\(\int \frac {c+d x}{(a+b (c+d x)^3)^2} \, dx\) [70]

Optimal result

Integrand size = 19, antiderivative size = 172 \[ \int \frac {c+d x}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {(c+d x)^2}{3 a d \left (a+b (c+d x)^3\right )}-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{2/3} d}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{4/3} b^{2/3} d}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{4/3} b^{2/3} d} \] Output:

1/3*(d*x+c)^2/a/d/(a+b*(d*x+c)^3)-1/9*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)/b^(2/3)/d-1/9*ln(a^(1/3)+b^(1/3)*(d*x+ 
c))/a^(4/3)/b^(2/3)/d+1/18*ln(a^(2/3)-a^(1/3)*b^(1/3)*(d*x+c)+b^(2/3)*(d*x 
+c)^2)/a^(4/3)/b^(2/3)/d
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.88 \[ \int \frac {c+d x}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {\frac {6 \sqrt [3]{a} (c+d x)^2}{a+b (c+d x)^3}+\frac {2 \sqrt {3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{b^{2/3}}-\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{b^{2/3}}+\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{b^{2/3}}}{18 a^{4/3} d} \] Input:

Integrate[(c + d*x)/(a + b*(c + d*x)^3)^2,x]
 

Output:

((6*a^(1/3)*(c + d*x)^2)/(a + b*(c + d*x)^3) + (2*Sqrt[3]*ArcTan[(-a^(1/3) 
 + 2*b^(1/3)*(c + d*x))/(Sqrt[3]*a^(1/3))])/b^(2/3) - (2*Log[a^(1/3) + b^( 
1/3)*(c + d*x)])/b^(2/3) + Log[a^(2/3) - a^(1/3)*b^(1/3)*(c + d*x) + b^(2/ 
3)*(c + d*x)^2]/b^(2/3))/(18*a^(4/3)*d)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {895, 819, 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.

\(\displaystyle \int \frac {c+d x}{\left (a+b (c+d x)^3\right )^2} \, dx\)

\(\Big \downarrow \) 895

\(\displaystyle \frac {\int \frac {c+d x}{\left (b (c+d x)^3+a\right )^2}d(c+d x)}{d}\)

\(\Big \downarrow \) 819

\(\displaystyle \frac {\frac {\int \frac {c+d x}{b (c+d x)^3+a}d(c+d x)}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}}{d}\)

\(\Big \downarrow \) 821

\(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}}{d}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {\frac {\int \frac {\sqrt [3]{b} (c+d x)+\sqrt [3]{a}}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}}{d}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)\right )}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}}{d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}}{d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{b^{2/3} (c+d x)^2-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+a^{2/3}}d(c+d x)-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}}{d}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} (c+d x)}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{3 \sqrt [3]{a} b^{2/3}}}{3 a}+\frac {(c+d x)^2}{3 a \left (a+b (c+d x)^3\right )}}{d}\)

Input:

Int[(c + d*x)/(a + b*(c + d*x)^3)^2,x]
 

Output:

((c + d*x)^2/(3*a*(a + b*(c + d*x)^3)) + (-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)))/(3*a))/d
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 819

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( 
c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 
1) + 1)/(a*n*(p + 1))   Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a 
, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p 
, x]
 

rule 821

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]
 

rule 895

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]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1142

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.25 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {\frac {d \,x^{2}}{3 a}+\frac {2 c x}{3 a}+\frac {c^{2}}{3 a d}}{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{9 a b d}\)	\(144\)
risch	\(\frac {\frac {d \,x^{2}}{3 a}+\frac {2 c x}{3 a}+\frac {c^{2}}{3 a d}}{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\left (\textit {\_R} d +c \right ) \ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{9 a b d}\)	\(144\)

Input:

int((d*x+c)/(a+b*(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
 

Output:

(1/3*d/a*x^2+2/3/a*c*x+1/3*c^2/a/d)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b 
*c^3+a)+1/9/a/b/d*sum((_R*d+c)/(_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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (133) = 266\).

