Integrand size = 17, antiderivative size = 169 \[ \int \frac {a+b x^3}{\left (c+d x^3\right )^2} \, dx=-\frac {(b c-a d) x}{3 c d \left (c+d x^3\right )}-\frac {(b c+2 a d) \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} c^{5/3} d^{4/3}}+\frac {(b c+2 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{4/3}}-\frac {(b c+2 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{18 c^{5/3} d^{4/3}} \] Output:
-1/3*(-a*d+b*c)*x/c/d/(d*x^3+c)-1/9*(2*a*d+b*c)*arctan(1/3*(c^(1/3)-2*d^(1 /3)*x)*3^(1/2)/c^(1/3))*3^(1/2)/c^(5/3)/d^(4/3)+1/9*(2*a*d+b*c)*ln(c^(1/3) +d^(1/3)*x)/c^(5/3)/d^(4/3)-1/18*(2*a*d+b*c)*ln(c^(2/3)-c^(1/3)*d^(1/3)*x+ d^(2/3)*x^2)/c^(5/3)/d^(4/3)
Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.86 \[ \int \frac {a+b x^3}{\left (c+d x^3\right )^2} \, dx=\frac {-\frac {6 c^{2/3} \sqrt [3]{d} (b c-a d) x}{c+d x^3}-2 \sqrt {3} (b c+2 a d) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )+2 (b c+2 a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-(b c+2 a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{18 c^{5/3} d^{4/3}} \] Input:
Integrate[(a + b*x^3)/(c + d*x^3)^2,x]
Output:
((-6*c^(2/3)*d^(1/3)*(b*c - a*d)*x)/(c + d*x^3) - 2*Sqrt[3]*(b*c + 2*a*d)* ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] + 2*(b*c + 2*a*d)*Log[c^(1/3) + d^(1/3)*x] - (b*c + 2*a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2 ])/(18*c^(5/3)*d^(4/3))
Time = 0.51 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {910, 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.
| \(\displaystyle \int \frac {a+b x^3}{\left (c+d x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 910 |
| \(\displaystyle \frac {(2 a d+b c) \int \frac {1}{d x^3+c}dx}{3 c d}-\frac {x (b c-a d)}{3 c d \left (c+d x^3\right )}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {(2 a d+b c) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{c}}dx}{3 c^{2/3}}\right )}{3 c d}-\frac {x (b c-a d)}{3 c d \left (c+d x^3\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(2 a d+b c) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{3 c d}-\frac {x (b c-a d)}{3 c d \left (c+d x^3\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {(2 a d+b c) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{3 c d}-\frac {x (b c-a d)}{3 c d \left (c+d x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(2 a d+b c) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{3 c d}-\frac {x (b c-a d)}{3 c d \left (c+d x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(2 a d+b c) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{3 c d}-\frac {x (b c-a d)}{3 c d \left (c+d x^3\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(2 a d+b c) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{3 c d}-\frac {x (b c-a d)}{3 c d \left (c+d x^3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(2 a d+b c) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{3 c d}-\frac {x (b c-a d)}{3 c d \left (c+d x^3\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(2 a d+b c) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{3 c d}-\frac {x (b c-a d)}{3 c d \left (c+d x^3\right )}\) |
Input:
Int[(a + b*x^3)/(c + d*x^3)^2,x]
Output:
-1/3*((b*c - a*d)*x)/(c*d*(c + d*x^3)) + ((b*c + 2*a*d)*(Log[c^(1/3) + d^( 1/3)*x]/(3*c^(2/3)*d^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3 ))/Sqrt[3]])/d^(1/3)) - Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2]/(2* d^(1/3)))/(3*c^(2/3))))/(3*c*d)
