\(\int \frac {x^2 (c+d x)^3}{a+b x} \, dx\) [22]

Optimal result

Integrand size = 18, antiderivative size = 129 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=-\frac {a (b c-a d)^3 x}{b^5}+\frac {(b c-a d)^3 x^2}{2 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^3}{3 b^3}+\frac {d^2 (3 b c-a d) x^4}{4 b^2}+\frac {d^3 x^5}{5 b}+\frac {a^2 (b c-a d)^3 \log (a+b x)}{b^6} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.96 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=\frac {60 a b (-b c+a d)^3 x+30 b^2 (b c-a d)^3 x^2+20 b^3 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^3+15 b^4 d^2 (3 b c-a d) x^4+12 b^5 d^3 x^5+60 a^2 (b c-a d)^3 \log (a+b x)}{60 b^6} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 129, 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.

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 0.18 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.54

method	result	size
norman	\(\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{b^{5}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{2 b^{4}}+\frac {d^{3} x^{5}}{5 b}-\frac {d^{2} \left (a d -3 b c \right ) x^{4}}{4 b^{2}}+\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{3}}{3 b^{3}}-\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{6}}\)	\(199\)
default	\(\frac {\frac {1}{5} d^{3} x^{5} b^{4}-\frac {1}{4} a \,b^{3} d^{3} x^{4}+\frac {3}{4} b^{4} c \,d^{2} x^{4}+\frac {1}{3} a^{2} b^{2} d^{3} x^{3}-a \,b^{3} c \,d^{2} x^{3}+b^{4} c^{2} d \,x^{3}-\frac {1}{2} a^{3} b \,d^{3} x^{2}+\frac {3}{2} a^{2} b^{2} c \,d^{2} x^{2}-\frac {3}{2} a \,b^{3} c^{2} d \,x^{2}+\frac {1}{2} b^{4} c^{3} x^{2}+a^{4} d^{3} x -3 a^{3} b c \,d^{2} x +3 a^{2} b^{2} c^{2} d x -a \,b^{3} c^{3} x}{b^{5}}-\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{6}}\)	\(223\)
risch	\(\frac {d^{3} x^{5}}{5 b}-\frac {a \,d^{3} x^{4}}{4 b^{2}}+\frac {3 c \,d^{2} x^{4}}{4 b}+\frac {a^{2} d^{3} x^{3}}{3 b^{3}}-\frac {a c \,d^{2} x^{3}}{b^{2}}+\frac {c^{2} d \,x^{3}}{b}-\frac {a^{3} d^{3} x^{2}}{2 b^{4}}+\frac {3 a^{2} c \,d^{2} x^{2}}{2 b^{3}}-\frac {3 a \,c^{2} d \,x^{2}}{2 b^{2}}+\frac {c^{3} x^{2}}{2 b}+\frac {a^{4} d^{3} x}{b^{5}}-\frac {3 a^{3} c \,d^{2} x}{b^{4}}+\frac {3 a^{2} c^{2} d x}{b^{3}}-\frac {a \,c^{3} x}{b^{2}}-\frac {a^{5} \ln \left (b x +a \right ) d^{3}}{b^{6}}+\frac {3 a^{4} \ln \left (b x +a \right ) c \,d^{2}}{b^{5}}-\frac {3 a^{3} \ln \left (b x +a \right ) c^{2} d}{b^{4}}+\frac {a^{2} \ln \left (b x +a \right ) c^{3}}{b^{3}}\)	\(244\)
parallelrisch	\(-\frac {-12 d^{3} x^{5} b^{5}+15 x^{4} a \,b^{4} d^{3}-45 x^{4} b^{5} c \,d^{2}-20 x^{3} a^{2} b^{3} d^{3}+60 x^{3} a \,b^{4} c \,d^{2}-60 x^{3} b^{5} c^{2} d +30 x^{2} a^{3} b^{2} d^{3}-90 x^{2} a^{2} b^{3} c \,d^{2}+90 x^{2} a \,b^{4} c^{2} d -30 x^{2} b^{5} c^{3}+60 \ln \left (b x +a \right ) a^{5} d^{3}-180 \ln \left (b x +a \right ) a^{4} b c \,d^{2}+180 \ln \left (b x +a \right ) a^{3} b^{2} c^{2} d -60 \ln \left (b x +a \right ) a^{2} b^{3} c^{3}-60 x \,a^{4} b \,d^{3}+180 x \,a^{3} b^{2} c \,d^{2}-180 x \,a^{2} b^{3} c^{2} d +60 x a \,b^{4} c^{3}}{60 b^{6}}\)	\(245\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.67 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=\frac {12 \, b^{5} d^{3} x^{5} + 15 \, {\left (3 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{4} + 20 \, {\left (3 \, b^{5} c^{2} d - 3 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{3} + 