Time = 0.10 (sec) , antiderivative size = 852, normalized size of antiderivative = 4.95 \[ \int \frac {c+d x}{\left (a+b (c+d x)^3\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)/(a+b*(d*x+c)^3)^2,x, algorithm="fricas")
 

Output:

[1/18*(6*a*b^2*d^2*x^2 + 12*a*b^2*c*d*x + 6*a*b^2*c^2 + 3*sqrt(1/3)*(a*b^2 
*d^3*x^3 + 3*a*b^2*c*d^2*x^2 + 3*a*b^2*c^2*d*x + a*b^2*c^3 + a^2*b)*sqrt(( 
-a*b^2)^(1/3)/a)*log((2*b^2*d^3*x^3 + 6*b^2*c*d^2*x^2 + 6*b^2*c^2*d*x + 2* 
b^2*c^3 - a*b + 3*sqrt(1/3)*(a*b*d*x + a*b*c + 2*(d^2*x^2 + 2*c*d*x + c^2) 
*(-a*b^2)^(2/3) + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2 
/3)*(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)) + (b 
*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*(-a*b^2)^(2/3)*log(b^2 
*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (-a*b^2)^(1/3)*(b*d*x + b*c) + (-a*b^2) 
^(2/3)) - 2*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*(-a*b^2) 
^(2/3)*log(b*d*x + b*c - (-a*b^2)^(1/3)))/(a^2*b^3*d^4*x^3 + 3*a^2*b^3*c*d 
^3*x^2 + 3*a^2*b^3*c^2*d^2*x + (a^2*b^3*c^3 + a^3*b^2)*d), 1/18*(6*a*b^2*d 
^2*x^2 + 12*a*b^2*c*d*x + 6*a*b^2*c^2 + 6*sqrt(1/3)*(a*b^2*d^3*x^3 + 3*a*b 
^2*c*d^2*x^2 + 3*a*b^2*c^2*d*x + a*b^2*c^3 + a^2*b)*sqrt(-(-a*b^2)^(1/3)/a 
)*arctan(sqrt(1/3)*(2*b*d*x + 2*b*c + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3) 
/a)/b) + (b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*(-a*b^2)^(2 
/3)*log(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2 + (-a*b^2)^(1/3)*(b*d*x + b*c) 
 + (-a*b^2)^(2/3)) - 2*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + 
a)*(-a*b^2)^(2/3)*log(b*d*x + b*c - (-a*b^2)^(1/3)))/(a^2*b^3*d^4*x^3 + 3* 
a^2*b^3*c*d^3*x^2 + 3*a^2*b^3*c^2*d^2*x + (a^2*b^3*c^3 + a^3*b^2)*d)]
 

Sympy [A] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.61 \[ \int \frac {c+d x}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {c^{2} + 2 c d x + d^{2} x^{2}}{3 a^{2} d + 3 a b c^{3} d + 9 a b c^{2} d^{2} x + 9 a b c d^{3} x^{2} + 3 a b d^{4} x^{3}} + \frac {\operatorname {RootSum} {\left (729 t^{3} a^{4} b^{2} + 1, \left ( t \mapsto t \log {\left (x + \frac {81 t^{2} a^{3} b + c}{d} \right )} \right )\right )}}{d} \] Input:

integrate((d*x+c)/(a+b*(d*x+c)**3)**2,x)
                                                                                    
                                                                                    
 

Output:

(c**2 + 2*c*d*x + d**2*x**2)/(3*a**2*d + 3*a*b*c**3*d + 9*a*b*c**2*d**2*x 
+ 9*a*b*c*d**3*x**2 + 3*a*b*d**4*x**3) + RootSum(729*_t**3*a**4*b**2 + 1, 
Lambda(_t, _t*log(x + (81*_t**2*a**3*b + c)/d)))/d
 

Maxima [F]

\[ \int \frac {c+d x}{\left (a+b (c+d x)^3\right )^2} \, dx=\int { \frac {d x + c}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{2}} \,d x } \] Input:

integrate((d*x+c)/(a+b*(d*x+c)^3)^2,x, algorithm="maxima")
 

Output:

1/3*(d^2*x^2 + 2*c*d*x + c^2)/(a*b*d^4*x^3 + 3*a*b*c*d^3*x^2 + 3*a*b*c^2*d 
^2*x + (a*b*c^3 + a^2)*d) + 1/3*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
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.19 \[ \int \frac {c+d x}{\left (a+b (c+d x)^3\right )^2} \, dx=-\frac {2 \, \sqrt {3} \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}}{3 \, \left (-a^{2} b\right )^{\frac {2}{3}}}\right ) + \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left (2 \, a b d x + 2 \, a b c - \left (-a^{2} b\right )^{\frac {2}{3}}\right )}^{2} + 3 \, \left (-a^{2} b\right )^{\frac {4}{3}}\right ) - 2 \, \left (-\frac {1}{a b^{2} d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | a b d x + a b c + \left (-a^{2} b\right )^{\frac {2}{3}} \right |}\right )}{18 \, a} + \frac {d^{2} x^{2} + 2 \, c d x + c^{2}}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} a d} \] Input:

integrate((d*x+c)/(a+b*(d*x+c)^3)^2,x, algorithm="giac")
 

