\(\int \frac {1}{(c+d x)^4 (a+b (c+d x)^3)^3} \, dx\) [84]

Optimal result

Integrand size = 21, antiderivative size = 101 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {1}{3 a^3 d (c+d x)^3}-\frac {b}{6 a^2 d \left (a+b (c+d x)^3\right )^2}-\frac {2 b}{3 a^3 d \left (a+b (c+d x)^3\right )}-\frac {3 b \log (c+d x)}{a^4 d}+\frac {b \log \left (a+b (c+d x)^3\right )}{a^4 d} \] Output:

-1/3/a^3/d/(d*x+c)^3-1/6*b/a^2/d/(a+b*(d*x+c)^3)^2-2/3*b/a^3/d/(a+b*(d*x+c 
)^3)-3*b*ln(d*x+c)/a^4/d+b*ln(a+b*(d*x+c)^3)/a^4/d
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {a \left (-\frac {2}{(c+d x)^3}-\frac {a b}{\left (a+b (c+d x)^3\right )^2}-\frac {4 b}{a+b (c+d x)^3}\right )-18 b \log (c+d x)+6 b \log \left (a+b (c+d x)^3\right )}{6 a^4 d} \] Input:

Integrate[1/((c + d*x)^4*(a + b*(c + d*x)^3)^3),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {895, 798, 54, 2009}

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.

\(\displaystyle \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^3} \, dx\)

\(\Big \downarrow \) 895

\(\displaystyle \frac {\int \frac {1}{(c+d x)^4 \left (b (c+d x)^3+a\right )^3}d(c+d x)}{d}\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {\int \frac {1}{(c+d x)^6 \left (b (c+d x)^3+a\right )^3}d(c+d x)^3}{3 d}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {\int \left (\frac {3 b^2}{a^4 \left (b (c+d x)^3+a\right )}+\frac {2 b^2}{a^3 \left (b (c+d x)^3+a\right )^2}+\frac {b^2}{a^2 \left (b (c+d x)^3+a\right )^3}-\frac {3 b}{a^4 (c+d x)^3}+\frac {1}{a^3 (c+d x)^6}\right )d(c+d x)^3}{3 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {3 b \log \left ((c+d x)^3\right )}{a^4}+\frac {3 b \log \left (a+b (c+d x)^3\right )}{a^4}-\frac {2 b}{a^3 \left (a+b (c+d x)^3\right )}-\frac {1}{a^3 (c+d x)^3}-\frac {b}{2 a^2 \left (a+b (c+d x)^3\right )^2}}{3 d}\)

Input:

Int[1/((c + d*x)^4*(a + b*(c + d*x)^3)^3),x]
 

Output:

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

Defintions of rubi rules used

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.59

method	result	size
default	\(-\frac {1}{3 a^{3} d \left (x d +c \right )^{3}}-\frac {3 b \ln \left (x d +c \right )}{a^{4} d}+\frac {b^{2} \left (\frac {-\frac {2 a \,d^{2} x^{3}}{3}-2 a c d \,x^{2}-2 a \,c^{2} x -\frac {a \left (4 c^{3} b +5 a \right )}{6 d b}}{\left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )^{2}}+\frac {\ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}{b d}\right )}{a^{4}}\)	\(161\)
risch	\(\frac {-\frac {b^{2} d^{5} x^{6}}{a^{3}}-\frac {6 b^{2} c \,d^{4} x^{5}}{a^{3}}-\frac {15 b^{2} c^{2} d^{3} x^{4}}{a^{3}}-\frac {d^{2} \left (40 c^{3} b +3 a \right ) b \,x^{3}}{2 a^{3}}-\frac {3 b c d \left (10 c^{3} b +3 a \right ) x^{2}}{2 a^{3}}-\frac {3 \left (4 c^{3} b +3 a \right ) b \,c^{2} x}{2 a^{3}}-\frac {6 b^{2} c^{6}+9 a b \,c^{3}+2 a^{2}}{6 d \,a^{3}}}{\left (x d +c \right )^{3} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )^{2}}-\frac {3 b \ln \left (x d +c \right )}{a^{4} d}+\frac {b \ln \left (-b \,d^{3} x^{3}-3 b c \,d^{2} x^{2}-3 d b x \,c^{2}-c^{3} b -a \right )}{a^{4} d}\)	\(245\)
norman	\(\frac {-\frac {b^{2} d^{5} x^{6}}{a^{3}}+\frac {-6 b^{4} c^{6} d^{8}-9 a \,b^{3} c^{3} d^{8}-2 a^{2} b^{2} d^{8}}{6 a^{3} d^{9} b^{2}}+\frac {\left (-40 b^{4} c^{3} d^{8}-3 a \,b^{3} d^{8}\right ) x^{3}}{2 a^{3} d^{6} b^{2}}+\frac {3 c \left (-10 b^{4} c^{3} d^{8}-3 a \,b^{3} d^{8}\right ) x^{2}}{2 a^{3} d^{7} b^{2}}+\frac {3 c^{2} \left (-4 b^{4} c^{3} d^{8}-3 a \,b^{3} d^{8}\right ) x}{2 a^{3} d^{8} b^{2}}-\frac {6 b^{2} c \,d^{4} x^{5}}{a^{3}}-\frac {15 b^{2} c^{2} d^{3} x^{4}}{a^{3}}}{\left (x d +c \right )^{3} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )^{2}}+\frac {b \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 d b x \,c^{2}+c^{3} b +a \right )}{a^{4} d}-\frac {3 b \ln \left (x d +c \right )}{a^{4} d}\)	\(302\)
parallelrisch	\(\text {Expression too large to display}\)	\(1625\)

