\(\int \frac {1}{(a+b (c+d x)^3)^2} \, dx\) [71]

Optimal result

Integrand size = 13, antiderivative size = 170 \[ \int \frac {1}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {c+d x}{3 a d \left (a+b (c+d x)^3\right )}-\frac {2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} \sqrt [3]{b} d}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{5/3} \sqrt [3]{b} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{9 a^{5/3} \sqrt [3]{b} d} \] Output:

1/3*(d*x+c)/a/d/(a+b*(d*x+c)^3)-2/9*arctan(1/3*(a^(1/3)-2*b^(1/3)*(d*x+c)) 
*3^(1/2)/a^(1/3))*3^(1/2)/a^(5/3)/b^(1/3)/d+2/9*ln(a^(1/3)+b^(1/3)*(d*x+c) 
)/a^(5/3)/b^(1/3)/d-1/9*ln(a^(2/3)-a^(1/3)*b^(1/3)*(d*x+c)+b^(2/3)*(d*x+c) 
^2)/a^(5/3)/b^(1/3)/d
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.89 \[ \int \frac {1}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {\frac {3 a^{2/3} (c+d x)}{a+b (c+d x)^3}+\frac {2 \sqrt {3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{\sqrt [3]{b}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{\sqrt [3]{b}}}{9 a^{5/3} d} \] Input:

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

Output:

((3*a^(2/3)*(c + d*x))/(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^(1/3) + (2*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]/b^(1/3))/(9*a^(5/3)*d)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {239, 749, 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.

\(\displaystyle \int \frac {1}{\left (a+b (c+d x)^3\right )^2} \, dx\)

\(\Big \downarrow \) 239

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

\(\Big \downarrow \) 749

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

\(\Big \downarrow \) 750

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

((c + d*x)/(3*a*(a + b*(c + d*x)^3)) + (2*(Log[a^(1/3) + b^(1/3)*(c + d*x) 
]/(3*a^(2/3)*b^(1/3)) + (-((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^(2/3))))/(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 239

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Simp[1/Coefficient[v, x, 1 
]   Subst[Int[(a + b*x^n)^p, x], x, v], x] /; FreeQ[{a, b, n, p}, x] && Lin 
earQ[v, x] && NeQ[v, x]
 

rule 749

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 
 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1))   Int[(a + b*x^ 
n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte 
gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
 

rule 750

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]
 

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 = 0.23 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {\frac {x}{3 a}+\frac {c}{3 d a}}{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{9 a b d}\)	\(127\)
risch	\(\frac {\frac {x}{3 a}+\frac {c}{3 d a}}{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}\right )}{9 a b d}\)	\(127\)

Input:

int(1/(a+b*(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
 

Output:

(1/3*x/a+1/3*c/d/a)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)+2/9/a/b/ 
d*sum(1/(_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. 344 vs. \(2 (131) = 262\).

Time = 0.12 (sec) , antiderivative size = 808, normalized size of antiderivative = 4.75 \[ \int \frac {1}{\left (a+b (c+d x)^3\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/(a+b*(d*x+c)^3)^2,x, algorithm="fricas")
 

Output:

[1/9*(3*a^2*b*d*x + 3*a^2*b*c + 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^2*b)^(1/3)/b)*log((2* 
a*b*d^3*x^3 + 6*a*b*c*d^2*x^2 + 6*a*b*c^2*d*x + 2*a*b*c^3 - a^2 + 3*sqrt(1 
/3)*(2*a*b*d^2*x^2 + 4*a*b*c*d*x + 2*a*b*c^2 + (a^2*b)^(2/3)*(d*x + c) - ( 
a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b) - 3*(a^2*b)^(1/3)*(a*d*x + a*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^2*b)^(2/3)*log(a*b*d^2*x^2 + 2*a*b*c 
*d*x + a*b*c^2 - (a^2*b)^(2/3)*(d*x + c) + (a^2*b)^(1/3)*a) + 2*(b*d^3*x^3 
 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*(a^2*b)^(2/3)*log(a*b*d*x + a* 
b*c + (a^2*b)^(2/3)))/(a^3*b^2*d^4*x^3 + 3*a^3*b^2*c*d^3*x^2 + 3*a^3*b^2*c 
^2*d^2*x + (a^3*b^2*c^3 + a^4*b)*d), 1/9*(3*a^2*b*d*x + 3*a^2*b*c + 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^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*(d*x + c) - 
(a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - (b*d^3*x^3 + 3*b*c*d^2*x^2 + 
 3*b*c^2*d*x + b*c^3 + a)*(a^2*b)^(2/3)*log(a*b*d^2*x^2 + 2*a*b*c*d*x + a* 
b*c^2 - (a^2*b)^(2/3)*(d*x + c) + (a^2*b)^(1/3)*a) + 2*(b*d^3*x^3 + 3*b*c* 
d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)*(a^2*b)^(2/3)*log(a*b*d*x + a*b*c + (a^ 
2*b)^(2/3)))/(a^3*b^2*d^4*x^3 + 3*a^3*b^2*c*d^3*x^2 + 3*a^3*b^2*c^2*d^2*x 
+ (a^3*b^2*c^3 + a^4*b)*d)]
 

Sympy [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.54 \[ \int \frac {1}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {c + d x}{3 a^{2} d + 3 a b c^{3} d + 9 a b c^{2} d^{2} x + 9 a b c d^{3} x^{2} + 3 a b d^{4} x^{3}} + \frac {\operatorname {RootSum} {\left (729 t^{3} a^{5} b - 8, \left ( t \mapsto t \log {\left (x + \frac {9 t a^{2} + 2 c}{2 d} \right )} \right )\right )}}{d} \] Input:

integrate(1/(a+b*(d*x+c)**3)**2,x)
                                                                                    
                                                                                    
 

Output:

(c + d*x)/(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**5*b - 8, Lambda(_t, _t*log(x + 
(9*_t*a**2 + 2*c)/(2*d))))/d
 

Maxima [F]

\[ \int \frac {1}{\left (a+b (c+d x)^3\right )^2} \, dx=\int { \frac {1}{{\left ({\left (d x + c\right )}^{3} b + a\right )}^{2}} \,d x } \] Input:

integrate(1/(a+b*(d*x+c)^3)^2,x, algorithm="maxima")
 

Output:

1/3*(d*x + c)/(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) + 2/3*integrate(1/(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 = 0.12 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {2 \, \sqrt {3} \left (\frac {1}{a^{2} b d^{3}}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c - \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}}\right ) - \left (\frac {1}{a^{2} b d^{3}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c - \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (\frac {1}{a^{2} b d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a} + \frac {d x + c}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} a d} \] Input:

integrate(1/(a+b*(d*x+c)^3)^2,x, algorithm="giac")
 

Output:

1/9*(2*sqrt(3)*(1/(a^2*b*d^3))^(1/3)*arctan(-(b*d*x + b*c + (a*b^2)^(1/3)) 
/(sqrt(3)*b*d*x + sqrt(3)*b*c - sqrt(3)*(a*b^2)^(1/3))) - (1/(a^2*b*d^3))^ 
(1/3)*log(4*(sqrt(3)*b*d*x + sqrt(3)*b*c - sqrt(3)*(a*b^2)^(1/3))^2 + 4*(b 
*d*x + b*c + (a*b^2)^(1/3))^2) + 2*(1/(a^2*b*d^3))^(1/3)*log(abs(b*d*x + b 
*c + (a*b^2)^(1/3))))/a + 1/3*(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)*a*d)
 

Mupad [B] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {\frac {x}{3\,a}+\frac {c}{3\,a\,d}}{b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a}+\frac {2\,\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{9\,a^{5/3}\,b^{1/3}\,d}+\frac {\ln \left (\frac {2\,b^2\,c\,d^5}{a}+\frac {2\,b^2\,d^6\,x}{a}+\frac {b^{5/3}\,d^5\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{a^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{5/3}\,b^{1/3}\,d}-\frac {\ln \left (\frac {2\,b^2\,c\,d^5}{a}+\frac {2\,b^2\,d^6\,x}{a}-\frac {b^{5/3}\,d^5\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{a^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{9\,a^{5/3}\,b^{1/3}\,d} \] Input:

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

Output:

(x/(3*a) + c/(3*a*d))/(a + b*c^3 + b*d^3*x^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2 
) + (2*log(b^(1/3)*c + a^(1/3) + b^(1/3)*d*x))/(9*a^(5/3)*b^(1/3)*d) + (lo 
g((2*b^2*c*d^5)/a + (2*b^2*d^6*x)/a + (b^(5/3)*d^5*(3^(1/2)*1i - 1))/a^(2/ 
3))*(3^(1/2)*1i - 1))/(9*a^(5/3)*b^(1/3)*d) - (log((2*b^2*c*d^5)/a + (2*b^ 
2*d^6*x)/a - (b^(5/3)*d^5*(3^(1/2)*1i + 1))/a^(2/3))*(3^(1/2)*1i + 1))/(9* 
a^(5/3)*b^(1/3)*d)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 677, normalized size of antiderivative = 3.98 \[ \int \frac {1}{\left (a+b (c+d x)^3\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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