\(\int \frac {x^3 (c+d x)^3}{a+b x} \, dx\) [21]

Optimal result

Integrand size = 18, antiderivative size = 152 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {a^2 (b c-a d)^3 x}{b^6}-\frac {a (b c-a d)^3 x^2}{2 b^5}+\frac {(b c-a d)^3 x^3}{3 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^6}{6 b}-\frac {a^3 (b c-a d)^3 \log (a+b x)}{b^7} \] Output:

a^2*(-a*d+b*c)^3*x/b^6-1/2*a*(-a*d+b*c)^3*x^2/b^5+1/3*(-a*d+b*c)^3*x^3/b^4 
+1/4*d*(a^2*d^2-3*a*b*c*d+3*b^2*c^2)*x^4/b^3+1/5*d^2*(-a*d+3*b*c)*x^5/b^2+ 
1/6*d^3*x^6/b-a^3*(-a*d+b*c)^3*ln(b*x+a)/b^7
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.95 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {-60 a^2 b (-b c+a d)^3 x+30 a b^2 (-b c+a d)^3 x^2+20 b^3 (b c-a d)^3 x^3+15 b^4 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^4+12 b^5 d^2 (3 b c-a d) x^5+10 b^6 d^3 x^6+60 a^3 (-b c+a d)^3 \log (a+b x)}{60 b^7} \] Input:

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

Output:

(-60*a^2*b*(-(b*c) + a*d)^3*x + 30*a*b^2*(-(b*c) + a*d)^3*x^2 + 20*b^3*(b* 
c - a*d)^3*x^3 + 15*b^4*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^4 + 12*b^5*d 
^2*(3*b*c - a*d)*x^5 + 10*b^6*d^3*x^6 + 60*a^3*(-(b*c) + a*d)^3*Log[a + b* 
x])/(60*b^7)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {99, 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.

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

\(\Big \downarrow \) 99

\(\displaystyle \int \left (\frac {a^3 (a d-b c)^3}{b^6 (a+b x)}-\frac {a^2 (a d-b c)^3}{b^6}+\frac {d x^3 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{b^3}+\frac {a x (a d-b c)^3}{b^5}+\frac {x^2 (b c-a d)^3}{b^4}+\frac {d^2 x^4 (3 b c-a d)}{b^2}+\frac {d^3 x^5}{b}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^3 (b c-a d)^3 \log (a+b x)}{b^7}+\frac {a^2 x (b c-a d)^3}{b^6}+\frac {d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{4 b^3}-\frac {a x^2 (b c-a d)^3}{2 b^5}+\frac {x^3 (b c-a d)^3}{3 b^4}+\frac {d^2 x^5 (3 b c-a d)}{5 b^2}+\frac {d^3 x^6}{6 b}\)

Input:

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

Output:

(a^2*(b*c - a*d)^3*x)/b^6 - (a*(b*c - a*d)^3*x^2)/(2*b^5) + ((b*c - a*d)^3 
*x^3)/(3*b^4) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^4)/(4*b^3) + (d^2*( 
3*b*c - a*d)*x^5)/(5*b^2) + (d^3*x^6)/(6*b) - (a^3*(b*c - a*d)^3*Log[a + b 
*x])/b^7
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 2009

