Integrand size = 16, antiderivative size = 157 \[ \int \frac {x (A+B x)}{\left (a+b x^3\right )^2} \, dx=-\frac {B}{3 b \left (a+b x^3\right )}+\frac {A x^2}{3 a \left (a+b x^3\right )}-\frac {A \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{2/3}}-\frac {A \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{2/3}}+\frac {A \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{2/3}} \] Output:
-1/3*B/b/(b*x^3+a)+1/3*A*x^2/a/(b*x^3+a)-1/9*A*arctan(1/3*(a^(1/3)-2*b^(1/ 3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(4/3)/b^(2/3)-1/9*A*ln(a^(1/3)+b^(1/3)*x) /a^(4/3)/b^(2/3)+1/18*A*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(4/3)/ b^(2/3)
Time = 0.07 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.85 \[ \int \frac {x (A+B x)}{\left (a+b x^3\right )^2} \, dx=\frac {\frac {6 \sqrt [3]{a} \left (-a B+A b x^2\right )}{a+b x^3}-2 \sqrt {3} A \sqrt [3]{b} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 A \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+A \sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b} \] Input:
Integrate[(x*(A + B*x))/(a + b*x^3)^2,x]
Output:
((6*a^(1/3)*(-(a*B) + A*b*x^2))/(a + b*x^3) - 2*Sqrt[3]*A*b^(1/3)*ArcTan[( 1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*A*b^(1/3)*Log[a^(1/3) + b^(1/3)*x] + A*b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(4/3)*b )
Time = 0.56 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {2393, 25, 27, 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.
| \(\displaystyle \int \frac {x (A+B x)}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle -\frac {\int -\frac {A x}{b x^3+a}dx}{3 a}-\frac {a B-A b x^2}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {A x}{b x^3+a}dx}{3 a}-\frac {a B-A b x^2}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \int \frac {x}{b x^3+a}dx}{3 a}-\frac {a B-A b x^2}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {A \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{3 a}-\frac {a B-A b x^2}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {A \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 a}-\frac {a B-A b x^2}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {A \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 a}-\frac {a B-A b x^2}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {A \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 a}-\frac {a B-A b x^2}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {A \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 a}-\frac {a B-A b x^2}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {A \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 a}-\frac {a B-A b x^2}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {A \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 a}-\frac {a B-A b x^2}{3 a b \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {A \left (\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{3 a}-\frac {a B-A b x^2}{3 a b \left (a+b x^3\right )}\) |
Input:
Int[(x*(A + B*x))/(a + b*x^3)^2,x]
Output:
-1/3*(a*B - A*b*x^2)/(a*b*(a + b*x^3)) + (A*(-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*a)
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[(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]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q , x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) In t[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^( p + 1), x], x] /; q == n - 1] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n , 0] && LtQ[p, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.37
| method | result | size |
| risch | \(\frac {\frac {x^{2} A}{3 a}-\frac {B}{3 b}}{b \,x^{3}+a}+\frac {A \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{9 a b}\) | \(58\) |
| default | \(A \left (\frac {x^{2}}{3 a \left (b \,x^{3}+a \right )}+\frac {-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{3 a}\right )-\frac {B}{3 b \left (b \,x^{3}+a \right )}\) | \(132\) |
Input:
int(x*(B*x+A)/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
(1/3/a*x^2*A-1/3*B/b)/(b*x^3+a)+1/9*A/a/b*sum(1/_R*ln(x-_R),_R=RootOf(_Z^3 *b+a))
Time = 0.09 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.76 \[ \int \frac {x (A+B x)}{\left (a+b x^3\right )^2} \, dx=\left [\frac {6 \, A a b^{2} x^{2} - 6 \, B a^{2} b + 3 \, \sqrt {\frac {1}{3}} {\left (A a b^{2} x^{3} + A a^{2} b\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) + {\left (A b x^{3} + A a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (A b x^{3} + A a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}}, \frac {6 \, A a b^{2} x^{2} - 6 \, B a^{2} b + 6 \, \sqrt {\frac {1}{3}} {\left (A a b^{2} x^{3} + A a^{2} b\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + {\left (A b x^{3} + A a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) - 2 \, {\left (A b x^{3} + A a\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}}\right ] \] Input:
integrate(x*(B*x+A)/(b*x^3+a)^2,x, algorithm="fricas")
Output:
[1/18*(6*A*a*b^2*x^2 - 6*B*a^2*b + 3*sqrt(1/3)*(A*a*b^2*x^3 + A*a^2*b)*sqr t((-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)) + (A*b*x^3 + A*a)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/ 3)*b*x + (-a*b^2)^(2/3)) - 2*(A*b*x^3 + A*a)*(-a*b^2)^(2/3)*log(b*x - (-a* b^2)^(1/3)))/(a^2*b^3*x^3 + a^3*b^2), 1/18*(6*A*a*b^2*x^2 - 6*B*a^2*b + 6* sqrt(1/3)*(A*a*b^2*x^3 + A*a^2*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) + (A*b*x^3 + A*a)*(-a *b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) - 2*(A*b*x^ 3 + A*a)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a^2*b^3*x^3 + a^3*b^2) ]
Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.36 \[ \int \frac {x (A+B x)}{\left (a+b x^3\right )^2} \, dx=A \operatorname {RootSum} {\left (729 t^{3} a^{4} b^{2} + 1, \left ( t \mapsto t \log {\left (81 t^{2} a^{3} b + x \right )} \right )\right )} + \frac {A b x^{2} - B a}{3 a^{2} b + 3 a b^{2} x^{3}} \] Input:
integrate(x*(B*x+A)/(b*x**3+a)**2,x)
Output:
A*RootSum(729*_t**3*a**4*b**2 + 1, Lambda(_t, _t*log(81*_t**2*a**3*b + x)) ) + (A*b*x**2 - B*a)/(3*a**2*b + 3*a*b**2*x**3)
Time = 0.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.89 \[ \int \frac {x (A+B x)}{\left (a+b x^3\right )^2} \, dx=\frac {\sqrt {3} A \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {A b x^{2} - B a}{3 \, {\left (a b^{2} x^{3} + a^{2} b\right )}} + \frac {A \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {A \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \] Input:
integrate(x*(B*x+A)/(b*x^3+a)^2,x, algorithm="maxima")
Output:
1/9*sqrt(3)*A*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b*(a/ b)^(1/3)) + 1/3*(A*b*x^2 - B*a)/(a*b^2*x^3 + a^2*b) + 1/18*A*log(x^2 - x*( a/b)^(1/3) + (a/b)^(2/3))/(a*b*(a/b)^(1/3)) - 1/9*A*log(x + (a/b)^(1/3))/( a*b*(a/b)^(1/3))
Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87 \[ \int \frac {x (A+B x)}{\left (a+b x^3\right )^2} \, dx=\frac {\sqrt {3} A \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {1}{3}} a} - \frac {A \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {1}{3}} a} - \frac {A \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2}} + \frac {A b x^{2} - B a}{3 \, {\left (b x^{3} + a\right )} a b} \] Input:
integrate(x*(B*x+A)/(b*x^3+a)^2,x, algorithm="giac")
Output:
1/9*sqrt(3)*A*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b ^2)^(1/3)*a) - 1/18*A*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^( 1/3)*a) - 1/9*A*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/a^2 + 1/3*(A*b*x^2 - B*a)/((b*x^3 + a)*a*b)
Time = 6.15 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.91 \[ \int \frac {x (A+B x)}{\left (a+b x^3\right )^2} \, dx=-\frac {\frac {B}{3\,b}-\frac {A\,x^2}{3\,a}}{b\,x^3+a}-\frac {A\,\ln \left (b^{1/3}\,x+a^{1/3}\right )}{9\,a^{4/3}\,b^{2/3}}+\frac {\ln \left (\frac {b^{2/3}\,{\left (A-\sqrt {3}\,A\,1{}\mathrm {i}\right )}^2}{36\,a^{5/3}}+\frac {A^2\,b\,x}{9\,a^2}\right )\,\left (A-\sqrt {3}\,A\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,b^{2/3}}+\frac {\ln \left (\frac {b^{2/3}\,{\left (A+\sqrt {3}\,A\,1{}\mathrm {i}\right )}^2}{36\,a^{5/3}}+\frac {A^2\,b\,x}{9\,a^2}\right )\,\left (A+\sqrt {3}\,A\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,b^{2/3}} \] Input:
int((x*(A + B*x))/(a + b*x^3)^2,x)
Output:
(log((b^(2/3)*(A - 3^(1/2)*A*1i)^2)/(36*a^(5/3)) + (A^2*b*x)/(9*a^2))*(A - 3^(1/2)*A*1i))/(18*a^(4/3)*b^(2/3)) - (A*log(b^(1/3)*x + a^(1/3)))/(9*a^( 4/3)*b^(2/3)) - (B/(3*b) - (A*x^2)/(3*a))/(a + b*x^3) + (log((b^(2/3)*(A + 3^(1/2)*A*1i)^2)/(36*a^(5/3)) + (A^2*b*x)/(9*a^2))*(A + 3^(1/2)*A*1i))/(1 8*a^(4/3)*b^(2/3))
Time = 0.15 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.13 \[ \int \frac {x (A+B x)}{\left (a+b x^3\right )^2} \, dx=\frac {-2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2}-2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a b \,x^{3}+6 b^{\frac {2}{3}} a^{\frac {4}{3}} x^{2}+6 b^{\frac {5}{3}} a^{\frac {1}{3}} x^{3}+\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2}+\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a b \,x^{3}-2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2}-2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a b \,x^{3}}{18 b^{\frac {2}{3}} a^{\frac {4}{3}} \left (b \,x^{3}+a \right )} \] Input:
int(x*(B*x+A)/(b*x^3+a)^2,x)
Output:
( - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2 - 2* sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b*x**3 + 6*b* *(2/3)*a**(1/3)*a*x**2 + 6*b**(2/3)*a**(1/3)*b*x**3 + log(a**(2/3) - b**(1 /3)*a**(1/3)*x + b**(2/3)*x**2)*a**2 + log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b*x**3 - 2*log(a**(1/3) + b**(1/3)*x)*a**2 - 2*log(a**( 1/3) + b**(1/3)*x)*a*b*x**3)/(18*b**(2/3)*a**(1/3)*a*(a + b*x**3))