Output:

-1/18*(2*sqrt(3)*(-1/(a*b^2*d^3))^(1/3)*arctan(1/3*sqrt(3)*(2*a*b*d*x + 2* 
a*b*c - (-a^2*b)^(2/3))/(-a^2*b)^(2/3)) + (-1/(a*b^2*d^3))^(1/3)*log((2*a* 
b*d*x + 2*a*b*c - (-a^2*b)^(2/3))^2 + 3*(-a^2*b)^(4/3)) - 2*(-1/(a*b^2*d^3 
))^(1/3)*log(abs(a*b*d*x + a*b*c + (-a^2*b)^(2/3))))/a + 1/3*(d^2*x^2 + 2* 
c*d*x + c^2)/((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*a*d)
 

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.26 \[ \int \frac {c+d x}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {\frac {d\,x^2}{3\,a}+\frac {c^2}{3\,a\,d}+\frac {2\,c\,x}{3\,a}}{b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a}-\frac {\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{9\,a^{4/3}\,b^{2/3}\,d}-\frac {\ln \left (\frac {b^{2/3}\,d^4\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{36\,a^{5/3}}+\frac {b\,c\,d^4}{9\,a^2}+\frac {b\,d^5\,x}{9\,a^2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,b^{2/3}\,d}+\frac {\ln \left (\frac {b^{2/3}\,d^4\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{36\,a^{5/3}}+\frac {b\,c\,d^4}{9\,a^2}+\frac {b\,d^5\,x}{9\,a^2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{4/3}\,b^{2/3}\,d} \] Input:

int((c + d*x)/(a + b*(c + d*x)^3)^2,x)
 

Output:

((d*x^2)/(3*a) + c^2/(3*a*d) + (2*c*x)/(3*a))/(a + b*c^3 + b*d^3*x^3 + 3*b 
*c^2*d*x + 3*b*c*d^2*x^2) - log(b^(1/3)*c + a^(1/3) + b^(1/3)*d*x)/(9*a^(4 
/3)*b^(2/3)*d) - (log((b^(2/3)*d^4*(3^(1/2)*1i - 1)^2)/(36*a^(5/3)) + (b*c 
*d^4)/(9*a^2) + (b*d^5*x)/(9*a^2))*(3^(1/2)*1i - 1))/(18*a^(4/3)*b^(2/3)*d 
) + (log((b^(2/3)*d^4*(3^(1/2)*1i + 1)^2)/(36*a^(5/3)) + (b*c*d^4)/(9*a^2) 
 + (b*d^5*x)/(9*a^2))*(3^(1/2)*1i + 1))/(18*a^(4/3)*b^(2/3)*d)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 705, normalized size of antiderivative = 4.10 \[ \int \frac {c+d x}{\left (a+b (c+d x)^3\right )^2} \, dx =\text {Too large to display} \] Input:

int((d*x+c)/(a+b*(d*x+c)^3)^2,x)
 

Output:

( - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sq 
rt(3)))*a*b*c - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/ 
(a**(1/3)*sqrt(3)))*b**2*c**4 - 6*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 
2*b**(1/3)*d*x)/(a**(1/3)*sqrt(3)))*b**2*c**3*d*x - 6*sqrt(3)*atan((a**(1/ 
3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sqrt(3)))*b**2*c**2*d**2*x** 
2 - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*c - 2*b**(1/3)*d*x)/(a**(1/3)*sq 
rt(3)))*b**2*c*d**3*x**3 - 2*b**(2/3)*a**(1/3)*a + 4*b**(2/3)*a**(1/3)*b*c 
**3 + 6*b**(2/3)*a**(1/3)*b*c**2*d*x - 2*b**(2/3)*a**(1/3)*b*d**3*x**3 + l 
og(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*b*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)*b**2*c**4 + 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)*b 
**2*c**3*d*x + 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)*b**2*c**2*d**2* 
x**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**2*c*d**3*x**3 - 2*log( 
a**(1/3) + b**(1/3)*c + b**(1/3)*d*x)*a*b*c - 2*log(a**(1/3) + b**(1/3)*c 
+ b**(1/3)*d*x)*b**2*c**4 - 6*log(a**(1/3) + b**(1/3)*c + b**(1/3)*d*x)*b* 
*2*c**3*d*x - 6*log(a**(1/3) + b**(1/3)*c + b**(1/3)*d*x)*b**2*c**2*d**...