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.62

method	result	size
norman	\(-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{3}}{3 b^{4}}+\frac {d^{3} x^{6}}{6 b}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{2 b^{5}}-\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{b^{6}}-\frac {d^{2} \left (a d -3 b c \right ) x^{5}}{5 b^{2}}+\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{4}}{4 b^{3}}+\frac {a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{7}}\)	\(246\)
risch	\(\frac {d^{3} x^{6}}{6 b}-\frac {x^{5} a \,d^{3}}{5 b^{2}}+\frac {3 x^{5} d^{2} c}{5 b}-\frac {3 x^{4} a c \,d^{2}}{4 b^{2}}+\frac {3 x^{4} c^{2} d}{4 b}+\frac {x^{4} a^{2} d^{3}}{4 b^{3}}-\frac {x^{3} a^{3} d^{3}}{3 b^{4}}+\frac {x^{3} a^{2} c \,d^{2}}{b^{3}}-\frac {x^{3} a \,c^{2} d}{b^{2}}+\frac {x^{3} c^{3}}{3 b}-\frac {3 x^{2} a^{3} c \,d^{2}}{2 b^{4}}+\frac {3 x^{2} a^{2} c^{2} d}{2 b^{3}}-\frac {x^{2} a \,c^{3}}{2 b^{2}}+\frac {x^{2} a^{4} d^{3}}{2 b^{5}}-\frac {a^{5} d^{3} x}{b^{6}}+\frac {3 a^{4} c \,d^{2} x}{b^{5}}-\frac {3 a^{3} c^{2} d x}{b^{4}}+\frac {a^{2} c^{3} x}{b^{3}}+\frac {a^{6} \ln \left (b x +a \right ) d^{3}}{b^{7}}-\frac {3 a^{5} \ln \left (b x +a \right ) c \,d^{2}}{b^{6}}+\frac {3 a^{4} \ln \left (b x +a \right ) c^{2} d}{b^{5}}-\frac {a^{3} \ln \left (b x +a \right ) c^{3}}{b^{4}}\)	\(302\)
parallelrisch	\(\frac {10 x^{6} d^{3} b^{6}-12 x^{5} a \,b^{5} d^{3}+36 x^{5} b^{6} c \,d^{2}+15 x^{4} a^{2} b^{4} d^{3}-45 x^{4} a \,b^{5} c \,d^{2}+45 x^{4} b^{6} c^{2} d -20 x^{3} a^{3} b^{3} d^{3}+60 x^{3} a^{2} b^{4} c \,d^{2}-60 x^{3} a \,b^{5} c^{2} d +20 x^{3} b^{6} c^{3}+30 x^{2} a^{4} b^{2} d^{3}-90 x^{2} a^{3} b^{3} c \,d^{2}+90 x^{2} a^{2} b^{4} c^{2} d -30 x^{2} a \,b^{5} c^{3}+60 \ln \left (b x +a \right ) a^{6} d^{3}-180 \ln \left (b x +a \right ) a^{5} b c \,d^{2}+180 \ln \left (b x +a \right ) a^{4} b^{2} c^{2} d -60 \ln \left (b x +a \right ) a^{3} b^{3} c^{3}-60 x \,a^{5} b \,d^{3}+180 x \,a^{4} b^{2} c \,d^{2}-180 x \,a^{3} b^{3} c^{2} d +60 x \,a^{2} b^{4} c^{3}}{60 b^{7}}\)	\(303\)
default	\(-\frac {-\frac {d^{3} x^{6} b^{5}}{6}+\frac {\left (\left (a d -b c \right ) b^{4} d^{2}-2 d^{2} b^{5} c \right ) x^{5}}{5}+\frac {\left (2 \left (a d -b c \right ) b^{4} c d -d b \left (a^{2} b^{2} d^{2}-d \,b^{3} c a +b^{4} c^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (a d -b c \right ) \left (a^{2} b^{2} d^{2}-d \,b^{3} c a +b^{4} c^{2}\right )-d b \left (a^{2} b^{2} c d -a \,b^{3} c^{2}\right )\right ) x^{3}}{3}+\frac {\left (\left (a d -b c \right ) \left (a^{2} b^{2} c d -a \,b^{3} c^{2}\right )-d b \left (d^{2} a^{4}-2 a^{3} b c d +a^{2} b^{2} c^{2}\right )\right ) x^{2}}{2}+\left (a d -b c \right ) \left (d^{2} a^{4}-2 a^{3} b c d +a^{2} b^{2} c^{2}\right ) x}{b^{6}}+\frac {a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{7}}\)	\(313\)

Input:

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

Output:

-1/3/b^4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*x^3+1/6*d^3*x^6/b+1 
/2*a/b^5*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*x^2-a^2*(a^3*d^3-3* 
a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^6*x-1/5/b^2*d^2*(a*d-3*b*c)*x^5+1/4*d 
*(a^2*d^2-3*a*b*c*d+3*b^2*c^2)*x^4/b^3+a^3*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2* 
c^2*d-b^3*c^3)/b^7*ln(b*x+a)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.76 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {10 \, b^{6} d^{3} x^{6} + 12 \, {\left (3 \, b^{6} c d^{2} - a b^{5} d^{3}\right )} x^{5} + 15 \, {\left (3 \, b^{6} c^{2} d - 3 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x^{4} + 20 \, {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} - 30 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 60 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x - 60 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \log \left (b x + a\right )}{60 \, b^{7}} \] Input:

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

Output:

1/60*(10*b^6*d^3*x^6 + 12*(3*b^6*c*d^2 - a*b^5*d^3)*x^5 + 15*(3*b^6*c^2*d 
- 3*a*b^5*c*d^2 + a^2*b^4*d^3)*x^4 + 20*(b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b 
^4*c*d^2 - a^3*b^3*d^3)*x^3 - 30*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3* 
c*d^2 - a^4*b^2*d^3)*x^2 + 60*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c 
*d^2 - a^5*b*d^3)*x - 60*(a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - 
a^6*d^3)*log(b*x + a))/b^7
 

