Integrand size = 21, antiderivative size = 154 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )} \, dx=-\frac {1}{a d (c+d x)}+\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 a^{4/3} d}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d} \] Output:
-1/a/d/(d*x+c)+1/3*b^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c))*3^(1/2)/ a^(1/3))*3^(1/2)/a^(4/3)/d+1/3*b^(1/3)*ln(a^(1/3)+b^(1/3)*(d*x+c))/a^(4/3) /d-1/6*b^(1/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*(d*x+c)+b^(2/3)*(d*x+c)^2)/a^(4/ 3)/d
Time = 0.06 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )} \, dx=\frac {-\frac {6 \sqrt [3]{a}}{c+d x}-2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )+2 \sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )-\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{6 a^{4/3} d} \] Input:
Integrate[1/((c + d*x)^2*(a + b*(c + d*x)^3)),x]
Output:
((-6*a^(1/3))/(c + d*x) - 2*Sqrt[3]*b^(1/3)*ArcTan[(-a^(1/3) + 2*b^(1/3)*( c + d*x))/(Sqrt[3]*a^(1/3))] + 2*b^(1/3)*Log[a^(1/3) + b^(1/3)*(c + d*x)] - b^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2])/ (6*a^(4/3)*d)
Time = 0.49 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {895, 847, 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.
| \(\displaystyle \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )} \, dx\) |
| \(\Big \downarrow \) 895 |
| \(\displaystyle \frac {\int \frac {1}{(c+d x)^2 \left (b (c+d x)^3+a\right )}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {-\frac {b \int \frac {c+d x}{b (c+d x)^3+a}d(c+d x)}{a}-\frac {1}{a (c+d x)}}{d}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {-\frac {b \left (\frac {\int \frac {\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{a}-\frac {1}{a (c+d x)}}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {-\frac {b \left (\frac {\int \frac {\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a (c+d x)}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a (c+d x)}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {-\frac {b \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a (c+d x)}}{d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {b \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a (c+d x)}}{d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {b \left (\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a (c+d x)}}{d}\) |
Input:
Int[1/((c + d*x)^2*(a + b*(c + d*x)^3)),x]
Output:
(-(1/(a*(c + d*x))) - (b*(-1/3*Log[a^(1/3) + b^(1/3)*(c + d*x)]/(a^(1/3)*b ^(2/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(c + d*x))/a^(1/3))/Sqrt[3]]) /b^(1/3)) + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/3)*(c + d*x)^2] /(2*b^(1/3)))/(3*a^(1/3)*b^(1/3))))/a)/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[(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[((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[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff icient[v, x, 1]*v^m) Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[ {a, b, m, n, p}, x] && LinearPairQ[u, v, 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.77 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {1}{a d \left (x d +c \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{4} d^{3} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left (\left (-4 a^{4} d^{4} \textit {\_R}^{3}+3 b d \right ) x -4 a^{4} c \,d^{3} \textit {\_R}^{3}-a^{3} d^{2} \textit {\_R}^{2}+3 b c \right )\right )}{3}\) | \(86\) |
| default | \(-\frac {1}{a d \left (x d +c \right )}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{3 d a}\) | \(92\) |
Input:
int(1/(d*x+c)^2/(a+b*(d*x+c)^3),x,method=_RETURNVERBOSE)
Output:
-1/a/d/(d*x+c)+1/3*sum(_R*ln((-4*_R^3*a^4*d^4+3*b*d)*x-4*a^4*c*d^3*_R^3-a^ 3*d^2*_R^2+3*b*c),_R=RootOf(_Z^3*a^4*d^3-b))
Time = 0.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )} \, dx=-\frac {2 \, \sqrt {3} {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} - {\left (a d x + a c\right )} \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, {\left (d x + c\right )} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b d x + b c + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 6}{6 \, {\left (a d^{2} x + a c d\right )}} \] Input:
integrate(1/(d*x+c)^2/(a+b*(d*x+c)^3),x, algorithm="fricas")
Output:
-1/6*(2*sqrt(3)*(d*x + c)*(b/a)^(1/3)*arctan(2/3*sqrt(3)*(d*x + c)*(b/a)^( 1/3) - 1/3*sqrt(3)) + (d*x + c)*(b/a)^(1/3)*log(b*d^2*x^2 + 2*b*c*d*x + b* c^2 - (a*d*x + a*c)*(b/a)^(2/3) + a*(b/a)^(1/3)) - 2*(d*x + c)*(b/a)^(1/3) *log(b*d*x + b*c + a*(b/a)^(2/3)) + 6)/(a*d^2*x + a*c*d)