30 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{2} - 60 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x + 60 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x + a\right )}{60 \, b^{6}} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.43 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=- \frac {a^{2} \left (a d - b c\right )^{3} \log {\left (a + b x \right )}}{b^{6}} + x^{4} \left (- \frac {a d^{3}}{4 b^{2}} + \frac {3 c d^{2}}{4 b}\right ) + x^{3} \left (\frac {a^{2} d^{3}}{3 b^{3}} - \frac {a c d^{2}}{b^{2}} + \frac {c^{2} d}{b}\right ) + x^{2} \left (- \frac {a^{3} d^{3}}{2 b^{4}} + \frac {3 a^{2} c d^{2}}{2 b^{3}} - \frac {3 a c^{2} d}{2 b^{2}} + \frac {c^{3}}{2 b}\right ) + x \left (\frac {a^{4} d^{3}}{b^{5}} - \frac {3 a^{3} c d^{2}}{b^{4}} + \frac {3 a^{2} c^{2} d}{b^{3}} - \frac {a c^{3}}{b^{2}}\right ) + \frac {d^{3} x^{5}}{5 b} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.66 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=\frac {12 \, b^{4} d^{3} x^{5} + 15 \, {\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{4} + 20 \, {\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{3} + 30 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2} - 60 \, {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{60 \, b^{5}} + \frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left (b x + a\right )}{b^{6}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.76 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=\frac {12 \, b^{4} d^{3} x^{5} + 45 \, b^{4} c d^{2} x^{4} - 15 \, a b^{3} d^{3} x^{4} + 60 \, b^{4} c^{2} d x^{3} - 60 \, a b^{3} c d^{2} x^{3} + 20 \, a^{2} b^{2} d^{3} x^{3} + 30 \, b^{4} c^{3} x^{2} - 90 \, a b^{3} c^{2} d x^{2} + 90 \, a^{2} b^{2} c d^{2} x^{2} - 30 \, a^{3} b d^{3} x^{2} - 60 \, a b^{3} c^{3} x + 180 \, a^{2} b^{2} c^{2} d x - 180 \, a^{3} b c d^{2} x + 60 \, a^{4} d^{3} x}{60 \, b^{5}} + \frac {{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{6}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.79 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=x^2\,\left (\frac {c^3}{2\,b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{2\,b}\right )-x^4\,\left (\frac {a\,d^3}{4\,b^2}-\frac {3\,c\,d^2}{4\,b}\right )+x^3\,\left (\frac {c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{3\,b}\right )-\frac {\ln \left (a+b\,x\right )\,\left (a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3\right )}{b^6}+\frac {d^3\,x^5}{5\,b}-\frac {a\,x\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{b} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.89 \[ \int \frac {x^2 (c+d x)^3}{a+b x} \, dx=\frac {-60 \,\mathrm {log}\left (b x +a \right ) a^{5} d^{3}+180 \,\mathrm {log}\left (b x +a \right ) a^{4} b c \,d^{2}-180 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} c^{2} d +60 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} c^{3}+60 a^{4} b \,d^{3} x -180 a^{3} b^{2} c \,d^{2} x -30 a^{3} b^{2} d^{3} x^{2}+180 a^{2} b^{3} c^{2} d x +90 a^{2} b^{3} c \,d^{2} x^{2}+20 a^{2} b^{3} d^{3} x^{3}-60 a \,b^{4} c^{3} x -90 a \,b^{4} c^{2} d \,x^{2}-60 a \,b^{4} c \,d^{2} x^{3}-15 a \,b^{4} d^{3} x^{4}+30 b^{5} c^{3} x^{2}+60 b^{5} c^{2} d \,x^{3}+45 b^{5} c \,d^{2} x^{4}+12 b^{5} d^{3} x^{5}}{60 b^{6}} \] Input:

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

Output:

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