Input:

int(1/(d*x+c)^4/(a+b*(d*x+c)^3)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/3/a^3/d/(d*x+c)^3-3*b*ln(d*x+c)/a^4/d+1/a^4*b^2*((-2/3*a*d^2*x^3-2*a*c* 
d*x^2-2*a*c^2*x-1/6*a/d*(4*b*c^3+5*a)/b)/(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2* 
d*x+b*c^3+a)^2+1/b/d*ln(b*d^3*x^3+3*b*c*d^2*x^2+3*b*c^2*d*x+b*c^3+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (95) = 190\).

Time = 0.12 (sec) , antiderivative size = 889, normalized size of antiderivative = 8.80 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3)^3,x, algorithm="fricas")
 

Output:

-1/6*(6*a*b^2*d^6*x^6 + 36*a*b^2*c*d^5*x^5 + 90*a*b^2*c^2*d^4*x^4 + 6*a*b^ 
2*c^6 + 3*(40*a*b^2*c^3 + 3*a^2*b)*d^3*x^3 + 9*a^2*b*c^3 + 9*(10*a*b^2*c^4 
 + 3*a^2*b*c)*d^2*x^2 + 2*a^3 + 9*(4*a*b^2*c^5 + 3*a^2*b*c^2)*d*x - 6*(b^3 
*d^9*x^9 + 9*b^3*c*d^8*x^8 + 36*b^3*c^2*d^7*x^7 + 2*(42*b^3*c^3 + a*b^2)*d 
^6*x^6 + b^3*c^9 + 6*(21*b^3*c^4 + 2*a*b^2*c)*d^5*x^5 + 2*a*b^2*c^6 + 6*(2 
1*b^3*c^5 + 5*a*b^2*c^2)*d^4*x^4 + (84*b^3*c^6 + 40*a*b^2*c^3 + a^2*b)*d^3 
*x^3 + a^2*b*c^3 + 3*(12*b^3*c^7 + 10*a*b^2*c^4 + a^2*b*c)*d^2*x^2 + 3*(3* 
b^3*c^8 + 4*a*b^2*c^5 + a^2*b*c^2)*d*x)*log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3* 
b*c^2*d*x + b*c^3 + a) + 18*(b^3*d^9*x^9 + 9*b^3*c*d^8*x^8 + 36*b^3*c^2*d^ 
7*x^7 + 2*(42*b^3*c^3 + a*b^2)*d^6*x^6 + b^3*c^9 + 6*(21*b^3*c^4 + 2*a*b^2 
*c)*d^5*x^5 + 2*a*b^2*c^6 + 6*(21*b^3*c^5 + 5*a*b^2*c^2)*d^4*x^4 + (84*b^3 
*c^6 + 40*a*b^2*c^3 + a^2*b)*d^3*x^3 + a^2*b*c^3 + 3*(12*b^3*c^7 + 10*a*b^ 
2*c^4 + a^2*b*c)*d^2*x^2 + 3*(3*b^3*c^8 + 4*a*b^2*c^5 + a^2*b*c^2)*d*x)*lo 
g(d*x + c))/(a^4*b^2*d^10*x^9 + 9*a^4*b^2*c*d^9*x^8 + 36*a^4*b^2*c^2*d^8*x 
^7 + 2*(42*a^4*b^2*c^3 + a^5*b)*d^7*x^6 + 6*(21*a^4*b^2*c^4 + 2*a^5*b*c)*d 
^6*x^5 + 6*(21*a^4*b^2*c^5 + 5*a^5*b*c^2)*d^5*x^4 + (84*a^4*b^2*c^6 + 40*a 
^5*b*c^3 + a^6)*d^4*x^3 + 3*(12*a^4*b^2*c^7 + 10*a^5*b*c^4 + a^6*c)*d^3*x^ 
2 + 3*(3*a^4*b^2*c^8 + 4*a^5*b*c^5 + a^6*c^2)*d^2*x + (a^4*b^2*c^9 + 2*a^5 
*b*c^6 + a^6*c^3)*d)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (88) = 176\).