Time = 0.41 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.29 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )} \, dx=- \frac {1}{a c d + a d^{2} x} + \frac {\operatorname {RootSum} {\left (27 t^{3} a^{4} - b, \left ( t \mapsto t \log {\left (x + \frac {9 t^{2} a^{3} + b c}{b d} \right )} \right )\right )}}{d} \] Input:
integrate(1/(d*x+c)**2/(a+b*(d*x+c)**3),x)
Output:
-1/(a*c*d + a*d**2*x) + RootSum(27*_t**3*a**4 - b, Lambda(_t, _t*log(x + ( 9*_t**2*a**3 + b*c)/(b*d))))/d
\[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{3} b + a\right )} {\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate(1/(d*x+c)^2/(a+b*(d*x+c)^3),x, algorithm="maxima")
Output:
-b*integrate((d*x + c)/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a), x)/a - 1/(a*d^2*x + a*c*d)
Time = 0.13 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )} \, dx=\frac {\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | -\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} - \frac {1}{{\left (d x + c\right )} d} \right |}\right )}{3 \, a} - \frac {\sqrt {3} \left (a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}} - \frac {2}{{\left (d x + c\right )} d}\right )}}{3 \, \left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}}}\right )}{3 \, a^{2} d} - \frac {\left (a^{2} b\right )^{\frac {1}{3}} \log \left (\left (\frac {b}{a d^{3}}\right )^{\frac {2}{3}} - \frac {\left (\frac {b}{a d^{3}}\right )^{\frac {1}{3}}}{{\left (d x + c\right )} d} + \frac {1}{{\left (d x + c\right )}^{2} d^{2}}\right )}{6 \, a^{2} d} - \frac {1}{{\left (d x + c\right )} a d} \] Input:
integrate(1/(d*x+c)^2/(a+b*(d*x+c)^3),x, algorithm="giac")
Output:
1/3*(b/(a*d^3))^(1/3)*log(abs(-(b/(a*d^3))^(1/3) - 1/((d*x + c)*d)))/a - 1 /3*sqrt(3)*(a^2*b)^(1/3)*arctan(1/3*sqrt(3)*((b/(a*d^3))^(1/3) - 2/((d*x + c)*d))/(b/(a*d^3))^(1/3))/(a^2*d) - 1/6*(a^2*b)^(1/3)*log((b/(a*d^3))^(2/ 3) - (b/(a*d^3))^(1/3)/((d*x + c)*d) + 1/((d*x + c)^2*d^2))/(a^2*d) - 1/(( d*x + c)*a*d)
Time = 0.43 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )} \, dx=\frac {b^{1/3}\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{3\,a^{4/3}\,d}-\frac {1}{a\,d\,\left (c+d\,x\right )}-\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,a^{4/3}\,d}+\frac {b^{1/3}\,\ln \left (2\,b^{1/3}\,c-a^{1/3}+2\,b^{1/3}\,d\,x-\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{a^{4/3}\,d} \] Input:
int(1/((a + b*(c + d*x)^3)*(c + d*x)^2),x)
Output:
(b^(1/3)*log(b^(1/3)*c + a^(1/3) + b^(1/3)*d*x))/(3*a^(4/3)*d) - 1/(a*d*(c + d*x)) - (b^(1/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*c - a^(1/3) + 2*b^( 1/3)*d*x)*((3^(1/2)*1i)/2 + 1/2))/(3*a^(4/3)*d) + (b^(1/3)*log(2*b^(1/3)*c - 3^(1/2)*a^(1/3)*1i - a^(1/3) + 2*b^(1/3)*d*x)*((3^(1/2)*1i)/6 - 1/6))/( a^(4/3)*d)
Time = 0.23 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.65 \[ \int \frac {1}{(c+d x)^2 \left (a+b (c+d x)^3\right )} \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} c -2 b^{\frac {1}{3}} d x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b \,c^{2}+2 \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} c -2 b^{\frac {1}{3}} d x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b c d x +6 b^{\frac {2}{3}} a^{\frac {1}{3}} d x -\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} c -b^{\frac {1}{3}} a^{\frac {1}{3}} d x +b^{\frac {2}{3}} c^{2}+2 b^{\frac {2}{3}} c d x +b^{\frac {2}{3}} d^{2} x^{2}\right ) b \,c^{2}-\mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} c -b^{\frac {1}{3}} a^{\frac {1}{3}} d x +b^{\frac {2}{3}} c^{2}+2 b^{\frac {2}{3}} c d x +b^{\frac {2}{3}} d^{2} x^{2}\right ) b c d x +2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} c +b^{\frac {1}{3}} d x \right ) b \,c^{2}+2 \,\mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} c +b^{\frac {1}{3}} d x \right ) b c d x}{6 b^{\frac {2}{3}} a^{\frac {4}{3}} c d \left (d x +c \right )} \] Input:
int(1/(d*x+c)^2/(a+b*(d*x+c)^3),x)
Output:
(2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sqrt( 3)))*b*c**2 + 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a **(1/3)*sqrt(3)))*b*c*d*x + 6*b**(2/3)*a**(1/3)*d*x - 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)*b*c**2 - 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)*b *c*d*x + 2*log(a**(1/3) + b**(1/3)*c + b**(1/3)*d*x)*b*c**2 + 2*log(a**(1/ 3) + b**(1/3)*c + b**(1/3)*d*x)*b*c*d*x)/(6*b**(2/3)*a**(1/3)*a*c*d*(c + d *x))