Integrand size = 13, antiderivative size = 155 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=-\frac {x^5}{6 b \left (a+b x^3\right )^2}-\frac {5 x^2}{18 b^2 \left (a+b x^3\right )}-\frac {5 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} \sqrt [3]{a} b^{8/3}}-\frac {5 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{8/3}}+\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{8/3}} \] Output:
-1/6*x^5/b/(b*x^3+a)^2-5/18*x^2/b^2/(b*x^3+a)-5/27*arctan(1/3*(a^(1/3)-2*b ^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(1/3)/b^(8/3)-5/27*ln(a^(1/3)+b^(1/3) *x)/a^(1/3)/b^(8/3)+5/54*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(1/3) /b^(8/3)
Time = 0.06 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.90 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=\frac {\frac {9 a b^{2/3} x^2}{\left (a+b x^3\right )^2}-\frac {24 b^{2/3} x^2}{a+b x^3}-\frac {10 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}-\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}}{54 b^{8/3}} \] Input:
Integrate[x^7/(a + b*x^3)^3,x]
Output:
((9*a*b^(2/3)*x^2)/(a + b*x^3)^2 - (24*b^(2/3)*x^2)/(a + b*x^3) - (10*Sqrt [3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) - (10*Log[a^(1/3) + b^(1/3)*x])/a^(1/3) + (5*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] )/a^(1/3))/(54*b^(8/3))
Time = 0.52 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {817, 817, 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.
| \(\displaystyle \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {5 \int \frac {x^4}{\left (b x^3+a\right )^2}dx}{6 b}-\frac {x^5}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {5 \left (\frac {2 \int \frac {x}{b x^3+a}dx}{3 b}-\frac {x^2}{3 b \left (a+b x^3\right )}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{3 b}-\frac {x^2}{3 b \left (a+b x^3\right )}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 b}-\frac {x^2}{3 b \left (a+b x^3\right )}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 b}-\frac {x^2}{3 b \left (a+b x^3\right )}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 b}-\frac {x^2}{3 b \left (a+b x^3\right )}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 b}-\frac {x^2}{3 b \left (a+b x^3\right )}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 b}-\frac {x^2}{3 b \left (a+b x^3\right )}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 b}-\frac {x^2}{3 b \left (a+b x^3\right )}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5 \left (\frac {2 \left (\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 b}-\frac {x^2}{3 b \left (a+b x^3\right )}\right )}{6 b}-\frac {x^5}{6 b \left (a+b x^3\right )^2}\) |
Input:
Int[x^7/(a + b*x^3)^3,x]
Output:
-1/6*x^5/(b*(a + b*x^3)^2) + (5*(-1/3*x^2/(b*(a + b*x^3)) + (2*(-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*b)))/(6*b)
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^( n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n *((m - n + 1)/(b*n*(p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x ] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && ! ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]
Int[(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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.47 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.36
| method | result | size |
| risch | \(\frac {-\frac {4 x^{5}}{9 b}-\frac {5 a \,x^{2}}{18 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{27 b^{3}}\) | \(56\) |
| default | \(\frac {-\frac {4 x^{5}}{9 b}-\frac {5 a \,x^{2}}{18 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {-\frac {5 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{b^{2}}\) | \(125\) |
Input:
int(x^7/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
(-4/9*x^5/b-5/18*a*x^2/b^2)/(b*x^3+a)^2+5/27/b^3*sum(1/_R*ln(x-_R),_R=Root Of(_Z^3*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (114) = 228\).
Time = 0.14 (sec) , antiderivative size = 510, normalized size of antiderivative = 3.29 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=\left [-\frac {24 \, a b^{3} x^{5} + 15 \, a^{2} b^{2} x^{2} - 15 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) - 5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 10 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a b^{6} x^{6} + 2 \, a^{2} b^{5} x^{3} + a^{3} b^{4}\right )}}, -\frac {24 \, a b^{3} x^{5} + 15 \, a^{2} b^{2} x^{2} - 30 \, \sqrt {\frac {1}{3}} {\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - 5 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 10 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a b^{6} x^{6} + 2 \, a^{2} b^{5} x^{3} + a^{3} b^{4}\right )}}\right ] \] Input:
integrate(x^7/(b*x^3+a)^3,x, algorithm="fricas")
Output:
[-1/54*(24*a*b^3*x^5 + 15*a^2*b^2*x^2 - 15*sqrt(1/3)*(a*b^3*x^6 + 2*a^2*b^ 2*x^3 + a^3*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)) - 5*(b^2*x^6 + 2*a*b*x^3 + a^2)*(-a*b^2)^ (2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) + 10*(b^2*x^6 + 2 *a*b*x^3 + a^2)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a*b^6*x^6 + 2*a ^2*b^5*x^3 + a^3*b^4), -1/54*(24*a*b^3*x^5 + 15*a^2*b^2*x^2 - 30*sqrt(1/3) *(a*b^3*x^6 + 2*a^2*b^2*x^3 + a^3*b)*sqrt(-(-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) - 5*(b^2*x^6 + 2*a *b*x^3 + a^2)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^( 2/3)) + 10*(b^2*x^6 + 2*a*b*x^3 + a^2)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^( 1/3)))/(a*b^6*x^6 + 2*a^2*b^5*x^3 + a^3*b^4)]
Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.45 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=\frac {- 5 a x^{2} - 8 b x^{5}}{18 a^{2} b^{2} + 36 a b^{3} x^{3} + 18 b^{4} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a b^{8} + 125, \left ( t \mapsto t \log {\left (\frac {729 t^{2} a b^{5}}{25} + x \right )} \right )\right )} \] Input:
integrate(x**7/(b*x**3+a)**3,x)
Output:
(-5*a*x**2 - 8*b*x**5)/(18*a**2*b**2 + 36*a*b**3*x**3 + 18*b**4*x**6) + Ro otSum(19683*_t**3*a*b**8 + 125, Lambda(_t, _t*log(729*_t**2*a*b**5/25 + x) ))
Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=-\frac {8 \, b x^{5} + 5 \, a x^{2}}{18 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} + \frac {5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {5 \, \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {5 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(x^7/(b*x^3+a)^3,x, algorithm="maxima")
Output:
-1/18*(8*b*x^5 + 5*a*x^2)/(b^4*x^6 + 2*a*b^3*x^3 + a^2*b^2) + 5/27*sqrt(3) *arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^3*(a/b)^(1/3)) + 5 /54*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^(1/3)) - 5/27*log(x + (a/b)^(1/3))/(b^3*(a/b)^(1/3))
Time = 0.13 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.92 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=-\frac {5 \, \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{2}} - \frac {5 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a b^{4}} - \frac {8 \, b x^{5} + 5 \, a x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} b^{2}} + \frac {5 \, \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a b^{4}} \] Input:
integrate(x^7/(b*x^3+a)^3,x, algorithm="giac")
Output:
-5/27*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^2) - 5/27*sqrt(3)*(-a*b ^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a*b^4) - 1/18*(8*b*x^5 + 5*a*x^2)/((b*x^3 + a)^2*b^2) + 5/54*(-a*b^2)^(2/3)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^4)
Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.04 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=\frac {5\,\ln \left (\frac {25\,x}{81\,b^3}-\frac {25\,{\left (-a\right )}^{1/3}}{81\,b^{10/3}}\right )}{27\,{\left (-a\right )}^{1/3}\,b^{8/3}}-\frac {\frac {4\,x^5}{9\,b}+\frac {5\,a\,x^2}{18\,b^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}+\frac {\ln \left (\frac {25\,x}{81\,b^3}-\frac {{\left (-a\right )}^{1/3}\,{\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}^2}{324\,b^{10/3}}\right )\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )}{54\,{\left (-a\right )}^{1/3}\,b^{8/3}}-\frac {\ln \left (\frac {25\,x}{81\,b^3}-\frac {{\left (-a\right )}^{1/3}\,{\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}^2}{324\,b^{10/3}}\right )\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )}{54\,{\left (-a\right )}^{1/3}\,b^{8/3}} \] Input:
int(x^7/(a + b*x^3)^3,x)
Output:
(5*log((25*x)/(81*b^3) - (25*(-a)^(1/3))/(81*b^(10/3))))/(27*(-a)^(1/3)*b^ (8/3)) - ((4*x^5)/(9*b) + (5*a*x^2)/(18*b^2))/(a^2 + b^2*x^6 + 2*a*b*x^3) + (log((25*x)/(81*b^3) - ((-a)^(1/3)*(3^(1/2)*5i - 5)^2)/(324*b^(10/3)))*( 3^(1/2)*5i - 5))/(54*(-a)^(1/3)*b^(8/3)) - (log((25*x)/(81*b^3) - ((-a)^(1 /3)*(3^(1/2)*5i + 5)^2)/(324*b^(10/3)))*(3^(1/2)*5i + 5))/(54*(-a)^(1/3)*b ^(8/3))
Time = 0.22 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.72 \[ \int \frac {x^7}{\left (a+b x^3\right )^3} \, dx=\frac {-10 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2}-20 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a b \,x^{3}-10 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} x^{6}-15 b^{\frac {2}{3}} a^{\frac {4}{3}} x^{2}-24 b^{\frac {5}{3}} a^{\frac {1}{3}} x^{5}+5 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2}+10 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a b \,x^{3}+5 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{2} x^{6}-10 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2}-20 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a b \,x^{3}-10 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{2} x^{6}}{54 b^{\frac {8}{3}} a^{\frac {1}{3}} \left (b^{2} x^{6}+2 a b \,x^{3}+a^{2}\right )} \] Input:
int(x^7/(b*x^3+a)^3,x)
Output:
( - 10*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2 - 2 0*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b*x**3 - 10 *sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**2*x**6 - 15 *b**(2/3)*a**(1/3)*a*x**2 - 24*b**(2/3)*a**(1/3)*b*x**5 + 5*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2 + 10*log(a**(2/3) - b**(1/3)*a* *(1/3)*x + b**(2/3)*x**2)*a*b*x**3 + 5*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**2*x**6 - 10*log(a**(1/3) + b**(1/3)*x)*a**2 - 20*log(a **(1/3) + b**(1/3)*x)*a*b*x**3 - 10*log(a**(1/3) + b**(1/3)*x)*b**2*x**6)/ (54*b**(2/3)*a**(1/3)*b**2*(a**2 + 2*a*b*x**3 + b**2*x**6))