Time = 3.65 (sec) , antiderivative size = 500, normalized size of antiderivative = 4.95 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {- 2 a^{2} - 9 a b c^{3} - 6 b^{2} c^{6} - 90 b^{2} c^{2} d^{4} x^{4} - 36 b^{2} c d^{5} x^{5} - 6 b^{2} d^{6} x^{6} + x^{3} \left (- 9 a b d^{3} - 120 b^{2} c^{3} d^{3}\right ) + x^{2} \left (- 27 a b c d^{2} - 90 b^{2} c^{4} d^{2}\right ) + x \left (- 27 a b c^{2} d - 36 b^{2} c^{5} d\right )}{6 a^{5} c^{3} d + 12 a^{4} b c^{6} d + 6 a^{3} b^{2} c^{9} d + 216 a^{3} b^{2} c^{2} d^{8} x^{7} + 54 a^{3} b^{2} c d^{9} x^{8} + 6 a^{3} b^{2} d^{10} x^{9} + x^{6} \cdot \left (12 a^{4} b d^{7} + 504 a^{3} b^{2} c^{3} d^{7}\right ) + x^{5} \cdot \left (72 a^{4} b c d^{6} + 756 a^{3} b^{2} c^{4} d^{6}\right ) + x^{4} \cdot \left (180 a^{4} b c^{2} d^{5} + 756 a^{3} b^{2} c^{5} d^{5}\right ) + x^{3} \cdot \left (6 a^{5} d^{4} + 240 a^{4} b c^{3} d^{4} + 504 a^{3} b^{2} c^{6} d^{4}\right ) + x^{2} \cdot \left (18 a^{5} c d^{3} + 180 a^{4} b c^{4} d^{3} + 216 a^{3} b^{2} c^{7} d^{3}\right ) + x \left (18 a^{5} c^{2} d^{2} + 72 a^{4} b c^{5} d^{2} + 54 a^{3} b^{2} c^{8} d^{2}\right )} - \frac {3 b \log {\left (\frac {c}{d} + x \right )}}{a^{4} d} + \frac {b \log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{a^{4} d} \] Input:

integrate(1/(d*x+c)**4/(a+b*(d*x+c)**3)**3,x)
 

Output:

(-2*a**2 - 9*a*b*c**3 - 6*b**2*c**6 - 90*b**2*c**2*d**4*x**4 - 36*b**2*c*d 
**5*x**5 - 6*b**2*d**6*x**6 + x**3*(-9*a*b*d**3 - 120*b**2*c**3*d**3) + x* 
*2*(-27*a*b*c*d**2 - 90*b**2*c**4*d**2) + x*(-27*a*b*c**2*d - 36*b**2*c**5 
*d))/(6*a**5*c**3*d + 12*a**4*b*c**6*d + 6*a**3*b**2*c**9*d + 216*a**3*b** 
2*c**2*d**8*x**7 + 54*a**3*b**2*c*d**9*x**8 + 6*a**3*b**2*d**10*x**9 + x** 
6*(12*a**4*b*d**7 + 504*a**3*b**2*c**3*d**7) + x**5*(72*a**4*b*c*d**6 + 75 
6*a**3*b**2*c**4*d**6) + x**4*(180*a**4*b*c**2*d**5 + 756*a**3*b**2*c**5*d 
**5) + x**3*(6*a**5*d**4 + 240*a**4*b*c**3*d**4 + 504*a**3*b**2*c**6*d**4) 
 + x**2*(18*a**5*c*d**3 + 180*a**4*b*c**4*d**3 + 216*a**3*b**2*c**7*d**3) 
+ x*(18*a**5*c**2*d**2 + 72*a**4*b*c**5*d**2 + 54*a**3*b**2*c**8*d**2)) - 
3*b*log(c/d + x)/(a**4*d) + b*log(3*c**2*x/d**2 + 3*c*x**2/d + x**3 + (a + 
 b*c**3)/(b*d**3))/(a**4*d)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (95) = 190\).

Time = 0.07 (sec) , antiderivative size = 438, normalized size of antiderivative = 4.34 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=-\frac {6 \, b^{2} d^{6} x^{6} + 36 \, b^{2} c d^{5} x^{5} + 90 \, b^{2} c^{2} d^{4} x^{4} + 6 \, b^{2} c^{6} + 3 \, {\left (40 \, b^{2} c^{3} + 3 \, a b\right )} d^{3} x^{3} + 9 \, a b c^{3} + 9 \, {\left (10 \, b^{2} c^{4} + 3 \, a b c\right )} d^{2} x^{2} + 9 \, {\left (4 \, b^{2} c^{5} + 3 \, a b c^{2}\right )} d x + 2 \, a^{2}}{6 \, {\left (a^{3} b^{2} d^{10} x^{9} + 9 \, a^{3} b^{2} c d^{9} x^{8} + 36 \, a^{3} b^{2} c^{2} d^{8} x^{7} + 2 \, {\left (42 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{7} x^{6} + 6 \, {\left (21 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{6} x^{5} + 6 \, {\left (21 \, a^{3} b^{2} c^{5} + 5 \, a^{4} b c^{2}\right )} d^{5} x^{4} + {\left (84 \, a^{3} b^{2} c^{6} + 40 \, a^{4} b c^{3} + a^{5}\right )} d^{4} x^{3} + 3 \, {\left (12 \, a^{3} b^{2} c^{7} + 10 \, a^{4} b c^{4} + a^{5} c\right )} d^{3} x^{2} + 3 \, {\left (3 \, a^{3} b^{2} c^{8} + 4 \, a^{4} b c^{5} + a^{5} c^{2}\right )} d^{2} x + {\left (a^{3} b^{2} c^{9} + 2 \, a^{4} b c^{6} + a^{5} c^{3}\right )} d\right )}} + \frac {b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{a^{4} d} - \frac {3 \, b \log \left (d x + c\right )}{a^{4} d} \] Input:

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3)^3,x, algorithm="maxima")
 

Output:

-1/6*(6*b^2*d^6*x^6 + 36*b^2*c*d^5*x^5 + 90*b^2*c^2*d^4*x^4 + 6*b^2*c^6 + 
3*(40*b^2*c^3 + 3*a*b)*d^3*x^3 + 9*a*b*c^3 + 9*(10*b^2*c^4 + 3*a*b*c)*d^2* 
x^2 + 9*(4*b^2*c^5 + 3*a*b*c^2)*d*x + 2*a^2)/(a^3*b^2*d^10*x^9 + 9*a^3*b^2 
*c*d^9*x^8 + 36*a^3*b^2*c^2*d^8*x^7 + 2*(42*a^3*b^2*c^3 + a^4*b)*d^7*x^6 + 
 6*(21*a^3*b^2*c^4 + 2*a^4*b*c)*d^6*x^5 + 6*(21*a^3*b^2*c^5 + 5*a^4*b*c^2) 
*d^5*x^4 + (84*a^3*b^2*c^6 + 40*a^4*b*c^3 + a^5)*d^4*x^3 + 3*(12*a^3*b^2*c 
^7 + 10*a^4*b*c^4 + a^5*c)*d^3*x^2 + 3*(3*a^3*b^2*c^8 + 4*a^4*b*c^5 + a^5* 
c^2)*d^2*x + (a^3*b^2*c^9 + 2*a^4*b*c^6 + a^5*c^3)*d) + b*log(b*d^3*x^3 + 
3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)/(a^4*d) - 3*b*log(d*x + c)/(a^4*d 
)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {b \log \left ({\left | -b - \frac {a}{{\left (d x + c\right )}^{3}} \right |}\right )}{a^{4} d} + \frac {5 \, b^{3} + \frac {6 \, a b^{2}}{{\left (d x + c\right )}^{3}}}{6 \, a^{4} {\left (b + \frac {a}{{\left (d x + c\right )}^{3}}\right )}^{2} d} - \frac {1}{3 \, {\left (d x + c\right )}^{3} a^{3} d} \] Input:

integrate(1/(d*x+c)^4/(a+b*(d*x+c)^3)^3,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 2.45 (sec) , antiderivative size = 438, normalized size of antiderivative = 4.34 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^3} \, dx=\frac {b\,\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{a^4\,d}-\frac {\frac {2\,a^2+9\,a\,b\,c^3+6\,b^2\,c^6}{6\,a^3\,d}+\frac {3\,x^2\,\left (10\,d\,b^2\,c^4+3\,a\,d\,b\,c\right )}{2\,a^3}+\frac {3\,x\,\left (4\,b^2\,c^5+3\,a\,b\,c^2\right )}{2\,a^3}+\frac {x^3\,\left (40\,b^2\,c^3\,d^2+3\,a\,b\,d^2\right )}{2\,a^3}+\frac {b^2\,d^5\,x^6}{a^3}+\frac {15\,b^2\,c^2\,d^3\,x^4}{a^3}+\frac {6\,b^2\,c\,d^4\,x^5}{a^3}}{x\,\left (3\,d\,a^2\,c^2+12\,d\,a\,b\,c^5+9\,d\,b^2\,c^8\right )+x^6\,\left (84\,b^2\,c^3\,d^6+2\,a\,b\,d^6\right )+x^2\,\left (3\,a^2\,c\,d^2+30\,a\,b\,c^4\,d^2+36\,b^2\,c^7\,d^2\right )+x^5\,\left (126\,b^2\,c^4\,d^5+12\,a\,b\,c\,d^5\right )+x^3\,\left (a^2\,d^3+40\,a\,b\,c^3\,d^3+84\,b^2\,c^6\,d^3\right )+a^2\,c^3+b^2\,c^9+x^4\,\left (126\,b^2\,c^5\,d^4+30\,a\,b\,c^2\,d^4\right )+b^2\,d^9\,x^9+2\,a\,b\,c^6+9\,b^2\,c\,d^8\,x^8+36\,b^2\,c^2\,d^7\,x^7}-\frac {3\,b\,\ln \left (c+d\,x\right )}{a^4\,d} \] Input:

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

Output:

(b*log(a + b*c^3 + b*d^3*x^3 + 3*b*c^2*d*x + 3*b*c*d^2*x^2))/(a^4*d) - ((2 
*a^2 + 6*b^2*c^6 + 9*a*b*c^3)/(6*a^3*d) + (3*x^2*(10*b^2*c^4*d + 3*a*b*c*d 
))/(2*a^3) + (3*x*(4*b^2*c^5 + 3*a*b*c^2))/(2*a^3) + (x^3*(40*b^2*c^3*d^2 
+ 3*a*b*d^2))/(2*a^3) + (b^2*d^5*x^6)/a^3 + (15*b^2*c^2*d^3*x^4)/a^3 + (6* 
b^2*c*d^4*x^5)/a^3)/(x*(3*a^2*c^2*d + 9*b^2*c^8*d + 12*a*b*c^5*d) + x^6*(8 
4*b^2*c^3*d^6 + 2*a*b*d^6) + x^2*(3*a^2*c*d^2 + 36*b^2*c^7*d^2 + 30*a*b*c^ 
4*d^2) + x^5*(126*b^2*c^4*d^5 + 12*a*b*c*d^5) + x^3*(a^2*d^3 + 84*b^2*c^6* 
d^3 + 40*a*b*c^3*d^3) + a^2*c^3 + b^2*c^9 + x^4*(126*b^2*c^5*d^4 + 30*a*b* 
c^2*d^4) + b^2*d^9*x^9 + 2*a*b*c^6 + 9*b^2*c*d^8*x^8 + 36*b^2*c^2*d^7*x^7) 
 - (3*b*log(c + d*x))/(a^4*d)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 2661, normalized size of antiderivative = 26.35 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*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*b*c**3 + 18*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*b*c**2*d*x + 18*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*b*c*d**2*x**2 + 6*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*b*d**3*x**3 + 12*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*b**2*c**6 + 72*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*b**2*c**5*d*x 
 + 180*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*b**2*c**4*d**2*x**2 + 2 
40*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*b**2*c**3*d**3*x**3 + 180*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**2*c**2*d**4*x**4 + 72*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*b**2*c*d**5*x**5 + 12*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/...