Integrand size = 15, antiderivative size = 189 \[ \int \frac {c+d x}{\left (a+b x^3\right )^2} \, dx=\frac {x (c+d x)}{3 a \left (a+b x^3\right )}-\frac {\left (2 \sqrt [3]{b} c+\sqrt [3]{a} d\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{2/3}}+\frac {\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{2/3}}-\frac {\left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{2/3}} \] Output:
1/3*x*(d*x+c)/a/(b*x^3+a)-1/9*(2*b^(1/3)*c+a^(1/3)*d)*arctan(1/3*(a^(1/3)- 2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(5/3)/b^(2/3)+1/9*(2*b^(1/3)*c-a^( 1/3)*d)*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/b^(2/3)-1/18*(2*b^(1/3)*c-a^(1/3)*d) *ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)/b^(2/3)
Time = 0.12 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x}{\left (a+b x^3\right )^2} \, dx=\frac {\frac {6 a x (c+d x)}{a+b x^3}-\frac {2 \sqrt {3} \sqrt [3]{a} \left (2 \sqrt [3]{b} c+\sqrt [3]{a} d\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}+\frac {2 \left (2 \sqrt [3]{a} \sqrt [3]{b} c-a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac {\left (-2 \sqrt [3]{a} \sqrt [3]{b} c+a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}}{18 a^2} \] Input:
Integrate[(c + d*x)/(a + b*x^3)^2,x]
Output:
((6*a*x*(c + d*x))/(a + b*x^3) - (2*Sqrt[3]*a^(1/3)*(2*b^(1/3)*c + a^(1/3) *d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (2*(2*a^(1/3)*b ^(1/3)*c - a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((-2*a^(1/3)*b^( 1/3)*c + a^(2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3 ))/(18*a^2)
Time = 0.73 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2394, 25, 2399, 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.
| \(\displaystyle \int \frac {c+d x}{\left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {x (c+d x)}{3 a \left (a+b x^3\right )}-\frac {\int -\frac {2 c+d x}{b x^3+a}dx}{3 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2 c+d x}{b x^3+a}dx}{3 a}+\frac {x (c+d x)}{3 a \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [3]{a} \left (4 \sqrt [3]{b} c+\sqrt [3]{a} d\right )-\sqrt [3]{b} \left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}}{3 a}+\frac {x (c+d x)}{3 a \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [3]{a} \left (4 \sqrt [3]{b} c+\sqrt [3]{a} d\right )-\sqrt [3]{b} \left (2 \sqrt [3]{b} c-\sqrt [3]{a} d\right ) x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 a}+\frac {x (c+d x)}{3 a \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 a}+\frac {x (c+d x)}{3 a \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 a}+\frac {x (c+d x)}{3 a \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \sqrt [3]{b} \left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 a}+\frac {x (c+d x)}{3 a \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt [3]{b} \left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 a}+\frac {x (c+d x)}{3 a \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \sqrt [3]{b} \left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 a}+\frac {x (c+d x)}{3 a \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} \left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (\sqrt [3]{a} d+2 \sqrt [3]{b} c\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}+\frac {\left (2 c-\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}}{3 a}+\frac {x (c+d x)}{3 a \left (a+b x^3\right )}\) |
Input:
Int[(c + d*x)/(a + b*x^3)^2,x]
Output:
(x*(c + d*x))/(3*a*(a + b*x^3)) + (((2*c - (a^(1/3)*d)/b^(1/3))*Log[a^(1/3 ) + b^(1/3)*x])/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*(2*b^(1/3)*c + a^(1/3)*d )*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3)) - ((2*c - (a^(1/3) *d)/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/2)/(3*a^(2/3) *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[((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] :> Simp[(-x)*Pq*((a + b *x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[ExpandToSum[n *(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x ] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]
Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numer ator[Rt[a/b, 3]], s = Denominator[Rt[a/b, 3]]}, Simp[(-r)*((B*r - A*s)/(3*a *s)) Int[1/(r + s*x), x], x] + Simp[r/(3*a*s) Int[(r*(B*r + 2*A*s) + s* (B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] & & NeQ[a*B^3 - b*A^3, 0] && PosQ[a/b]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.34
| method | result | size |
| risch | \(\frac {\frac {d \,x^{2}}{3 a}+\frac {c x}{3 a}}{b \,x^{3}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (d \textit {\_R} +2 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 b a}\) | \(65\) |
| default | \(c \left (\frac {x}{3 a \left (b \,x^{3}+a \right )}+\frac {\frac {2 \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}}{a}\right )+d \left (\frac {x^{2}}{3 a \left (b \,x^{3}+a \right )}+\frac {-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}}{3 a}\right )\) | \(230\) |
Input:
int((d*x+c)/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
(1/3*d/a*x^2+1/3*c/a*x)/(b*x^3+a)+1/9/b/a*sum((_R*d+2*c)/_R^2*ln(x-_R),_R= RootOf(_Z^3*b+a))
Result contains complex when optimal does not.
Time = 0.91 (sec) , antiderivative size = 2088, normalized size of antiderivative = 11.05 \[ \int \frac {c+d x}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)/(b*x^3+a)^2,x, algorithm="fricas")
Output:
1/36*(12*d*x^2 - 2*(a*b*x^3 + a^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b*c^3 + a*d^3)/(a^5*b^2) + (8*b*c^3 - a*d^3)/(a^5*b^2))^(1/3) + 4*(1/2)^(2/3)*c* d*(I*sqrt(3) - 1)/(a^3*b*((8*b*c^3 + a*d^3)/(a^5*b^2) + (8*b*c^3 - a*d^3)/ (a^5*b^2))^(1/3)))*log(1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b*c^3 + a*d^3) /(a^5*b^2) + (8*b*c^3 - a*d^3)/(a^5*b^2))^(1/3) + 4*(1/2)^(2/3)*c*d*(I*sqr t(3) - 1)/(a^3*b*((8*b*c^3 + a*d^3)/(a^5*b^2) + (8*b*c^3 - a*d^3)/(a^5*b^2 ))^(1/3)))^2*a^4*b*d - 2*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b*c^3 + a*d^3)/( a^5*b^2) + (8*b*c^3 - a*d^3)/(a^5*b^2))^(1/3) + 4*(1/2)^(2/3)*c*d*(I*sqrt( 3) - 1)/(a^3*b*((8*b*c^3 + a*d^3)/(a^5*b^2) + (8*b*c^3 - a*d^3)/(a^5*b^2)) ^(1/3)))*a^2*b*c^2 + 4*a*c*d^2 + (8*b*c^3 + a*d^3)*x) + 12*c*x + ((a*b*x^3 + a^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b*c^3 + a*d^3)/(a^5*b^2) + (8*b*c ^3 - a*d^3)/(a^5*b^2))^(1/3) + 4*(1/2)^(2/3)*c*d*(I*sqrt(3) - 1)/(a^3*b*(( 8*b*c^3 + a*d^3)/(a^5*b^2) + (8*b*c^3 - a*d^3)/(a^5*b^2))^(1/3))) + 3*sqrt (1/3)*(a*b*x^3 + a^2)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b*c^3 + a*d^ 3)/(a^5*b^2) + (8*b*c^3 - a*d^3)/(a^5*b^2))^(1/3) + 4*(1/2)^(2/3)*c*d*(I*s qrt(3) - 1)/(a^3*b*((8*b*c^3 + a*d^3)/(a^5*b^2) + (8*b*c^3 - a*d^3)/(a^5*b ^2))^(1/3)))^2*a^3*b + 32*c*d)/(a^3*b)))*log(-1/4*((1/2)^(1/3)*(I*sqrt(3) + 1)*((8*b*c^3 + a*d^3)/(a^5*b^2) + (8*b*c^3 - a*d^3)/(a^5*b^2))^(1/3) + 4 *(1/2)^(2/3)*c*d*(I*sqrt(3) - 1)/(a^3*b*((8*b*c^3 + a*d^3)/(a^5*b^2) + (8* b*c^3 - a*d^3)/(a^5*b^2))^(1/3)))^2*a^4*b*d + 2*((1/2)^(1/3)*(I*sqrt(3)...
Time = 0.58 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.56 \[ \int \frac {c+d x}{\left (a+b x^3\right )^2} \, dx=\operatorname {RootSum} {\left (729 t^{3} a^{5} b^{2} + 54 t a^{2} b c d + a d^{3} - 8 b c^{3}, \left ( t \mapsto t \log {\left (x + \frac {81 t^{2} a^{4} b d + 36 t a^{2} b c^{2} + 4 a c d^{2}}{a d^{3} + 8 b c^{3}} \right )} \right )\right )} + \frac {c x + d x^{2}}{3 a^{2} + 3 a b x^{3}} \] Input:
integrate((d*x+c)/(b*x**3+a)**2,x)
Output:
RootSum(729*_t**3*a**5*b**2 + 54*_t*a**2*b*c*d + a*d**3 - 8*b*c**3, Lambda (_t, _t*log(x + (81*_t**2*a**4*b*d + 36*_t*a**2*b*c**2 + 4*a*c*d**2)/(a*d* *3 + 8*b*c**3)))) + (c*x + d*x**2)/(3*a**2 + 3*a*b*x**3)
Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.89 \[ \int \frac {c+d x}{\left (a+b x^3\right )^2} \, dx=\frac {d x^{2} + c x}{3 \, {\left (a b x^{3} + a^{2}\right )}} + \frac {\sqrt {3} {\left (d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, c\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, c\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((d*x+c)/(b*x^3+a)^2,x, algorithm="maxima")
Output:
1/3*(d*x^2 + c*x)/(a*b*x^3 + a^2) + 1/9*sqrt(3)*(d*(a/b)^(1/3) + 2*c)*arct an(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b*(a/b)^(2/3)) + 1/18*( d*(a/b)^(1/3) - 2*c)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b*(a/b)^(2/ 3)) - 1/9*(d*(a/b)^(1/3) - 2*c)*log(x + (a/b)^(1/3))/(a*b*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.92 \[ \int \frac {c+d x}{\left (a+b x^3\right )^2} \, dx=-\frac {\sqrt {3} {\left (2 \, b c - \left (-a b^{2}\right )^{\frac {1}{3}} d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} - \frac {{\left (2 \, b c + \left (-a b^{2}\right )^{\frac {1}{3}} d\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} - \frac {{\left (d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, c\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{2}} + \frac {d x^{2} + c x}{3 \, {\left (b x^{3} + a\right )} a} \] Input:
integrate((d*x+c)/(b*x^3+a)^2,x, algorithm="giac")
Output:
-1/9*sqrt(3)*(2*b*c - (-a*b^2)^(1/3)*d)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^( 1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a) - 1/18*(2*b*c + (-a*b^2)^(1/3)*d)*l og(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a) - 1/9*(d*(-a/b) ^(1/3) + 2*c)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^2 + 1/3*(d*x^2 + c *x)/((b*x^3 + a)*a)
Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.89 \[ \int \frac {c+d x}{\left (a+b x^3\right )^2} \, dx=\left (\sum _{k=1}^3\ln \left (\frac {b\,\left (2\,c\,d+d^2\,x+{\mathrm {root}\left (729\,a^5\,b^2\,z^3+54\,a^2\,b\,c\,d\,z-8\,b\,c^3+a\,d^3,z,k\right )}^2\,a^3\,b\,81+\mathrm {root}\left (729\,a^5\,b^2\,z^3+54\,a^2\,b\,c\,d\,z-8\,b\,c^3+a\,d^3,z,k\right )\,a\,b\,c\,x\,18\right )}{a^2\,9}\right )\,\mathrm {root}\left (729\,a^5\,b^2\,z^3+54\,a^2\,b\,c\,d\,z-8\,b\,c^3+a\,d^3,z,k\right )\right )+\frac {\frac {d\,x^2}{3\,a}+\frac {c\,x}{3\,a}}{b\,x^3+a} \] Input:
int((c + d*x)/(a + b*x^3)^2,x)
Output:
symsum(log((b*(2*c*d + d^2*x + 81*root(729*a^5*b^2*z^3 + 54*a^2*b*c*d*z - 8*b*c^3 + a*d^3, z, k)^2*a^3*b + 18*root(729*a^5*b^2*z^3 + 54*a^2*b*c*d*z - 8*b*c^3 + a*d^3, z, k)*a*b*c*x))/(9*a^2))*root(729*a^5*b^2*z^3 + 54*a^2* b*c*d*z - 8*b*c^3 + a*d^3, z, k), k, 1, 3) + ((d*x^2)/(3*a) + (c*x)/(3*a)) /(a + b*x^3)
Time = 0.16 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.86 \[ \int \frac {c+d x}{\left (a+b x^3\right )^2} \, dx=\frac {-4 b^{\frac {1}{3}} a^{\frac {5}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) c -4 b^{\frac {4}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) c \,x^{3}-2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2} d -2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a b d \,x^{3}-2 b^{\frac {1}{3}} a^{\frac {5}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) c -2 b^{\frac {4}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) c \,x^{3}+4 b^{\frac {1}{3}} a^{\frac {5}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) c +4 b^{\frac {4}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) c \,x^{3}+6 b^{\frac {2}{3}} a^{\frac {4}{3}} c x +6 b^{\frac {2}{3}} a^{\frac {4}{3}} d \,x^{2}+\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2} d +\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a b d \,x^{3}-2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2} d -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a b d \,x^{3}}{18 b^{\frac {2}{3}} a^{\frac {7}{3}} \left (b \,x^{3}+a \right )} \] Input:
int((d*x+c)/(b*x^3+a)^2,x)
Output:
( - 4*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s qrt(3)))*a*c - 4*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/ (a**(1/3)*sqrt(3)))*b*c*x**3 - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a **(1/3)*sqrt(3)))*a**2*d - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1 /3)*sqrt(3)))*a*b*d*x**3 - 2*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a** (1/3)*x + b**(2/3)*x**2)*a*c - 2*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3) *a**(1/3)*x + b**(2/3)*x**2)*b*c*x**3 + 4*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*a*c + 4*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*b*c*x**3 + 6*b**(2/3)*a**(1/3)*a*c*x + 6*b**(2/3)*a**(1/3)*a*d*x**2 + log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*d + log(a**(2/3) - b**(1/3)*a **(1/3)*x + b**(2/3)*x**2)*a*b*d*x**3 - 2*log(a**(1/3) + b**(1/3)*x)*a**2* d - 2*log(a**(1/3) + b**(1/3)*x)*a*b*d*x**3)/(18*b**(2/3)*a**(1/3)*a**2*(a + b*x**3))