Integrand size = 26, antiderivative size = 394 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {527 b x}{486 a^5 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {b x}{12 a^2 \left (a+b x^3\right )^3 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {23 b x}{108 a^3 \left (a+b x^3\right )^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {73 b x}{162 a^4 \left (a+b x^3\right ) \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {a+b x^3}{2 a^5 x^2 \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {770 b^{2/3} \left (a+b x^3\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{243 \sqrt {3} a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}-\frac {770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}}+\frac {385 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{17/3} \sqrt {a^2+2 a b x^3+b^2 x^6}} \] Output:
-527/486*b*x/a^5/((b*x^3+a)^2)^(1/2)-1/12*b*x/a^2/(b*x^3+a)^3/((b*x^3+a)^2 )^(1/2)-23/108*b*x/a^3/(b*x^3+a)^2/((b*x^3+a)^2)^(1/2)-73/162*b*x/a^4/(b*x ^3+a)/((b*x^3+a)^2)^(1/2)-1/2*(b*x^3+a)/a^5/x^2/((b*x^3+a)^2)^(1/2)+770/72 9*b^(2/3)*(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^(17/3)/((b*x^3+a)^2)^(1/2)-770/729*b^(2/3)*(b*x^3+a)*ln(a^(1/3)+b^(1 /3)*x)/a^(17/3)/((b*x^3+a)^2)^(1/2)+385/729*b^(2/3)*(b*x^3+a)*ln(a^(2/3)-a ^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(17/3)/((b*x^3+a)^2)^(1/2)
Time = 1.18 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.59 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {\left (a+b x^3\right ) \left (-243 a^{11/3} b x-621 a^{8/3} b x \left (a+b x^3\right )-1314 a^{5/3} b x \left (a+b x^3\right )^2-3162 a^{2/3} b x \left (a+b x^3\right )^3-\frac {1458 a^{2/3} \left (a+b x^3\right )^4}{x^2}-3080 \sqrt {3} b^{2/3} \left (a+b x^3\right )^4 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )-3080 b^{2/3} \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+1540 b^{2/3} \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )\right )}{2916 a^{17/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \] Input:
Integrate[1/(x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]
Output:
((a + b*x^3)*(-243*a^(11/3)*b*x - 621*a^(8/3)*b*x*(a + b*x^3) - 1314*a^(5/ 3)*b*x*(a + b*x^3)^2 - 3162*a^(2/3)*b*x*(a + b*x^3)^3 - (1458*a^(2/3)*(a + b*x^3)^4)/x^2 - 3080*Sqrt[3]*b^(2/3)*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2*b ^(1/3)*x)/(Sqrt[3]*a^(1/3))] - 3080*b^(2/3)*(a + b*x^3)^4*Log[a^(1/3) + b^ (1/3)*x] + 1540*b^(2/3)*(a + b*x^3)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^ (2/3)*x^2]))/(2916*a^(17/3)*((a + b*x^3)^2)^(5/2))
Time = 0.47 (sec) , antiderivative size = 266, 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, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1384 |
| \(\displaystyle \frac {b^5 \left (a+b x^3\right ) \int \frac {1}{b^5 x^3 \left (b x^3+a\right )^5}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \int \frac {1}{x^3 \left (b x^3+a\right )^5}dx}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \int \frac {1}{x^3 \left (b x^3+a\right )^4}dx}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \int \frac {1}{x^3 \left (b x^3+a\right )^3}dx}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \left (\frac {4 \int \frac {1}{x^3 \left (b x^3+a\right )^2}dx}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \left (\frac {4 \left (\frac {5 \int \frac {1}{x^3 \left (b x^3+a\right )}dx}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \left (\frac {4 \left (\frac {5 \left (-\frac {b \int \frac {1}{b x^3+a}dx}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\left (a+b x^3\right ) \left (\frac {7 \left (\frac {11 \left (\frac {4 \left (\frac {5 \left (-\frac {b \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{3 a}+\frac {1}{3 a x^2 \left (a+b x^3\right )}\right )}{3 a}+\frac {1}{6 a x^2 \left (a+b x^3\right )^2}\right )}{9 a}+\frac {1}{9 a x^2 \left (a+b x^3\right )^3}\right )}{6 a}+\frac {1}{12 a x^2 \left (a+b x^3\right )^4}\right )}{\sqrt {a^2+2 a b x^3+b^2 x^6}}\) |
Input:
Int[1/(x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]
Output:
((a + b*x^3)*(1/(12*a*x^2*(a + b*x^3)^4) + (7*(1/(9*a*x^2*(a + b*x^3)^3) + (11*(1/(6*a*x^2*(a + b*x^3)^2) + (4*(1/(3*a*x^2*(a + b*x^3)) + (5*(-1/2*1 /(a*x^2) - (b*(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))/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^(2/3))))/a))/(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[((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*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[((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 = 1.82 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.35
| method | result | size |
| risch | \(\frac {\sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (-\frac {385 b^{4} x^{12}}{243 a^{5}}-\frac {154 b^{3} x^{9}}{27 a^{4}}-\frac {2387 b^{2} x^{6}}{324 a^{3}}-\frac {931 b \,x^{3}}{243 a^{2}}-\frac {1}{2 a}\right )}{\left (b \,x^{3}+a \right )^{5} x^{2}}+\frac {770 \sqrt {\left (b \,x^{3}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{17} \textit {\_Z}^{3}+b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{17}-3 b^{2}\right ) x -a^{6} b \textit {\_R} \right )\right )}{729 \left (b \,x^{3}+a \right )}\) | \(138\) |
| default | \(-\frac {\left (-3080 \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^{14}+3080 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) b^{4} x^{14}-1540 \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^{14}+4620 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4} x^{12}-12320 \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^{11}+12320 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a \,b^{3} x^{11}-6160 \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^{11}+16632 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{3} x^{9}-18480 \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^{8}+18480 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{2} b^{2} x^{8}-9240 \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^{8}+21483 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b^{2} x^{6}-12320 \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^{5}+12320 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{3} b \,x^{5}-6160 \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^{5}+11172 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3} b \,x^{3}-3080 \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^{2}+3080 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) a^{4} x^{2}-1540 \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^{2}+1458 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}\right ) \left (b \,x^{3}+a \right )}{2916 x^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{5} {\left (\left (b \,x^{3}+a \right )^{2}\right )}^{\frac {5}{2}}}\) | \(542\) |
Input:
int(1/x^3/(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*(-385/243/a^5*b^4*x^12-154/27/a^4*b^3*x^9- 2387/324/a^3*b^2*x^6-931/243/a^2*b*x^3-1/2/a)/x^2+770/729*((b*x^3+a)^2)^(1 /2)/(b*x^3+a)*sum(_R*ln((-4*_R^3*a^17-3*b^2)*x-a^6*b*_R),_R=RootOf(_Z^3*a^ 17+b^2))
Time = 0.09 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {4620 \, b^{4} x^{12} + 16632 \, a b^{3} x^{9} + 21483 \, a^{2} b^{2} x^{6} + 11172 \, a^{3} b x^{3} + 1458 \, a^{4} - 3080 \, \sqrt {3} {\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 1540 \, {\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - 3080 \, {\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x - a \left (-\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right )}{2916 \, {\left (a^{5} b^{4} x^{14} + 4 \, a^{6} b^{3} x^{11} + 6 \, a^{7} b^{2} x^{8} + 4 \, a^{8} b x^{5} + a^{9} x^{2}\right )}} \] Input:
integrate(1/x^3/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")
Output:
-1/2916*(4620*b^4*x^12 + 16632*a*b^3*x^9 + 21483*a^2*b^2*x^6 + 11172*a^3*b *x^3 + 1458*a^4 - 3080*sqrt(3)*(b^4*x^14 + 4*a*b^3*x^11 + 6*a^2*b^2*x^8 + 4*a^3*b*x^5 + a^4*x^2)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b^2/a^ 2)^(2/3) - sqrt(3)*b)/b) + 1540*(b^4*x^14 + 4*a*b^3*x^11 + 6*a^2*b^2*x^8 + 4*a^3*b*x^5 + a^4*x^2)*(-b^2/a^2)^(1/3)*log(b^2*x^2 + a*b*x*(-b^2/a^2)^(1 /3) + a^2*(-b^2/a^2)^(2/3)) - 3080*(b^4*x^14 + 4*a*b^3*x^11 + 6*a^2*b^2*x^ 8 + 4*a^3*b*x^5 + a^4*x^2)*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/3))) /(a^5*b^4*x^14 + 4*a^6*b^3*x^11 + 6*a^7*b^2*x^8 + 4*a^8*b*x^5 + a^9*x^2)
\[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {1}{x^{3} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/x**3/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)
Output:
Integral(1/(x**3*((a + b*x**3)**2)**(5/2)), x)
Time = 0.13 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.49 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=-\frac {1540 \, b^{4} x^{12} + 5544 \, a b^{3} x^{9} + 7161 \, a^{2} b^{2} x^{6} + 3724 \, a^{3} b x^{3} + 486 \, a^{4}}{972 \, {\left (a^{5} b^{4} x^{14} + 4 \, a^{6} b^{3} x^{11} + 6 \, a^{7} b^{2} x^{8} + 4 \, a^{8} b x^{5} + a^{9} x^{2}\right )}} - \frac {770 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {385 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {770 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{729 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate(1/x^3/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")
Output:
-1/972*(1540*b^4*x^12 + 5544*a*b^3*x^9 + 7161*a^2*b^2*x^6 + 3724*a^3*b*x^3 + 486*a^4)/(a^5*b^4*x^14 + 4*a^6*b^3*x^11 + 6*a^7*b^2*x^8 + 4*a^8*b*x^5 + a^9*x^2) - 770/729*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^( 1/3))/(a^5*(a/b)^(2/3)) + 385/729*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/( a^5*(a/b)^(2/3)) - 770/729*log(x + (a/b)^(1/3))/(a^5*(a/b)^(2/3))
Time = 0.15 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.55 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\frac {770 \, b \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{729 \, a^{6} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {770 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{729 \, a^{6} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {385 \, \left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{729 \, a^{6} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {1}{2 \, a^{5} x^{2} \mathrm {sgn}\left (b x^{3} + a\right )} - \frac {1054 \, b^{4} x^{10} + 3600 \, a b^{3} x^{7} + 4245 \, a^{2} b^{2} x^{4} + 1780 \, a^{3} b x}{972 \, {\left (b x^{3} + a\right )}^{4} a^{5} \mathrm {sgn}\left (b x^{3} + a\right )} \] Input:
integrate(1/x^3/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")
Output:
770/729*b*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^6*sgn(b*x^3 + a)) - 7 70/729*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/ b)^(1/3))/(a^6*sgn(b*x^3 + a)) - 385/729*(-a*b^2)^(1/3)*log(x^2 + x*(-a/b) ^(1/3) + (-a/b)^(2/3))/(a^6*sgn(b*x^3 + a)) - 1/2/(a^5*x^2*sgn(b*x^3 + a)) - 1/972*(1054*b^4*x^10 + 3600*a*b^3*x^7 + 4245*a^2*b^2*x^4 + 1780*a^3*b*x )/((b*x^3 + a)^4*a^5*sgn(b*x^3 + a))
Timed out. \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx=\int \frac {1}{x^3\,{\left (a^2+2\,a\,b\,x^3+b^2\,x^6\right )}^{5/2}} \,d x \] Input:
int(1/(x^3*(a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)),x)
Output:
int(1/(x^3*(a^2 + b^2*x^6 + 2*a*b*x^3)^(5/2)), x)
Time = 0.19 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.33 \[ \int \frac {1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
int(1/x^3/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)
Output:
(3080*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))* a**4*b*x**2 + 12320*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1 /3)*sqrt(3)))*a**3*b**2*x**5 + 18480*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b **(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**3*x**8 + 12320*a**(1/3)*sqrt(3)*ata n((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**4*x**11 + 3080*a**(1/ 3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**5*x**14 + 1540*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**4*b* x**2 + 6160*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a **3*b**2*x**5 + 9240*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3 )*x**2)*a**2*b**3*x**8 + 6160*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b**4*x**11 + 1540*a**(1/3)*log(a**(2/3) - b**(1/3)*a**( 1/3)*x + b**(2/3)*x**2)*b**5*x**14 - 3080*a**(1/3)*log(a**(1/3) + b**(1/3) *x)*a**4*b*x**2 - 12320*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a**3*b**2*x**5 - 18480*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a**2*b**3*x**8 - 12320*a**(1/ 3)*log(a**(1/3) + b**(1/3)*x)*a*b**4*x**11 - 3080*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*b**5*x**14 - 1458*b**(1/3)*a**5 - 11172*b**(1/3)*a**4*b*x**3 - 21483*b**(1/3)*a**3*b**2*x**6 - 16632*b**(1/3)*a**2*b**3*x**9 - 4620*b**( 1/3)*a*b**4*x**12)/(2916*b**(1/3)*a**6*x**2*(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))