\(\int \frac {(c+d x)^3}{(a+b (c+d x)^3)^2} \, dx\) [68]

Optimal result

Integrand size = 21, antiderivative size = 170 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^2} \, dx=-\frac {c+d x}{3 b d \left (a+b (c+d x)^3\right )}-\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3} b^{4/3} d}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{9 a^{2/3} b^{4/3} d}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{18 a^{2/3} b^{4/3} d} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.89 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {-\frac {6 \sqrt [3]{b} (c+d x)}{a+b (c+d x)^3}+\frac {2 \sqrt {3} \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} (c+d x)}{\sqrt {3} \sqrt [3]{a}}\right )}{a^{2/3}}+\frac {2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} (c+d x)\right )}{a^{2/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} (c+d x)+b^{2/3} (c+d x)^2\right )}{a^{2/3}}}{18 b^{4/3} d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {895, 817, 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.

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

\(\Big \downarrow \) 895

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

\(\Big \downarrow \) 817

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

\(\Big \downarrow \) 750

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1142

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1103

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

Input:

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

Output:

(-1/3*(c + d*x)/(b*(a + b*(c + d*x)^3)) + (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*b))/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 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 817

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^( 
n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[c^n 
*((m - n + 1)/(b*n*(p + 1)))   Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x], x 
] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  ! 
ILtQ[(m + n*(p + 1) + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, 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 = 0.56 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.73

method	result	size
default	\(\frac {-\frac {x}{3 b}-\frac {c}{3 d b}}{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{9 b^{2} d}\)	\(124\)
risch	\(\frac {-\frac {x}{3 b}-\frac {c}{3 d b}}{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{3} \textit {\_Z}^{3}+3 b c \,d^{2} \textit {\_Z}^{2}+3 b \,c^{2} d \textit {\_Z} +c^{3} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{d^{2} \textit {\_R}^{2}+2 c d \textit {\_R} +c^{2}}}{9 b^{2} d}\)	\(124\)

Input:

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

Output:

(-1/3*x/b-1/3*c/d/b)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a)+1/9/b^2 
/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. 345 vs. \(2 (131) = 262\).

Time = 0.09 (sec) , antiderivative size = 810, normalized size of antiderivative = 4.76 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/18*(6*a^2*b*d*x + 6*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^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^2*b*d*x + 6*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^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 = 0.77 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.54 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {- c - d x}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} + \frac {\operatorname {RootSum} {\left (729 t^{3} a^{2} b^{4} - 1, \left ( t \mapsto t \log {\left (x + \frac {9 t a b + c}{d} \right )} \right )\right )}}{d} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.26 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {2 \, \sqrt {3} \left (\frac {1}{a^{2} b d^{3}}\right )^{\frac {1}{3}} \arctan \left (-\frac {b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}}}{\sqrt {3} b d x + \sqrt {3} b c - \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}}\right ) - \left (\frac {1}{a^{2} b d^{3}}\right )^{\frac {1}{3}} \log \left (4 \, {\left (\sqrt {3} b d x + \sqrt {3} b c - \sqrt {3} \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2} + 4 \, {\left (b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}}\right )}^{2}\right ) + 2 \, \left (\frac {1}{a^{2} b d^{3}}\right )^{\frac {1}{3}} \log \left ({\left | b d x + b c + \left (a b^{2}\right )^{\frac {1}{3}} \right |}\right )}{18 \, b} - \frac {d x + c}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} b d} \] Input:

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

Output:

1/18*(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))))/b - 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)*b*d)
 

Mupad [B] (verification not implemented)

Time = 0.71 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^2} \, dx=\frac {\ln \left (b^{1/3}\,c+a^{1/3}+b^{1/3}\,d\,x\right )}{9\,a^{2/3}\,b^{4/3}\,d}-\frac {\frac {x}{3\,b}+\frac {c}{3\,b\,d}}{b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a}+\frac {\ln \left (b\,c\,d^5+b\,d^6\,x+\frac {a^{1/3}\,b^{2/3}\,d^5\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{2/3}\,b^{4/3}\,d}-\frac {\ln \left (b\,c\,d^5+b\,d^6\,x-\frac {a^{1/3}\,b^{2/3}\,d^5\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{18\,a^{2/3}\,b^{4/3}\,d} \] Input:

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

Output:

log(b^(1/3)*c + a^(1/3) + b^(1/3)*d*x)/(9*a^(2/3)*b^(4/3)*d) - (x/(3*b) + 
c/(3*b*d))/(a + b*c^3 + b*d^3*x^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2) + (log(b* 
c*d^5 + b*d^6*x + (a^(1/3)*b^(2/3)*d^5*(3^(1/2)*1i - 1))/2)*(3^(1/2)*1i - 
1))/(18*a^(2/3)*b^(4/3)*d) - (log(b*c*d^5 + b*d^6*x - (a^(1/3)*b^(2/3)*d^5 
*(3^(1/2)*1i + 1))/2)*(3^(1/2)*1i + 1))/(18*a^(2/3)*b^(4/3)*d)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 677, normalized size of antiderivative = 3.98 \[ \int \frac {(c+d x)^3}{\left (a+b (c+d x)^3\right )^2} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^3/(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**(...