Integrand size = 20, antiderivative size = 200 \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx=-\frac {A}{5 a^2 x^5}+\frac {2 A b-a B}{2 a^3 x^2}+\frac {b (A b-a B) x}{3 a^3 \left (a+b x^3\right )}-\frac {b^{2/3} (8 A b-5 a B) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{11/3}}+\frac {b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac {b^{2/3} (8 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3}} \] Output:
-1/5*A/a^2/x^5+1/2*(2*A*b-B*a)/a^3/x^2+1/3*b*(A*b-B*a)*x/a^3/(b*x^3+a)-1/9 *b^(2/3)*(8*A*b-5*B*a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3 ^(1/2)/a^(11/3)+1/9*b^(2/3)*(8*A*b-5*B*a)*ln(a^(1/3)+b^(1/3)*x)/a^(11/3)-1 /18*b^(2/3)*(8*A*b-5*B*a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(11/ 3)
Time = 0.16 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx=\frac {-\frac {18 a^{5/3} A}{x^5}-\frac {45 a^{2/3} (-2 A b+a B)}{x^2}-\frac {30 a^{2/3} b (-A b+a B) x}{a+b x^3}-10 \sqrt {3} b^{2/3} (8 A b-5 a B) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+10 b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+5 b^{2/3} (-8 A b+5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{90 a^{11/3}} \] Input:
Integrate[(A + B*x^3)/(x^6*(a + b*x^3)^2),x]
Output:
((-18*a^(5/3)*A)/x^5 - (45*a^(2/3)*(-2*A*b + a*B))/x^2 - (30*a^(2/3)*b*(-( A*b) + a*B)*x)/(a + b*x^3) - 10*Sqrt[3]*b^(2/3)*(8*A*b - 5*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 10*b^(2/3)*(8*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x] + 5*b^(2/3)*(-8*A*b + 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(90*a^(11/3))
Time = 0.65 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {957, 847, 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.
| \(\displaystyle \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 957 |
| \(\displaystyle \frac {(8 A b-5 a B) \int \frac {1}{x^6 \left (b x^3+a\right )}dx}{3 a b}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {(8 A b-5 a B) \left (-\frac {b \int \frac {1}{x^3 \left (b x^3+a\right )}dx}{a}-\frac {1}{5 a x^5}\right )}{3 a b}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {(8 A b-5 a B) \left (-\frac {b \left (-\frac {b \int \frac {1}{b x^3+a}dx}{a}-\frac {1}{2 a x^2}\right )}{a}-\frac {1}{5 a x^5}\right )}{3 a b}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {(8 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{a}-\frac {1}{2 a x^2}\right )}{a}-\frac {1}{5 a x^5}\right )}{3 a b}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(8 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{a}-\frac {1}{5 a x^5}\right )}{3 a b}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {(8 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{a}-\frac {1}{5 a x^5}\right )}{3 a b}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(8 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{a}-\frac {1}{5 a x^5}\right )}{3 a b}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(8 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{a}-\frac {1}{5 a x^5}\right )}{3 a b}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(8 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{a}-\frac {1}{5 a x^5}\right )}{3 a b}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(8 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{a}-\frac {1}{5 a x^5}\right )}{3 a b}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(8 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{a}-\frac {1}{2 a x^2}\right )}{a}-\frac {1}{5 a x^5}\right )}{3 a b}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}\) |
Input:
Int[(A + B*x^3)/(x^6*(a + b*x^3)^2),x]
Output:
(A*b - a*B)/(3*a*b*x^5*(a + b*x^3)) + ((8*A*b - 5*a*B)*(-1/5*1/(a*x^5) - ( b*(-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))/a))/( 3*a*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[((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*(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[((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)]))
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]
Time = 0.83 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.77
| method | result | size |
| default | \(-\frac {A}{5 a^{2} x^{5}}-\frac {-2 A b +B a}{2 x^{2} a^{3}}+\frac {b \left (\frac {\left (\frac {A b}{3}-\frac {B a}{3}\right ) x}{b \,x^{3}+a}+\frac {\left (8 A b -5 B a \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}\right )}{a^{3}}\) | \(154\) |
| risch | \(\frac {\frac {b \left (8 A b -5 B a \right ) x^{6}}{6 a^{3}}+\frac {\left (8 A b -5 B a \right ) x^{3}}{10 a^{2}}-\frac {A}{5 a}}{x^{5} \left (b \,x^{3}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} \textit {\_Z}^{3}-512 A^{3} b^{5}+960 A^{2} B a \,b^{4}-600 A \,B^{2} a^{2} b^{3}+125 B^{3} a^{3} b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{11}+1536 A^{3} b^{5}-2880 A^{2} B a \,b^{4}+1800 A \,B^{2} a^{2} b^{3}-375 B^{3} a^{3} b^{2}\right ) x +\left (-64 A^{2} a^{4} b^{3}+80 A B \,a^{5} b^{2}-25 B^{2} a^{6} b \right ) \textit {\_R} \right )\right )}{9}\) | \(201\) |
Input:
int((B*x^3+A)/x^6/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/5*A/a^2/x^5-1/2*(-2*A*b+B*a)/x^2/a^3+1/a^3*b*((1/3*A*b-1/3*B*a)*x/(b*x^ 3+a)+1/3*(8*A*b-5*B*a)*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2 /3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3 *3^(1/2)*(2/(a/b)^(1/3)*x-1))))
Time = 0.09 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.38 \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx=-\frac {15 \, {\left (5 \, B a b - 8 \, A b^{2}\right )} x^{6} + 9 \, {\left (5 \, B a^{2} - 8 \, A a b\right )} x^{3} + 18 \, A a^{2} + 10 \, \sqrt {3} {\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} + {\left (5 \, B a^{2} - 8 \, A a b\right )} x^{5}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 5 \, {\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} + {\left (5 \, B a^{2} - 8 \, A a b\right )} x^{5}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 10 \, {\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} + {\left (5 \, B a^{2} - 8 \, A a b\right )} x^{5}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right )}{90 \, {\left (a^{3} b x^{8} + a^{4} x^{5}\right )}} \] Input:
integrate((B*x^3+A)/x^6/(b*x^3+a)^2,x, algorithm="fricas")
Output:
-1/90*(15*(5*B*a*b - 8*A*b^2)*x^6 + 9*(5*B*a^2 - 8*A*a*b)*x^3 + 18*A*a^2 + 10*sqrt(3)*((5*B*a*b - 8*A*b^2)*x^8 + (5*B*a^2 - 8*A*a*b)*x^5)*(b^2/a^2)^ (1/3)*arctan(1/3*(2*sqrt(3)*a*x*(b^2/a^2)^(2/3) - sqrt(3)*b)/b) - 5*((5*B* a*b - 8*A*b^2)*x^8 + (5*B*a^2 - 8*A*a*b)*x^5)*(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)) + 10*((5*B*a*b - 8*A*b^2)*x ^8 + (5*B*a^2 - 8*A*a*b)*x^5)*(b^2/a^2)^(1/3)*log(b*x + a*(b^2/a^2)^(1/3)) )/(a^3*b*x^8 + a^4*x^5)
Time = 0.49 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx=\operatorname {RootSum} {\left (729 t^{3} a^{11} - 512 A^{3} b^{5} + 960 A^{2} B a b^{4} - 600 A B^{2} a^{2} b^{3} + 125 B^{3} a^{3} b^{2}, \left ( t \mapsto t \log {\left (- \frac {9 t a^{4}}{- 8 A b^{2} + 5 B a b} + x \right )} \right )\right )} + \frac {- 6 A a^{2} + x^{6} \cdot \left (40 A b^{2} - 25 B a b\right ) + x^{3} \cdot \left (24 A a b - 15 B a^{2}\right )}{30 a^{4} x^{5} + 30 a^{3} b x^{8}} \] Input:
integrate((B*x**3+A)/x**6/(b*x**3+a)**2,x)
Output:
RootSum(729*_t**3*a**11 - 512*A**3*b**5 + 960*A**2*B*a*b**4 - 600*A*B**2*a **2*b**3 + 125*B**3*a**3*b**2, Lambda(_t, _t*log(-9*_t*a**4/(-8*A*b**2 + 5 *B*a*b) + x))) + (-6*A*a**2 + x**6*(40*A*b**2 - 25*B*a*b) + x**3*(24*A*a*b - 15*B*a**2))/(30*a**4*x**5 + 30*a**3*b*x**8)