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.60 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {a^{3} \left (a d - b c\right )^{3} \log {\left (a + b x \right )}}{b^{7}} + x^{5} \left (- \frac {a d^{3}}{5 b^{2}} + \frac {3 c d^{2}}{5 b}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4 b^{3}} - \frac {3 a c d^{2}}{4 b^{2}} + \frac {3 c^{2} d}{4 b}\right ) + x^{3} \left (- \frac {a^{3} d^{3}}{3 b^{4}} + \frac {a^{2} c d^{2}}{b^{3}} - \frac {a c^{2} d}{b^{2}} + \frac {c^{3}}{3 b}\right ) + x^{2} \left (\frac {a^{4} d^{3}}{2 b^{5}} - \frac {3 a^{3} c d^{2}}{2 b^{4}} + \frac {3 a^{2} c^{2} d}{2 b^{3}} - \frac {a c^{3}}{2 b^{2}}\right ) + x \left (- \frac {a^{5} d^{3}}{b^{6}} + \frac {3 a^{4} c d^{2}}{b^{5}} - \frac {3 a^{3} c^{2} d}{b^{4}} + \frac {a^{2} c^{3}}{b^{3}}\right ) + \frac {d^{3} x^{6}}{6 b} \] Input:

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

Output:

a**3*(a*d - b*c)**3*log(a + b*x)/b**7 + x**5*(-a*d**3/(5*b**2) + 3*c*d**2/ 
(5*b)) + x**4*(a**2*d**3/(4*b**3) - 3*a*c*d**2/(4*b**2) + 3*c**2*d/(4*b)) 
+ x**3*(-a**3*d**3/(3*b**4) + a**2*c*d**2/b**3 - a*c**2*d/b**2 + c**3/(3*b 
)) + x**2*(a**4*d**3/(2*b**5) - 3*a**3*c*d**2/(2*b**4) + 3*a**2*c**2*d/(2* 
b**3) - a*c**3/(2*b**2)) + x*(-a**5*d**3/b**6 + 3*a**4*c*d**2/b**5 - 3*a** 
3*c**2*d/b**4 + a**2*c**3/b**3) + d**3*x**6/(6*b)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.75 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {10 \, b^{5} d^{3} x^{6} + 12 \, {\left (3 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{5} + 15 \, {\left (3 \, b^{5} c^{2} d - 3 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{4} + 20 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{3} - 30 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{2} + 60 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x}{60 \, b^{6}} - \frac {{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \log \left (b x + a\right )}{b^{7}} \] Input:

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

Output:

1/60*(10*b^5*d^3*x^6 + 12*(3*b^5*c*d^2 - a*b^4*d^3)*x^5 + 15*(3*b^5*c^2*d 
- 3*a*b^4*c*d^2 + a^2*b^3*d^3)*x^4 + 20*(b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b 
^3*c*d^2 - a^3*b^2*d^3)*x^3 - 30*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2* 
c*d^2 - a^4*b*d^3)*x^2 + 60*(a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 
 - a^5*d^3)*x)/b^6 - (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6* 
d^3)*log(b*x + a)/b^7
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (142) = 284\).

Time = 0.12 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.88 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {10 \, b^{5} d^{3} x^{6} + 36 \, b^{5} c d^{2} x^{5} - 12 \, a b^{4} d^{3} x^{5} + 45 \, b^{5} c^{2} d x^{4} - 45 \, a b^{4} c d^{2} x^{4} + 15 \, a^{2} b^{3} d^{3} x^{4} + 20 \, b^{5} c^{3} x^{3} - 60 \, a b^{4} c^{2} d x^{3} + 60 \, a^{2} b^{3} c d^{2} x^{3} - 20 \, a^{3} b^{2} d^{3} x^{3} - 30 \, a b^{4} c^{3} x^{2} + 90 \, a^{2} b^{3} c^{2} d x^{2} - 90 \, a^{3} b^{2} c d^{2} x^{2} + 30 \, a^{4} b d^{3} x^{2} + 60 \, a^{2} b^{3} c^{3} x - 180 \, a^{3} b^{2} c^{2} d x + 180 \, a^{4} b c d^{2} x - 60 \, a^{5} d^{3} x}{60 \, b^{6}} - \frac {{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} \] Input:

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

Output:

1/60*(10*b^5*d^3*x^6 + 36*b^5*c*d^2*x^5 - 12*a*b^4*d^3*x^5 + 45*b^5*c^2*d* 
x^4 - 45*a*b^4*c*d^2*x^4 + 15*a^2*b^3*d^3*x^4 + 20*b^5*c^3*x^3 - 60*a*b^4* 
c^2*d*x^3 + 60*a^2*b^3*c*d^2*x^3 - 20*a^3*b^2*d^3*x^3 - 30*a*b^4*c^3*x^2 + 
 90*a^2*b^3*c^2*d*x^2 - 90*a^3*b^2*c*d^2*x^2 + 30*a^4*b*d^3*x^2 + 60*a^2*b 
^3*c^3*x - 180*a^3*b^2*c^2*d*x + 180*a^4*b*c*d^2*x - 60*a^5*d^3*x)/b^6 - ( 
a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*log(abs(b*x + a)) 
/b^7
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.89 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=x^3\,\left (\frac {c^3}{3\,b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{3\,b}\right )-x^5\,\left (\frac {a\,d^3}{5\,b^2}-\frac {3\,c\,d^2}{5\,b}\right )+x^4\,\left (\frac {3\,c^2\,d}{4\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{4\,b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3\right )}{b^7}+\frac {d^3\,x^6}{6\,b}-\frac {a\,x^2\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{2\,b}+\frac {a^2\,x\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{b^2} \] Input:

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

Output:

x^3*(c^3/(3*b) - (a*((3*c^2*d)/b + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/b))/(3* 
b)) - x^5*((a*d^3)/(5*b^2) - (3*c*d^2)/(5*b)) + x^4*((3*c^2*d)/(4*b) + (a* 
((a*d^3)/b^2 - (3*c*d^2)/b))/(4*b)) + (log(a + b*x)*(a^6*d^3 - a^3*b^3*c^3 
 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2))/b^7 + (d^3*x^6)/(6*b) - (a*x^2*(c^3/b 
 - (a*((3*c^2*d)/b + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/b))/b))/(2*b) + (a^2* 
x*(c^3/b - (a*((3*c^2*d)/b + (a*((a*d^3)/b^2 - (3*c*d^2)/b))/b))/b))/b^2
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.99 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {60 \,\mathrm {log}\left (b x +a \right ) a^{6} d^{3}-180 \,\mathrm {log}\left (b x +a \right ) a^{5} b c \,d^{2}+180 \,\mathrm {log}\left (b x +a \right ) a^{4} b^{2} c^{2} d -60 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{3} c^{3}-60 a^{5} b \,d^{3} x +180 a^{4} b^{2} c \,d^{2} x +30 a^{4} b^{2} d^{3} x^{2}-180 a^{3} b^{3} c^{2} d x -90 a^{3} b^{3} c \,d^{2} x^{2}-20 a^{3} b^{3} d^{3} x^{3}+60 a^{2} b^{4} c^{3} x +90 a^{2} b^{4} c^{2} d \,x^{2}+60 a^{2} b^{4} c \,d^{2} x^{3}+15 a^{2} b^{4} d^{3} x^{4}-30 a \,b^{5} c^{3} x^{2}-60 a \,b^{5} c^{2} d \,x^{3}-45 a \,b^{5} c \,d^{2} x^{4}-12 a \,b^{5} d^{3} x^{5}+20 b^{6} c^{3} x^{3}+45 b^{6} c^{2} d \,x^{4}+36 b^{6} c \,d^{2} x^{5}+10 b^{6} d^{3} x^{6}}{60 b^{7}} \] Input:

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

Output:

(60*log(a + b*x)*a**6*d**3 - 180*log(a + b*x)*a**5*b*c*d**2 + 180*log(a + 
b*x)*a**4*b**2*c**2*d - 60*log(a + b*x)*a**3*b**3*c**3 - 60*a**5*b*d**3*x 
+ 180*a**4*b**2*c*d**2*x + 30*a**4*b**2*d**3*x**2 - 180*a**3*b**3*c**2*d*x 
 - 90*a**3*b**3*c*d**2*x**2 - 20*a**3*b**3*d**3*x**3 + 60*a**2*b**4*c**3*x 
 + 90*a**2*b**4*c**2*d*x**2 + 60*a**2*b**4*c*d**2*x**3 + 15*a**2*b**4*d**3 
*x**4 - 30*a*b**5*c**3*x**2 - 60*a*b**5*c**2*d*x**3 - 45*a*b**5*c*d**2*x** 
4 - 12*a*b**5*d**3*x**5 + 20*b**6*c**3*x**3 + 45*b**6*c**2*d*x**4 + 36*b** 
6*c*d**2*x**5 + 10*b**6*d**3*x**6)/(60*b**7)