Integrand size = 26, antiderivative size = 398 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {455}{972 a^4 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {1}{12 a x \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {13}{108 a^2 x \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {65}{324 a^3 x \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \left (a+b x^3\right )}{243 a^5 x \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {455 \sqrt [3]{b} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{1458 a^{16/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \]
455/972/a^4/x/((b*x^3+a)^2)^(1/2)+1/12/a/x/(b*x^3+a)^3/((b*x^3+a)^2)^(1/2) +13/108/a^2/x/(b*x^3+a)^2/((b*x^3+a)^2)^(1/2)+65/324/a^3/x/(b*x^3+a)/((b*x ^3+a)^2)^(1/2)-455/243*(b*x^3+a)/a^5/x/((b*x^3+a)^2)^(1/2)+455/729*b^(1/3) *(b*x^3+a)*ln(a^(1/3)+b^(1/3)*x)/a^(16/3)/((b*x^3+a)^2)^(1/2)-455/1458*b^( 1/3)*(b*x^3+a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(16/3)/((b*x^3+ a)^2)^(1/2)+455/729*b^(1/3)*(b*x^3+a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^( 1/3)*3^(1/2))/a^(16/3)*3^(1/2)/((b*x^3+a)^2)^(1/2)
Time = 1.09 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.61 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (-243 a^{10/3} b x^2-594 a^{7/3} b x^2 \left (a+b x^3\right )-1179 a^{4/3} b x^2 \left (a+b x^3\right )^2-2544 \sqrt [3]{a} b x^2 \left (a+b x^3\right )^3-\frac {2916 \sqrt [3]{a} \left (a+b x^3\right )^4}{x}-1820 \sqrt {3} \sqrt [3]{b} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )+1820 \sqrt [3]{b} \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-910 \sqrt [3]{b} \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{2916 a^{16/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]
((a + b*x^3)*(-243*a^(10/3)*b*x^2 - 594*a^(7/3)*b*x^2*(a + b*x^3) - 1179*a ^(4/3)*b*x^2*(a + b*x^3)^2 - 2544*a^(1/3)*b*x^2*(a + b*x^3)^3 - (2916*a^(1 /3)*(a + b*x^3)^4)/x - 1820*Sqrt[3]*b^(1/3)*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] + 1820*b^(1/3)*(a + b*x^3)^4*Log[a^(1/3 ) + b^(1/3)*x] - 910*b^(1/3)*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(16/3)*((a + b*x^3)^2)^(5/2))
Time = 0.48 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.68, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {1384, 27, 819, 819, 819, 819, 847, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \int \frac {1}{b^5 x^2 \left (b x^3+a\right )^5}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \int \frac {1}{x^2 \left (b x^3+a\right )^5}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \int \frac {1}{x^2 \left (b x^3+a\right )^4}dx}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \int \frac {1}{x^2 \left (b x^3+a\right )^3}dx}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \left (\frac {7 \int \frac {1}{x^2 \left (b x^3+a\right )^2}dx}{6 a}+\frac {1}{6 a x \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \left (\frac {7 \left (\frac {4 \int \frac {1}{x^2 \left (b x^3+a\right )}dx}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \left (\frac {7 \left (\frac {4 \left (-\frac {b \int \frac {x}{b x^3+a}dx}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \left (\frac {7 \left (\frac {4 \left (-\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \left (\frac {7 \left (\frac {4 \left (-\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \left (\frac {7 \left (\frac {4 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \left (\frac {7 \left (\frac {4 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \left (\frac {7 \left (\frac {4 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \left (\frac {7 \left (\frac {4 \left (-\frac {b \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \left (\frac {7 \left (\frac {4 \left (-\frac {b \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {13 \left (\frac {10 \left (\frac {7 \left (\frac {4 \left (-\frac {b \left (\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{3 a}+\frac {1}{3 a x \left (a+b x^3\right )}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x \left (a+b x^3\right )^3}\right )}{12 a}+\frac {1}{12 a x \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
((a + b*x^3)*(1/(12*a*x*(a + b*x^3)^4) + (13*(1/(9*a*x*(a + b*x^3)^3) + (1 0*(1/(6*a*x*(a + b*x^3)^2) + (7*(1/(3*a*x*(a + b*x^3)) + (4*(-(1/(a*x)) - (b*(-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))))/a))/(3*a)))/(6*a)) )/(9*a)))/(12*a)))/Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]
3.2.15.3.1 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[((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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.34 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.34
| method | result | size |
| risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {455 b^{4} x^{12}}{243 a^{5}}-\frac {2275 b^{3} x^{9}}{324 a^{4}}-\frac {260 b^{2} x^{6}}{27 a^{3}}-\frac {1352 b \,x^{3}}{243 a^{2}}-\frac {1}{a}\right )}{\left (b \,x^{3}+a \right )^{5} x}+\frac {455 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{16} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{16}+3 b \right ) x -a^{11} \textit {\_R}^{2}\right )\right )}{729 \left (b \,x^{3}+a \right )}\) | \(137\) |
