^2)^(5/2)+110/729*( b*x^3+a)^5*ln(a^(1/3)+b^(1/3)*x)/a^(14/3)/b^(1/3)/(b^2*x^6+2*a*b*x^3+a^2)^ (5/2)-55/729*(b*x^3+a)^5*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(14/3 )/b^(1/3)/(b^2*x^6+2*a*b*x^3+a^2)^(5/2)-110/729*(b*x^3+a)^5*arctan(1/3*(a^ (1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(14/3)/b^(1/3)/(b^2*x^6+2*a*b*x^3+a^ 2)^(5/2)*3^(1/2)
Time = 1.09 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (243 a^{11/3} x+297 a^{8/3} x \left (a+b x^3\right )+396 a^{5/3} x \left (a+b x^3\right )^2+660 a^{2/3} x \left (a+b x^3\right )^3+\frac {440 \sqrt {3} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}+\frac {440 \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {220 \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}\right )}{2916 a^{14/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]
((a + b*x^3)*(243*a^(11/3)*x + 297*a^(8/3)*x*(a + b*x^3) + 396*a^(5/3)*x*( a + b*x^3)^2 + 660*a^(2/3)*x*(a + b*x^3)^3 + (440*Sqrt[3]*(a + b*x^3)^4*Ar cTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/b^(1/3) + (440*(a + b*x^ 3)^4*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (220*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3)))/(2916*a^(14/3)*((a + b*x^3)^2) ^(5/2))
Time = 0.46 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.74, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {1384, 749, 749, 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.
| \(\displaystyle \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \int \frac {1}{\left (b^2 x^3+a b\right )^5}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \int \frac {1}{\left (b^2 x^3+a b\right )^4}dx}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \int \frac {1}{\left (b^2 x^3+a b\right )^3}dx}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \left (\frac {5 \int \frac {1}{\left (b^2 x^3+a b\right )^2}dx}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \left (\frac {5 \left (\frac {2 \int \frac {1}{b^2 x^3+a b}dx}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\int \frac {\sqrt [3]{b} \left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right )}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{3 a^{2/3} b^{2/3}}+\frac {\int \frac {1}{b^{2/3} x+\sqrt [3]{a} \sqrt [3]{b}}dx}{3 a^{2/3} b^{2/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\int \frac {\sqrt [3]{b} \left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right )}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{3 a^{2/3} b^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx-\frac {\int -\frac {b \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{2 b}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx+\frac {\int \frac {b \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{2 b}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{4/3} x^2-\sqrt [3]{a} b x+a^{2/3} b^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \left (\frac {11 \left (\frac {8 \left (\frac {5 \left (\frac {2 \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 b}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}\right )}{3 a b}+\frac {x}{3 a b^2 \left (a+b x^3\right )}\right )}{6 a b}+\frac {x}{6 a b^3 \left (a+b x^3\right )^2}\right )}{9 a b}+\frac {x}{9 a b^4 \left (a+b x^3\right )^3}\right )}{12 a b}+\frac {x}{12 a b^5 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
(b^5*(a + b*x^3)*(x/(12*a*b^5*(a + b*x^3)^4) + (11*(x/(9*a*b^4*(a + b*x^3) ^3) + (8*(x/(6*a*b^3*(a + b*x^3)^2) + (5*(x/(3*a*b^2*(a + b*x^3)) + (2*(Lo g[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)))/(9*a*b)))/(12*a*b)) )/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]
3.2.13.3.1 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[((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 = 1.53 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.30
| method | result | size |
| risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\frac {55 b^{3} x^{10}}{243 a^{4}}+\frac {22 b^{2} x^{7}}{27 a^{3}}+\frac {341 b \,x^{4}}{324 a^{2}}+\frac {133 x}{243 a}\right )}{\left (b \,x^{3}+a \right )^{5}}+\frac {110 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}\right )}{729 \left (b \,x^{3}+a \right ) b \,a^{4}}\) | \(110\) |
| default | \(\frac {\left (-440 \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}+440 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{12}-220 \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}+660 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4} x^{10}-1760 \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}+1760 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{9}-880 \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}+2376 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{3} x^{7}-2640 \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}+2640 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{6}-1320 \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}+3069 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b^{2} x^{4}-1760 \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}+1760 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{3}-880 \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}+1596 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b x -440 \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}+440 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4}-220 \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 \,a^{4} {\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*(55/243/a^4*b^3*x^10+22/27*b^2/a^3*x^7+341 /324*b/a^2*x^4+133/243*x/a)+110/729*((b*x^3+a)^2)^(1/2)/(b*x^3+a)/b/a^4*su m(1/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Time = 0.28 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.98 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\left [\frac {660 \, a^{2} b^{4} x^{10} + 2376 \, a^{3} b^{3} x^{7} + 3069 \, a^{4} b^{2} x^{4} + 1596 \, a^{5} b x + 660 \, \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 ) - 220 \, {\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 ) + 440 \, {\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^{6} b^{5} x^{12} + 4 \, a^{7} b^{4} x^{9} + 6 \, a^{8} b^{3} x^{6} + 4 \, a^{9} b^{2} x^{3} + a^{10} b\right )}}, \frac {660 \, a^{2} b^{4} x^{10} + 2376 \, a^{3} b^{3} x^{7} + 3069 \, a^{4} b^{2} x^{4} + 1596 \, a^{5} b x + 1320 \, \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 ) - 220 \, {\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 ) + 440 \, {\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^{6} b^{5} x^{12} + 4 \, a^{7} b^{4} x^{9} + 6 \, a^{8} b^{3} x^{6} + 4 \, a^{9} b^{2} x^{3} + a^{10} b\right )}}\right ] \]
[1/2916*(660*a^2*b^4*x^10 + 2376*a^3*b^3*x^7 + 3069*a^4*b^2*x^4 + 1596*a^5 *b*x + 660*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)) - 220*(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) + 440*(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^6*b^5*x^12 + 4*a^7*b^ 4*x^9 + 6*a^8*b^3*x^6 + 4*a^9*b^2*x^3 + a^10*b), 1/2916*(660*a^2*b^4*x^10 + 2376*a^3*b^3*x^7 + 3069*a^4*b^2*x^4 + 1596*a^5*b*x + 1320*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) - 220*(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) + 440*(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^6*b^5*x^12 + 4*a^7*b^4*x^9 + 6*a^ 8*b^3*x^6 + 4*a^9*b^2*x^3 + a^10*b)]
\[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {1}{\left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {220 \, b^{3} x^{10} + 792 \, a b^{2} x^{7} + 1023 \, a^{2} b x^{4} + 532 \, a^{3} x}{972 \, {\left (a^{4} b^{4} x^{12} + 4 \, a^{5} b^{3} x^{9} + 6 \, a^{6} b^{2} x^{6} + 4 \, a^{7} b x^{3} + a^{8}\right )}} + \frac {110 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {55 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {110 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
1/972*(220*b^3*x^10 + 792*a*b^2*x^7 + 1023*a^2*b*x^4 + 532*a^3*x)/(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) + 110/729*sqrt( 3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^4*b*(a/b)^(2/3)) - 55/729*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^4*b*(a/b)^(2/3)) + 110 /729*log(x + (a/b)^(1/3))/(a^4*b*(a/b)^(2/3))
Time = 0.33 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {110 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{5} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {110 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{5} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {55 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{5} b \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {220 \, b^{3} x^{10} + 792 \, a b^{2} x^{7} + 1023 \, a^{2} b x^{4} + 532 \, a^{3} x}{972 \, {\left (b x^{3} + a\right )}^{4} a^{4} \mathrm {sgn}\left (b x^{3} + a\right )} \]
-110/729*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^5*sgn(b*x^3 + a)) + 11 0/729*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b )^(1/3))/(a^5*b*sgn(b*x^3 + a)) + 55/729*(-a*b^2)^(1/3)*log(x^2 + x*(-a/b) ^(1/3) + (-a/b)^(2/3))/(a^5*b*sgn(b*x^3 + a)) + 1/972*(220*b^3*x^10 + 792* a*b^2*x^7 + 1023*a^2*b*x^4 + 532*a^3*x)/((b*x^3 + a)^4*a^4*sgn(b*x^3 + a))
Timed out. \[ \int \frac {1}{\left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {1}{{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \]