Integrand size = 20, antiderivative size = 232 \[ \int \frac {x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {(A b-3 a B) x^2}{2 b^4}+\frac {B x^5}{5 b^3}-\frac {a^2 (A b-a B) x^2}{6 b^4 \left (a+b x^3\right )^2}+\frac {a (7 A b-10 a B) x^2}{9 b^4 \left (a+b x^3\right )}+\frac {4 a^{2/3} (5 A b-11 a B) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} b^{14/3}}+\frac {4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}-\frac {2 a^{2/3} (5 A b-11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 b^{14/3}} \] Output:
1/2*(A*b-3*B*a)*x^2/b^4+1/5*B*x^5/b^3-1/6*a^2*(A*b-B*a)*x^2/b^4/(b*x^3+a)^ 2+1/9*a*(7*A*b-10*B*a)*x^2/b^4/(b*x^3+a)+4/27*a^(2/3)*(5*A*b-11*B*a)*arcta n(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/b^(14/3)+4/27*a^(2/3) *(5*A*b-11*B*a)*ln(a^(1/3)+b^(1/3)*x)/b^(14/3)-2/27*a^(2/3)*(5*A*b-11*B*a) *ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(14/3)
Time = 0.20 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.93 \[ \int \frac {x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {135 b^{2/3} (A b-3 a B) x^2+54 b^{5/3} B x^5+\frac {45 a^2 b^{2/3} (-A b+a B) x^2}{\left (a+b x^3\right )^2}+\frac {30 a b^{2/3} (7 A b-10 a B) x^2}{a+b x^3}-40 \sqrt {3} a^{2/3} (-5 A b+11 a B) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-40 a^{2/3} (-5 A b+11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+20 a^{2/3} (-5 A b+11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{270 b^{14/3}} \] Input:
Integrate[(x^10*(A + B*x^3))/(a + b*x^3)^3,x]
Output:
(135*b^(2/3)*(A*b - 3*a*B)*x^2 + 54*b^(5/3)*B*x^5 + (45*a^2*b^(2/3)*(-(A*b ) + a*B)*x^2)/(a + b*x^3)^2 + (30*a*b^(2/3)*(7*A*b - 10*a*B)*x^2)/(a + b*x ^3) - 40*Sqrt[3]*a^(2/3)*(-5*A*b + 11*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/ 3))/Sqrt[3]] - 40*a^(2/3)*(-5*A*b + 11*a*B)*Log[a^(1/3) + b^(1/3)*x] + 20* a^(2/3)*(-5*A*b + 11*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/ (270*b^(14/3))
Time = 0.59 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {957, 817, 831, 2009}
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.
| \(\displaystyle \int \frac {x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 957 |
| \(\displaystyle \frac {x^{11} (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(5 A b-11 a B) \int \frac {x^{10}}{\left (b x^3+a\right )^2}dx}{6 a b}\) |
| \(\Big \downarrow \) 817 |
| \(\displaystyle \frac {x^{11} (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(5 A b-11 a B) \left (\frac {8 \int \frac {x^7}{b x^3+a}dx}{3 b}-\frac {x^8}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
| \(\Big \downarrow \) 831 |
| \(\displaystyle \frac {x^{11} (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(5 A b-11 a B) \left (\frac {8 \int \left (\frac {x^4}{b}+\frac {a^2 x}{b^2 \left (b x^3+a\right )}-\frac {a x}{b^2}\right )dx}{3 b}-\frac {x^8}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^{11} (A b-a B)}{6 a b \left (a+b x^3\right )^2}-\frac {(5 A b-11 a B) \left (\frac {8 \left (-\frac {a^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{8/3}}+\frac {a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}-\frac {a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}-\frac {a x^2}{2 b^2}+\frac {x^5}{5 b}\right )}{3 b}-\frac {x^8}{3 b \left (a+b x^3\right )}\right )}{6 a b}\) |
Input:
Int[(x^10*(A + B*x^3))/(a + b*x^3)^3,x]
Output:
((A*b - a*B)*x^11)/(6*a*b*(a + b*x^3)^2) - ((5*A*b - 11*a*B)*(-1/3*x^8/(b* (a + b*x^3)) + (8*(-1/2*(a*x^2)/b^2 + x^5/(5*b) - (a^(5/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b^(8/3)) - (a^(5/3)*Log[a^(1/ 3) + b^(1/3)*x])/(3*b^(8/3)) + (a^(5/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(8/3))))/(3*b)))/(6*a*b)
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_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x ^m, a + b*x^n, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && Gt Q[m, 2*n - 1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a *b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* (p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 1)]))
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.97 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.50
| method | result | size |
| risch | \(\frac {B \,x^{5}}{5 b^{3}}+\frac {A \,x^{2}}{2 b^{3}}-\frac {3 B a \,x^{2}}{2 b^{4}}+\frac {\left (\frac {7}{9} a \,b^{2} A -\frac {10}{9} a^{2} b B \right ) x^{5}+\frac {a^{2} \left (11 A b -17 B a \right ) x^{2}}{18}}{b^{4} \left (b \,x^{3}+a \right )^{2}}-\frac {4 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (5 A b -11 B a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{27 b^{5}}\) | \(116\) |
| default | \(\frac {\frac {b B \,x^{5}}{5}+\frac {\left (A b -3 B a \right ) x^{2}}{2}}{b^{4}}-\frac {a \left (\frac {\left (-\frac {7}{9} b^{2} A +\frac {10}{9} a b B \right ) x^{5}-\frac {a \left (11 A b -17 B a \right ) x^{2}}{18}}{\left (b \,x^{3}+a \right )^{2}}+\left (\frac {20 A b}{9}-\frac {44 B a}{9}\right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )\right )}{b^{4}}\) | \(176\) |
Input:
int(x^10*(B*x^3+A)/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
1/5*B*x^5/b^3+1/2/b^3*A*x^2-3/2/b^4*B*a*x^2+((7/9*a*b^2*A-10/9*a^2*b*B)*x^ 5+1/18*a^2*(11*A*b-17*B*a)*x^2)/b^4/(b*x^3+a)^2-4/27/b^5*a*sum((5*A*b-11*B *a)/_R*ln(x-_R),_R=RootOf(_Z^3*b+a))
Time = 0.10 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.57 \[ \int \frac {x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {54 \, B b^{3} x^{11} - 27 \, {\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{8} - 96 \, {\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{5} - 60 \, {\left (11 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2} + 40 \, \sqrt {3} {\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \, {\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + 20 \, {\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \, {\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 40 \, {\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \, {\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x + b \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{270 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \] Input:
integrate(x^10*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="fricas")
Output:
1/270*(54*B*b^3*x^11 - 27*(11*B*a*b^2 - 5*A*b^3)*x^8 - 96*(11*B*a^2*b - 5* A*a*b^2)*x^5 - 60*(11*B*a^3 - 5*A*a^2*b)*x^2 + 40*sqrt(3)*((11*B*a*b^2 - 5 *A*b^3)*x^6 + 11*B*a^3 - 5*A*a^2*b + 2*(11*B*a^2*b - 5*A*a*b^2)*x^3)*(a^2/ b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(a^2/b^2)^(1/3) - sqrt(3)*a)/a) + 20* ((11*B*a*b^2 - 5*A*b^3)*x^6 + 11*B*a^3 - 5*A*a^2*b + 2*(11*B*a^2*b - 5*A*a *b^2)*x^3)*(a^2/b^2)^(1/3)*log(a*x^2 - b*x*(a^2/b^2)^(2/3) + a*(a^2/b^2)^( 1/3)) - 40*((11*B*a*b^2 - 5*A*b^3)*x^6 + 11*B*a^3 - 5*A*a^2*b + 2*(11*B*a^ 2*b - 5*A*a*b^2)*x^3)*(a^2/b^2)^(1/3)*log(a*x + b*(a^2/b^2)^(2/3)))/(b^6*x ^6 + 2*a*b^5*x^3 + a^2*b^4)
Time = 1.72 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.83 \[ \int \frac {x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {B x^{5}}{5 b^{3}} + x^{2} \left (\frac {A}{2 b^{3}} - \frac {3 B a}{2 b^{4}}\right ) + \frac {x^{5} \cdot \left (14 A a b^{2} - 20 B a^{2} b\right ) + x^{2} \cdot \left (11 A a^{2} b - 17 B a^{3}\right )}{18 a^{2} b^{4} + 36 a b^{5} x^{3} + 18 b^{6} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} b^{14} - 8000 A^{3} a^{2} b^{3} + 52800 A^{2} B a^{3} b^{2} - 116160 A B^{2} a^{4} b + 85184 B^{3} a^{5}, \left ( t \mapsto t \log {\left (\frac {729 t^{2} b^{9}}{400 A^{2} a b^{2} - 1760 A B a^{2} b + 1936 B^{2} a^{3}} + x \right )} \right )\right )} \] Input:
integrate(x**10*(B*x**3+A)/(b*x**3+a)**3,x)
Output:
B*x**5/(5*b**3) + x**2*(A/(2*b**3) - 3*B*a/(2*b**4)) + (x**5*(14*A*a*b**2 - 20*B*a**2*b) + x**2*(11*A*a**2*b - 17*B*a**3))/(18*a**2*b**4 + 36*a*b**5 *x**3 + 18*b**6*x**6) + RootSum(19683*_t**3*b**14 - 8000*A**3*a**2*b**3 + 52800*A**2*B*a**3*b**2 - 116160*A*B**2*a**4*b + 85184*B**3*a**5, Lambda(_t , _t*log(729*_t**2*b**9/(400*A**2*a*b**2 - 1760*A*B*a**2*b + 1936*B**2*a** 3) + x)))
Time = 0.11 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.98 \[ \int \frac {x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {2 \, {\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5} + {\left (17 \, B a^{3} - 11 \, A a^{2} b\right )} x^{2}}{18 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac {4 \, \sqrt {3} {\left (11 \, B a^{2} - 5 \, A a b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {2 \, B b x^{5} - 5 \, {\left (3 \, B a - A b\right )} x^{2}}{10 \, b^{4}} + \frac {2 \, {\left (11 \, B a^{2} - 5 \, A a b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {4 \, {\left (11 \, B a^{2} - 5 \, A a b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(x^10*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="maxima")
Output:
-1/18*(2*(10*B*a^2*b - 7*A*a*b^2)*x^5 + (17*B*a^3 - 11*A*a^2*b)*x^2)/(b^6* x^6 + 2*a*b^5*x^3 + a^2*b^4) + 4/27*sqrt(3)*(11*B*a^2 - 5*A*a*b)*arctan(1/ 3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^5*(a/b)^(1/3)) + 1/10*(2*B*b *x^5 - 5*(3*B*a - A*b)*x^2)/b^4 + 2/27*(11*B*a^2 - 5*A*a*b)*log(x^2 - x*(a /b)^(1/3) + (a/b)^(2/3))/(b^5*(a/b)^(1/3)) - 4/27*(11*B*a^2 - 5*A*a*b)*log (x + (a/b)^(1/3))/(b^5*(a/b)^(1/3))
Time = 0.13 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.12 \[ \int \frac {x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=-\frac {4 \, {\left (11 \, B a^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, A a b \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{4}} - \frac {4 \, \sqrt {3} {\left (11 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac {2}{3}} A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{6}} + \frac {2 \, {\left (11 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac {2}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, b^{6}} - \frac {20 \, B a^{2} b x^{5} - 14 \, A a b^{2} x^{5} + 17 \, B a^{3} x^{2} - 11 \, A a^{2} b x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} b^{4}} + \frac {2 \, B b^{12} x^{5} - 15 \, B a b^{11} x^{2} + 5 \, A b^{12} x^{2}}{10 \, b^{15}} \] Input:
integrate(x^10*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="giac")
Output:
-4/27*(11*B*a^2*(-a/b)^(1/3) - 5*A*a*b*(-a/b)^(1/3))*(-a/b)^(1/3)*log(abs( x - (-a/b)^(1/3)))/(a*b^4) - 4/27*sqrt(3)*(11*(-a*b^2)^(2/3)*B*a - 5*(-a*b ^2)^(2/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^6 + 2/27*(11*(-a*b^2)^(2/3)*B*a - 5*(-a*b^2)^(2/3)*A*b)*log(x^2 + x*(-a/b)^(1 /3) + (-a/b)^(2/3))/b^6 - 1/18*(20*B*a^2*b*x^5 - 14*A*a*b^2*x^5 + 17*B*a^3 *x^2 - 11*A*a^2*b*x^2)/((b*x^3 + a)^2*b^4) + 1/10*(2*B*b^12*x^5 - 15*B*a*b ^11*x^2 + 5*A*b^12*x^2)/b^15
Time = 0.96 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.92 \[ \int \frac {x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {x^5\,\left (\frac {7\,A\,a\,b^2}{9}-\frac {10\,B\,a^2\,b}{9}\right )-x^2\,\left (\frac {17\,B\,a^3}{18}-\frac {11\,A\,a^2\,b}{18}\right )}{a^2\,b^4+2\,a\,b^5\,x^3+b^6\,x^6}+x^2\,\left (\frac {A}{2\,b^3}-\frac {3\,B\,a}{2\,b^4}\right )+\frac {B\,x^5}{5\,b^3}+\frac {4\,a^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (5\,A\,b-11\,B\,a\right )}{27\,b^{14/3}}+\frac {4\,a^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,A\,b-11\,B\,a\right )}{27\,b^{14/3}}-\frac {4\,a^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,A\,b-11\,B\,a\right )}{27\,b^{14/3}} \] Input:
int((x^10*(A + B*x^3))/(a + b*x^3)^3,x)
Output:
(x^5*((7*A*a*b^2)/9 - (10*B*a^2*b)/9) - x^2*((17*B*a^3)/18 - (11*A*a^2*b)/ 18))/(a^2*b^4 + b^6*x^6 + 2*a*b^5*x^3) + x^2*(A/(2*b^3) - (3*B*a)/(2*b^4)) + (B*x^5)/(5*b^3) + (4*a^(2/3)*log(b^(1/3)*x + a^(1/3))*(5*A*b - 11*B*a)) /(27*b^(14/3)) + (4*a^(2/3)*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3) )*((3^(1/2)*1i)/2 - 1/2)*(5*A*b - 11*B*a))/(27*b^(14/3)) - (4*a^(2/3)*log( 3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(5*A*b - 11*B*a))/(27*b^(14/3))
Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.85 \[ \int \frac {x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx=\frac {-40 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{3}-40 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2} b \,x^{3}-60 b^{\frac {2}{3}} a^{\frac {7}{3}} x^{2}-36 b^{\frac {5}{3}} a^{\frac {4}{3}} x^{5}+9 b^{\frac {8}{3}} a^{\frac {1}{3}} x^{8}+20 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{3}+20 \,\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2} b \,x^{3}-40 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{3}-40 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2} b \,x^{3}}{45 b^{\frac {11}{3}} a^{\frac {1}{3}} \left (b \,x^{3}+a \right )} \] Input:
int(x^10*(B*x^3+A)/(b*x^3+a)^3,x)
Output:
( - 40*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3 - 4 0*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b*x**3 - 60*b**(2/3)*a**(1/3)*a**2*x**2 - 36*b**(2/3)*a**(1/3)*a*b*x**5 + 9*b**(2/ 3)*a**(1/3)*b**2*x**8 + 20*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x **2)*a**3 + 20*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b* x**3 - 40*log(a**(1/3) + b**(1/3)*x)*a**3 - 40*log(a**(1/3) + b**(1/3)*x)* a**2*b*x**3)/(45*b**(2/3)*a**(1/3)*b**3*(a + b*x**3))