| default | \(-\frac {\left (-1820 \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^{13}-1820 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{13}+910 \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^{13}+5460 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4} x^{12}-7280 \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^{10}-7280 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{10}+3640 \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^{10}+20475 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{3} x^{9}-10920 \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^{7}-10920 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{7}+5460 \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^{7}+28080 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b^{2} x^{6}-7280 \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^{4}-7280 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{4}+3640 \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^{4}+16224 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3} b \,x^{3}-1820 \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} x -1820 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4} x +910 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) a^{4} x +2916 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}\right ) \left (b \,x^{3}+a \right )}{2916 \left (\frac {a}{b}\right )^{\frac {1}{3}} x \,a^{5} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}\) | \(536\) |
((b*x^3+a)^2)^(1/2)/(b*x^3+a)^5*(-455/243/a^5*b^4*x^12-2275/324*b^3/a^4*x^ 9-260/27*b^2/a^3*x^6-1352/243*b/a^2*x^3-1/a)/x+455/729*((b*x^3+a)^2)^(1/2) /(b*x^3+a)*sum(_R*ln((-4*_R^3*a^16+3*b)*x-a^11*_R^2),_R=RootOf(_Z^3*a^16-b ))
Time = 0.28 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {5460 \, b^{4} x^{12} + 20475 \, a b^{3} x^{9} + 28080 \, a^{2} b^{2} x^{6} + 16224 \, a^{3} b x^{3} + 2916 \, a^{4} + 1820 \, \sqrt {3} {\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 910 \, {\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 1820 \, {\left (b^{4} x^{13} + 4 \, a b^{3} x^{10} + 6 \, a^{2} b^{2} x^{7} + 4 \, a^{3} b x^{4} + a^{4} x\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{2916 \, {\left (a^{5} b^{4} x^{13} + 4 \, a^{6} b^{3} x^{10} + 6 \, a^{7} b^{2} x^{7} + 4 \, a^{8} b x^{4} + a^{9} x\right )}} \]
-1/2916*(5460*b^4*x^12 + 20475*a*b^3*x^9 + 28080*a^2*b^2*x^6 + 16224*a^3*b *x^3 + 2916*a^4 + 1820*sqrt(3)*(b^4*x^13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^3*b*x^4 + a^4*x)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(b/a)^(1/3) - 1/3*sq rt(3)) + 910*(b^4*x^13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^3*b*x^4 + a^4* x)*(b/a)^(1/3)*log(b*x^2 - a*x*(b/a)^(2/3) + a*(b/a)^(1/3)) - 1820*(b^4*x^ 13 + 4*a*b^3*x^10 + 6*a^2*b^2*x^7 + 4*a^3*b*x^4 + a^4*x)*(b/a)^(1/3)*log(b *x + a*(b/a)^(2/3)))/(a^5*b^4*x^13 + 4*a^6*b^3*x^10 + 6*a^7*b^2*x^7 + 4*a^ 8*b*x^4 + a^9*x)
\[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.48 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {1820 \, b^{4} x^{12} + 6825 \, a b^{3} x^{9} + 9360 \, a^{2} b^{2} x^{6} + 5408 \, a^{3} b x^{3} + 972 \, a^{4}}{972 \, {\left (a^{5} b^{4} x^{13} + 4 \, a^{6} b^{3} x^{10} + 6 \, a^{7} b^{2} x^{7} + 4 \, a^{8} b x^{4} + a^{9} x\right )}} - \frac {455 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {455 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {455 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
-1/972*(1820*b^4*x^12 + 6825*a*b^3*x^9 + 9360*a^2*b^2*x^6 + 5408*a^3*b*x^3 + 972*a^4)/(a^5*b^4*x^13 + 4*a^6*b^3*x^10 + 6*a^7*b^2*x^7 + 4*a^8*b*x^4 + a^9*x) - 455/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/ 3))/(a^5*(a/b)^(1/3)) - 455/1458*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a ^5*(a/b)^(1/3)) + 455/729*log(x + (a/b)^(1/3))/(a^5*(a/b)^(1/3))
Time = 0.32 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {455 \, b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{6} \mathrm {sgn}\left (b x^{3} + a\right )} + \frac {455 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{6} b \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {455 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{1458 \, a^{6} b \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {1}{a^{5} x \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {848 \, b^{4} x^{11} + 2937 \, a b^{3} x^{8} + 3528 \, a^{2} b^{2} x^{5} + 1520 \, a^{3} b x^{2}}{972 \, {\left (b x^{3} + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x^{3} + a\right )} \]
455/729*b*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a^6*sgn(b*x^3 + a)) + 4 55/729*sqrt(3)*(-a*b^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/ b)^(1/3))/(a^6*b*sgn(b*x^3 + a)) - 455/1458*(-a*b^2)^(2/3)*log(x^2 + x*(-a /b)^(1/3) + (-a/b)^(2/3))/(a^6*b*sgn(b*x^3 + a)) - 1/(a^5*x*sgn(b*x^3 + a) ) - 1/972*(848*b^4*x^11 + 2937*a*b^3*x^8 + 3528*a^2*b^2*x^5 + 1520*a^3*b*x ^2)/((b*x^3 + a)^4*a^5*sgn(b*x^3 + a))
Timed out. \[ \int \frac {1}{x^2 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {1}{x^2\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \]