Integrand size = 15, antiderivative size = 180 \[ \int \frac {x}{a+b (c+d x)^3} \, dx=-\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{2/3} d^2}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3} d^2}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{2/3} b^{2/3} d^2} \] Output:
-1/3*(a^(1/3)-b^(1/3)*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))*3^(1/2)/a^ (1/3))*3^(1/2)/a^(2/3)/b^(2/3)/d^2-1/3*(a^(1/3)+b^(1/3)*c)*ln(a^(1/3)+b^(1 /3)*(d*x+c))/a^(2/3)/b^(2/3)/d^2+1/6*(a^(1/3)+b^(1/3)*c)*ln(a^(2/3)-a^(1/3 )*b^(1/3)*(d*x+c)+b^(2/3)*(d*x+c)^2)/a^(2/3)/b^(2/3)/d^2
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.44 \[ \int \frac {x}{a+b (c+d x)^3} \, dx=\frac {\text {RootSum}\left [a+b c^3+3 b c^2 d \text {$\#$1}+3 b c d^2 \text {$\#$1}^2+b d^3 \text {$\#$1}^3\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}}{c^2+2 c d \text {$\#$1}+d^2 \text {$\#$1}^2}\&\right ]}{3 b d} \] Input:
Integrate[x/(a + b*(c + d*x)^3),x]
Output:
RootSum[a + b*c^3 + 3*b*c^2*d*#1 + 3*b*c*d^2*#1^2 + b*d^3*#1^3 & , (Log[x - #1]*#1)/(c^2 + 2*c*d*#1 + d^2*#1^2) & ]/(3*b*d)
Time = 0.62 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {896, 25, 2399, 16, 25, 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.
| \(\displaystyle \int \frac {x}{a+b (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 896 |
| \(\displaystyle \frac {\int \frac {d x}{b (c+d x)^3+a}d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int -\frac {d x}{b (c+d x)^3+a}d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 2399 |
| \(\displaystyle \frac {-\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-2 \sqrt [3]{b} c\right )+\sqrt [3]{b} \left (\sqrt [3]{b} c+\sqrt [3]{a}\right ) (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \int \frac {1}{\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}}{d^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {-\frac {\int -\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-2 \sqrt [3]{b} c\right )+\sqrt [3]{b} \left (\sqrt [3]{b} c+\sqrt [3]{a}\right ) (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt [3]{a} \left (\sqrt [3]{a}-2 \sqrt [3]{b} c\right )+\sqrt [3]{b} \left (\sqrt [3]{b} c+\sqrt [3]{a}\right ) (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}}{d^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {1}{2} \left (\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}}{d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {1}{2} \left (\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}}{d^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3}{2} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {1}{2} \sqrt [3]{b} \left (\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}}{d^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \sqrt [3]{b} \left (\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}}{d^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {1}{2} \sqrt [3]{b} \left (\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\sqrt {3} \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}}{d^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\frac {1}{2} \left (\frac {\sqrt [3]{a}}{\sqrt [3]{b}}+c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )-\frac {\sqrt {3} \left (\sqrt [3]{a}-\sqrt [3]{b} c\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3} \sqrt [3]{b}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{2/3} b^{2/3}}}{d^2}\) |
Input:
Int[x/(a + b*(c + d*x)^3),x]
Output:
(-1/3*((a^(1/3) + b^(1/3)*c)*Log[a^(1/3) + b^(1/3)*(c + d*x)])/(a^(2/3)*b^ (2/3)) + (-((Sqrt[3]*(a^(1/3) - b^(1/3)*c)*ArcTan[(1 - (2*b^(1/3)*(c + d*x ))/a^(1/3))/Sqrt[3]])/b^(1/3)) + ((a^(1/3)/b^(1/3) + c)*Log[a^(2/3) - a^(1 /3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/2)/(3*a^(2/3)*b^(1/3)))/d^2
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_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
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[((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 = 0.19 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.40
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2} d +c^{3} b +a \right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{3 b d}\) | \(72\) |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 \textit {\_Z} b \,c^{2} d +c^{3} b +a \right )}{\sum }\frac {\textit {\_R} \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{3 b d}\) | \(72\) |
Input:
int(x/(a+b*(d*x+c)^3),x,method=_RETURNVERBOSE)
Output:
1/3/b/d*sum(_R/(_R^2*d^2+2*_R*c*d+c^2)*ln(x-_R),_R=RootOf(_Z^3*b*d^3+3*_Z^ 2*b*c*d^2+3*_Z*b*c^2*d+b*c^3+a))
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 1950, normalized size of antiderivative = 10.83 \[ \int \frac {x}{a+b (c+d x)^3} \, dx=\text {Too large to display} \] Input:
integrate(x/(a+b*(d*x+c)^3),x, algorithm="fricas")
Output:
1/36*(9*(I*sqrt(3) + 1)*(-1/54*(b*c^3 + a)/(a^2*b^2*d^6) + 1/54*(b*c^3 - a )/(a^2*b^2*d^6))^(1/3) - 3*sqrt(1/3)*sqrt(-((9*(I*sqrt(3) + 1)*(-1/54*(b*c ^3 + a)/(a^2*b^2*d^6) + 1/54*(b*c^3 - a)/(a^2*b^2*d^6))^(1/3) + c*(-I*sqrt (3) + 1)/(a*b*d^4*(-1/54*(b*c^3 + a)/(a^2*b^2*d^6) + 1/54*(b*c^3 - a)/(a^2 *b^2*d^6))^(1/3)))^2*a*b*d^4 - 144*c)/(a*b*d^4)) + c*(-I*sqrt(3) + 1)/(a*b *d^4*(-1/54*(b*c^3 + a)/(a^2*b^2*d^6) + 1/54*(b*c^3 - a)/(a^2*b^2*d^6))^(1 /3)))*log(1/36*(9*(I*sqrt(3) + 1)*(-1/54*(b*c^3 + a)/(a^2*b^2*d^6) + 1/54* (b*c^3 - a)/(a^2*b^2*d^6))^(1/3) + c*(-I*sqrt(3) + 1)/(a*b*d^4*(-1/54*(b*c ^3 + a)/(a^2*b^2*d^6) + 1/54*(b*c^3 - a)/(a^2*b^2*d^6))^(1/3)))^2*a^2*b*d^ 4 - 1/6*(9*(I*sqrt(3) + 1)*(-1/54*(b*c^3 + a)/(a^2*b^2*d^6) + 1/54*(b*c^3 - a)/(a^2*b^2*d^6))^(1/3) + c*(-I*sqrt(3) + 1)/(a*b*d^4*(-1/54*(b*c^3 + a) /(a^2*b^2*d^6) + 1/54*(b*c^3 - a)/(a^2*b^2*d^6))^(1/3)))*a*b*c^2*d^2 + 2*b *c^4 + 2*(b*c^3 - a)*d*x - 4*a*c + 1/12*sqrt(1/3)*((9*(I*sqrt(3) + 1)*(-1/ 54*(b*c^3 + a)/(a^2*b^2*d^6) + 1/54*(b*c^3 - a)/(a^2*b^2*d^6))^(1/3) + c*( -I*sqrt(3) + 1)/(a*b*d^4*(-1/54*(b*c^3 + a)/(a^2*b^2*d^6) + 1/54*(b*c^3 - a)/(a^2*b^2*d^6))^(1/3)))*a^2*b*d^4 + 6*a*b*c^2*d^2)*sqrt(-((9*(I*sqrt(3) + 1)*(-1/54*(b*c^3 + a)/(a^2*b^2*d^6) + 1/54*(b*c^3 - a)/(a^2*b^2*d^6))^(1 /3) + c*(-I*sqrt(3) + 1)/(a*b*d^4*(-1/54*(b*c^3 + a)/(a^2*b^2*d^6) + 1/54* (b*c^3 - a)/(a^2*b^2*d^6))^(1/3)))^2*a*b*d^4 - 144*c)/(a*b*d^4))) + 1/36*( 9*(I*sqrt(3) + 1)*(-1/54*(b*c^3 + a)/(a^2*b^2*d^6) + 1/54*(b*c^3 - a)/(...
Time = 0.42 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.46 \[ \int \frac {x}{a+b (c+d x)^3} \, dx=\operatorname {RootSum} {\left (27 t^{3} a^{2} b^{2} d^{6} - 9 t a b c d^{2} + a + b c^{3}, \left ( t \mapsto t \log {\left (x + \frac {9 t^{2} a^{2} b d^{4} + 3 t a b c^{2} d^{2} - a c - b c^{4}}{a d - b c^{3} d} \right )} \right )\right )} \] Input:
integrate(x/(a+b*(d*x+c)**3),x)
Output:
RootSum(27*_t**3*a**2*b**2*d**6 - 9*_t*a*b*c*d**2 + a + b*c**3, Lambda(_t, _t*log(x + (9*_t**2*a**2*b*d**4 + 3*_t*a*b*c**2*d**2 - a*c - b*c**4)/(a*d - b*c**3*d))))
\[ \int \frac {x}{a+b (c+d x)^3} \, dx=\int { \frac {x}{{\left (d x + c\right )}^{3} b + a} \,d x } \] Input:
integrate(x/(a+b*(d*x+c)^3),x, algorithm="maxima")
Output:
integrate(x/((d*x + c)^3*b + a), x)
\[ \int \frac {x}{a+b (c+d x)^3} \, dx=\int { \frac {x}{{\left (d x + c\right )}^{3} b + a} \,d x } \] Input:
integrate(x/(a+b*(d*x+c)^3),x, algorithm="giac")
Output:
integrate(x/((d*x + c)^3*b + a), x)
Time = 0.66 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.81 \[ \int \frac {x}{a+b (c+d x)^3} \, dx=\sum _{k=1}^3\ln \left (-\mathrm {root}\left (27\,a^2\,b^2\,d^6\,z^3-9\,a\,b\,c\,d^2\,z+b\,c^3+a,z,k\right )\,\left (3\,b^2\,c^2\,d^4-\mathrm {root}\left (27\,a^2\,b^2\,d^6\,z^3-9\,a\,b\,c\,d^2\,z+b\,c^3+a,z,k\right )\,a\,b^2\,d^6\,9+3\,b^2\,c\,d^5\,x\right )+b\,d^3\,x\right )\,\mathrm {root}\left (27\,a^2\,b^2\,d^6\,z^3-9\,a\,b\,c\,d^2\,z+b\,c^3+a,z,k\right ) \] Input:
int(x/(a + b*(c + d*x)^3),x)
Output:
symsum(log(b*d^3*x - root(27*a^2*b^2*d^6*z^3 - 9*a*b*c*d^2*z + b*c^3 + a, z, k)*(3*b^2*c^2*d^4 - 9*root(27*a^2*b^2*d^6*z^3 - 9*a*b*c*d^2*z + b*c^3 + a, z, k)*a*b^2*d^6 + 3*b^2*c*d^5*x))*root(27*a^2*b^2*d^6*z^3 - 9*a*b*c*d^ 2*z + b*c^3 + a, z, k), k, 1, 3)
Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.29 \[ \int \frac {x}{a+b (c+d x)^3} \, dx=\frac {2 b^{\frac {1}{3}} a^{\frac {2}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} c -2 b^{\frac {1}{3}} d x}{a^{\frac {1}{3}} \sqrt {3}}\right ) c -2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} c -2 b^{\frac {1}{3}} d x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a +b^{\frac {1}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} c -b^{\frac {1}{3}} a^{\frac {1}{3}} d x +b^{\frac {2}{3}} c^{2}+2 b^{\frac {2}{3}} c d x +b^{\frac {2}{3}} d^{2} x^{2}\right ) c -2 b^{\frac {1}{3}} a^{\frac {2}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} c +b^{\frac {1}{3}} d x \right ) c +\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} c -b^{\frac {1}{3}} a^{\frac {1}{3}} d x +b^{\frac {2}{3}} c^{2}+2 b^{\frac {2}{3}} c d x +b^{\frac {2}{3}} d^{2} x^{2}\right ) a -2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} c +b^{\frac {1}{3}} d x \right ) a}{6 b^{\frac {2}{3}} a^{\frac {4}{3}} d^{2}} \] Input:
int(x/(a+b*(d*x+c)^3),x)
Output:
(2*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d* x)/(a**(1/3)*sqrt(3)))*c - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b** (1/3)*d*x)/(a**(1/3)*sqrt(3)))*a + b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/ 3)*a**(1/3)*c - b**(1/3)*a**(1/3)*d*x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*c - 2*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*c + b **(1/3)*d*x)*c + log(a**(2/3) - b**(1/3)*a**(1/3)*c - b**(1/3)*a**(1/3)*d* x + b**(2/3)*c**2 + 2*b**(2/3)*c*d*x + b**(2/3)*d**2*x**2)*a - 2*log(a**(1 /3) + b**(1/3)*c + b**(1/3)*d*x)*a)/(6*b**(2/3)*a**(1/3)*a*d**2)