Time = 0.11 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx=-\frac {5 \, {\left (5 \, B a b - 8 \, A b^{2}\right )} x^{6} + 3 \, {\left (5 \, B a^{2} - 8 \, A a b\right )} x^{3} + 6 \, A a^{2}}{30 \, {\left (a^{3} b x^{8} + a^{4} x^{5}\right )}} - \frac {\sqrt {3} {\left (5 \, B a - 8 \, A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (5 \, B a - 8 \, A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (5 \, B a - 8 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((B*x^3+A)/x^6/(b*x^3+a)^2,x, algorithm="maxima")
Output:
-1/30*(5*(5*B*a*b - 8*A*b^2)*x^6 + 3*(5*B*a^2 - 8*A*a*b)*x^3 + 6*A*a^2)/(a ^3*b*x^8 + a^4*x^5) - 1/9*sqrt(3)*(5*B*a - 8*A*b)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*(a/b)^(2/3)) + 1/18*(5*B*a - 8*A*b)*log(x ^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*(a/b)^(2/3)) - 1/9*(5*B*a - 8*A*b)* log(x + (a/b)^(1/3))/(a^3*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.03 \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx=-\frac {\sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 8 \, \left (-a b^{2}\right )^{\frac {1}{3}} A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4}} + \frac {{\left (5 \, B a b - 8 \, A b^{2}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{4}} - \frac {{\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 8 \, \left (-a b^{2}\right )^{\frac {1}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{4}} - \frac {B a b x - A b^{2} x}{3 \, {\left (b x^{3} + a\right )} a^{3}} - \frac {5 \, B a x^{3} - 10 \, A b x^{3} + 2 \, A a}{10 \, a^{3} x^{5}} \] Input:
integrate((B*x^3+A)/x^6/(b*x^3+a)^2,x, algorithm="giac")
Output:
-1/9*sqrt(3)*(5*(-a*b^2)^(1/3)*B*a - 8*(-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt (3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/a^4 + 1/9*(5*B*a*b - 8*A*b^2)*(-a/b )^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^4 - 1/18*(5*(-a*b^2)^(1/3)*B*a - 8*(- a*b^2)^(1/3)*A*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^4 - 1/3*(B*a* b*x - A*b^2*x)/((b*x^3 + a)*a^3) - 1/10*(5*B*a*x^3 - 10*A*b*x^3 + 2*A*a)/( a^3*x^5)
Time = 1.06 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.88 \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx=\frac {\frac {x^3\,\left (8\,A\,b-5\,B\,a\right )}{10\,a^2}-\frac {A}{5\,a}+\frac {b\,x^6\,\left (8\,A\,b-5\,B\,a\right )}{6\,a^3}}{b\,x^8+a\,x^5}+\frac {b^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (8\,A\,b-5\,B\,a\right )}{9\,a^{11/3}}-\frac {b^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (8\,A\,b-5\,B\,a\right )}{9\,a^{11/3}}+\frac {b^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (8\,A\,b-5\,B\,a\right )}{9\,a^{11/3}} \] Input:
int((A + B*x^3)/(x^6*(a + b*x^3)^2),x)
Output:
((x^3*(8*A*b - 5*B*a))/(10*a^2) - A/(5*a) + (b*x^6*(8*A*b - 5*B*a))/(6*a^3 ))/(a*x^5 + b*x^8) + (b^(2/3)*log(b^(1/3)*x + a^(1/3))*(8*A*b - 5*B*a))/(9 *a^(11/3)) - (b^(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)*(8*A*b - 5*B*a))/(9*a^(11/3)) + (b^(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)*(8*A*b - 5*B*a))/ (9*a^(11/3))
Time = 0.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.57 \[ \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx=\frac {-10 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} x^{5}-5 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{2} x^{5}+10 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{2} x^{5}-6 b^{\frac {1}{3}} a^{2}+15 b^{\frac {4}{3}} a \,x^{3}}{30 b^{\frac {1}{3}} a^{3} x^{5}} \] Input:
int((B*x^3+A)/x^6/(b*x^3+a)^2,x)
Output:
( - 10*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3))) *b**2*x**5 - 5*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2 )*b**2*x**5 + 10*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*b**2*x**5 - 6*b**(1/3 )*a**2 + 15*b**(1/3)*a*b*x**3)/(30*b**(1/3)*a**3*x**5)