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[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 0])
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 = 0.88 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.38
| method | result | size |
| risch | \(\frac {\left (a d -b c \right ) x}{3 d c \left (d \,x^{3}+c \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+c \right )}{\sum }\frac {\left (2 a d +b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 c \,d^{2}}\) | \(65\) |
| default | \(\frac {\left (a d -b c \right ) x}{3 d c \left (d \,x^{3}+c \right )}+\frac {\left (2 a d +b c \right ) \left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right )}{3 c d}\) | \(134\) |
Input:
int((b*x^3+a)/(d*x^3+c)^2,x,method=_RETURNVERBOSE)
Output:
1/3*(a*d-b*c)/d/c*x/(d*x^3+c)+1/9/c/d^2*sum((2*a*d+b*c)/_R^2*ln(x-_R),_R=R ootOf(_Z^3*d+c))
Time = 0.09 (sec) , antiderivative size = 537, normalized size of antiderivative = 3.18 \[ \int \frac {a+b x^3}{\left (c+d x^3\right )^2} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (b c^{3} d + 2 \, a c^{2} d^{2} + {\left (b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{3}\right )} \sqrt {-\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (\frac {2 \, c d x^{3} - 3 \, \left (c^{2} d\right )^{\frac {1}{3}} c x - c^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, c d x^{2} + \left (c^{2} d\right )^{\frac {2}{3}} x - \left (c^{2} d\right )^{\frac {1}{3}} c\right )} \sqrt {-\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}}}{d x^{3} + c}\right ) - {\left ({\left (b c d + 2 \, a d^{2}\right )} x^{3} + b c^{2} + 2 \, a c d\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x^{2} - \left (c^{2} d\right )^{\frac {2}{3}} x + \left (c^{2} d\right )^{\frac {1}{3}} c\right ) + 2 \, {\left ({\left (b c d + 2 \, a d^{2}\right )} x^{3} + b c^{2} + 2 \, a c d\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x + \left (c^{2} d\right )^{\frac {2}{3}}\right ) - 6 \, {\left (b c^{3} d - a c^{2} d^{2}\right )} x}{18 \, {\left (c^{3} d^{3} x^{3} + c^{4} d^{2}\right )}}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (b c^{3} d + 2 \, a c^{2} d^{2} + {\left (b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{3}\right )} \sqrt {\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (c^{2} d\right )^{\frac {2}{3}} x - \left (c^{2} d\right )^{\frac {1}{3}} c\right )} \sqrt {\frac {\left (c^{2} d\right )^{\frac {1}{3}}}{d}}}{c^{2}}\right ) - {\left ({\left (b c d + 2 \, a d^{2}\right )} x^{3} + b c^{2} + 2 \, a c d\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x^{2} - \left (c^{2} d\right )^{\frac {2}{3}} x + \left (c^{2} d\right )^{\frac {1}{3}} c\right ) + 2 \, {\left ({\left (b c d + 2 \, a d^{2}\right )} x^{3} + b c^{2} + 2 \, a c d\right )} \left (c^{2} d\right )^{\frac {2}{3}} \log \left (c d x + \left (c^{2} d\right )^{\frac {2}{3}}\right ) - 6 \, {\left (b c^{3} d - a c^{2} d^{2}\right )} x}{18 \, {\left (c^{3} d^{3} x^{3} + c^{4} d^{2}\right )}}\right ] \] Input:
integrate((b*x^3+a)/(d*x^3+c)^2,x, algorithm="fricas")
Output:
[1/18*(3*sqrt(1/3)*(b*c^3*d + 2*a*c^2*d^2 + (b*c^2*d^2 + 2*a*c*d^3)*x^3)*s qrt(-(c^2*d)^(1/3)/d)*log((2*c*d*x^3 - 3*(c^2*d)^(1/3)*c*x - c^2 + 3*sqrt( 1/3)*(2*c*d*x^2 + (c^2*d)^(2/3)*x - (c^2*d)^(1/3)*c)*sqrt(-(c^2*d)^(1/3)/d ))/(d*x^3 + c)) - ((b*c*d + 2*a*d^2)*x^3 + b*c^2 + 2*a*c*d)*(c^2*d)^(2/3)* log(c*d*x^2 - (c^2*d)^(2/3)*x + (c^2*d)^(1/3)*c) + 2*((b*c*d + 2*a*d^2)*x^ 3 + b*c^2 + 2*a*c*d)*(c^2*d)^(2/3)*log(c*d*x + (c^2*d)^(2/3)) - 6*(b*c^3*d - a*c^2*d^2)*x)/(c^3*d^3*x^3 + c^4*d^2), 1/18*(6*sqrt(1/3)*(b*c^3*d + 2*a *c^2*d^2 + (b*c^2*d^2 + 2*a*c*d^3)*x^3)*sqrt((c^2*d)^(1/3)/d)*arctan(sqrt( 1/3)*(2*(c^2*d)^(2/3)*x - (c^2*d)^(1/3)*c)*sqrt((c^2*d)^(1/3)/d)/c^2) - (( b*c*d + 2*a*d^2)*x^3 + b*c^2 + 2*a*c*d)*(c^2*d)^(2/3)*log(c*d*x^2 - (c^2*d )^(2/3)*x + (c^2*d)^(1/3)*c) + 2*((b*c*d + 2*a*d^2)*x^3 + b*c^2 + 2*a*c*d) *(c^2*d)^(2/3)*log(c*d*x + (c^2*d)^(2/3)) - 6*(b*c^3*d - a*c^2*d^2)*x)/(c^ 3*d^3*x^3 + c^4*d^2)]
Time = 0.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.57 \[ \int \frac {a+b x^3}{\left (c+d x^3\right )^2} \, dx=\frac {x \left (a d - b c\right )}{3 c^{2} d + 3 c d^{2} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} c^{5} d^{4} - 8 a^{3} d^{3} - 12 a^{2} b c d^{2} - 6 a b^{2} c^{2} d - b^{3} c^{3}, \left ( t \mapsto t \log {\left (\frac {9 t c^{2} d}{2 a d + b c} + x \right )} \right )\right )} \] Input:
integrate((b*x**3+a)/(d*x**3+c)**2,x)
Output:
x*(a*d - b*c)/(3*c**2*d + 3*c*d**2*x**3) + RootSum(729*_t**3*c**5*d**4 - 8 *a**3*d**3 - 12*a**2*b*c*d**2 - 6*a*b**2*c**2*d - b**3*c**3, Lambda(_t, _t *log(9*_t*c**2*d/(2*a*d + b*c) + x)))
Time = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.93 \[ \int \frac {a+b x^3}{\left (c+d x^3\right )^2} \, dx=-\frac {{\left (b c - a d\right )} x}{3 \, {\left (c d^{2} x^{3} + c^{2} d\right )}} + \frac {\sqrt {3} {\left (b c + 2 \, a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{9 \, c d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} - \frac {{\left (b c + 2 \, a d\right )} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{18 \, c d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} + \frac {{\left (b c + 2 \, a d\right )} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{9 \, c d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}} \] Input:
integrate((b*x^3+a)/(d*x^3+c)^2,x, algorithm="maxima")
Output:
-1/3*(b*c - a*d)*x/(c*d^2*x^3 + c^2*d) + 1/9*sqrt(3)*(b*c + 2*a*d)*arctan( 1/3*sqrt(3)*(2*x - (c/d)^(1/3))/(c/d)^(1/3))/(c*d^2*(c/d)^(2/3)) - 1/18*(b *c + 2*a*d)*log(x^2 - x*(c/d)^(1/3) + (c/d)^(2/3))/(c*d^2*(c/d)^(2/3)) + 1 /9*(b*c + 2*a*d)*log(x + (c/d)^(1/3))/(c*d^2*(c/d)^(2/3))
Time = 0.13 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.95 \[ \int \frac {a+b x^3}{\left (c+d x^3\right )^2} \, dx=-\frac {\sqrt {3} {\left (b c + 2 \, a d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-c d^{2}\right )^{\frac {2}{3}} c} - \frac {{\left (b c + 2 \, a d\right )} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{18 \, \left (-c d^{2}\right )^{\frac {2}{3}} c} - \frac {{\left (b c + 2 \, a d\right )} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{9 \, c^{2} d} - \frac {b c x - a d x}{3 \, {\left (d x^{3} + c\right )} c d} \] Input:
integrate((b*x^3+a)/(d*x^3+c)^2,x, algorithm="giac")
Output:
-1/9*sqrt(3)*(b*c + 2*a*d)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^ (1/3))/((-c*d^2)^(2/3)*c) - 1/18*(b*c + 2*a*d)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/((-c*d^2)^(2/3)*c) - 1/9*(b*c + 2*a*d)*(-c/d)^(1/3)*log(abs( x - (-c/d)^(1/3)))/(c^2*d) - 1/3*(b*c*x - a*d*x)/((d*x^3 + c)*c*d)
Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.85 \[ \int \frac {a+b x^3}{\left (c+d x^3\right )^2} \, dx=\frac {\ln \left (d^{1/3}\,x+c^{1/3}\right )\,\left (2\,a\,d+b\,c\right )}{9\,c^{5/3}\,d^{4/3}}-\frac {\ln \left (c^{1/3}-2\,d^{1/3}\,x+\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,a\,d+b\,c\right )}{9\,c^{5/3}\,d^{4/3}}+\frac {\ln \left (2\,d^{1/3}\,x-c^{1/3}+\sqrt {3}\,c^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,a\,d+b\,c\right )}{9\,c^{5/3}\,d^{4/3}}+\frac {x\,\left (a\,d-b\,c\right )}{3\,c\,d\,\left (d\,x^3+c\right )} \] Input:
int((a + b*x^3)/(c + d*x^3)^2,x)
Output:
(log(d^(1/3)*x + c^(1/3))*(2*a*d + b*c))/(9*c^(5/3)*d^(4/3)) - (log(3^(1/2 )*c^(1/3)*1i - 2*d^(1/3)*x + c^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(2*a*d + b*c) )/(9*c^(5/3)*d^(4/3)) + (log(3^(1/2)*c^(1/3)*1i + 2*d^(1/3)*x - c^(1/3))*( (3^(1/2)*1i)/2 - 1/2)*(2*a*d + b*c))/(9*c^(5/3)*d^(4/3)) + (x*(a*d - b*c)) /(3*c*d*(c + d*x^3))
Time = 0.23 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.07 \[ \int \frac {a+b x^3}{\left (c+d x^3\right )^2} \, dx=\frac {-4 c^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x}{c^{\frac {1}{3}} \sqrt {3}}\right ) a d -4 c^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x}{c^{\frac {1}{3}} \sqrt {3}}\right ) a \,d^{2} x^{3}-2 c^{\frac {7}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x}{c^{\frac {1}{3}} \sqrt {3}}\right ) b -2 c^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x}{c^{\frac {1}{3}} \sqrt {3}}\right ) b d \,x^{3}-2 c^{\frac {4}{3}} \mathrm {log}\left (c^{\frac {2}{3}}-d^{\frac {1}{3}} c^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right ) a d -2 c^{\frac {1}{3}} \mathrm {log}\left (c^{\frac {2}{3}}-d^{\frac {1}{3}} c^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right ) a \,d^{2} x^{3}-c^{\frac {7}{3}} \mathrm {log}\left (c^{\frac {2}{3}}-d^{\frac {1}{3}} c^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right ) b -c^{\frac {4}{3}} \mathrm {log}\left (c^{\frac {2}{3}}-d^{\frac {1}{3}} c^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right ) b d \,x^{3}+4 c^{\frac {4}{3}} \mathrm {log}\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) a d +4 c^{\frac {1}{3}} \mathrm {log}\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) a \,d^{2} x^{3}+2 c^{\frac {7}{3}} \mathrm {log}\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) b +2 c^{\frac {4}{3}} \mathrm {log}\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) b d \,x^{3}+6 d^{\frac {4}{3}} a c x -6 d^{\frac {1}{3}} b \,c^{2} x}{18 d^{\frac {4}{3}} c^{2} \left (d \,x^{3}+c \right )} \] Input:
int((b*x^3+a)/(d*x^3+c)^2,x)
Output:
( - 4*c**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)*sqrt(3)))* a*c*d - 4*c**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)*sqrt(3 )))*a*d**2*x**3 - 2*c**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1 /3)*sqrt(3)))*b*c**2 - 2*c**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/( c**(1/3)*sqrt(3)))*b*c*d*x**3 - 2*c**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3 )*x + d**(2/3)*x**2)*a*c*d - 2*c**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)*a*d**2*x**3 - c**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)* x + d**(2/3)*x**2)*b*c**2 - c**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)*b*c*d*x**3 + 4*c**(1/3)*log(c**(1/3) + d**(1/3)*x)*a*c*d + 4*c**(1/3)*log(c**(1/3) + d**(1/3)*x)*a*d**2*x**3 + 2*c**(1/3)*log(c**(1/3 ) + d**(1/3)*x)*b*c**2 + 2*c**(1/3)*log(c**(1/3) + d**(1/3)*x)*b*c*d*x**3 + 6*d**(1/3)*a*c*d*x - 6*d**(1/3)*b*c**2*x)/(18*d**(1/3)*c**2*d*(c + d*x** 3))