\(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^5} \, dx\) [38]

Optimal result

Integrand size = 24, antiderivative size = 107 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {b x}{64 c^3 d^5}+\frac {x^2}{64 c^2 d^5}+\frac {\left (b^2-4 a c\right )^3}{512 c^4 d^5 (b+2 c x)^4}-\frac {3 \left (b^2-4 a c\right )^2}{256 c^4 d^5 (b+2 c x)^2}-\frac {3 \left (b^2-4 a c\right ) \log (b+2 c x)}{128 c^4 d^5} \] Output:

1/64*b*x/c^3/d^5+1/64*x^2/c^2/d^5+1/512*(-4*a*c+b^2)^3/c^4/d^5/(2*c*x+b)^4 
-3/256*(-4*a*c+b^2)^2/c^4/d^5/(2*c*x+b)^2-3/128*(-4*a*c+b^2)*ln(2*c*x+b)/c 
^4/d^5
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {8 b c x+8 c^2 x^2+\frac {\left (b^2-4 a c\right )^3}{(b+2 c x)^4}-\frac {6 \left (b^2-4 a c\right )^2}{(b+2 c x)^2}-12 \left (b^2-4 a c\right ) \log (b+2 c x)}{512 c^4 d^5} \] Input:

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^5,x]
 

Output:

(8*b*c*x + 8*c^2*x^2 + (b^2 - 4*a*c)^3/(b + 2*c*x)^4 - (6*(b^2 - 4*a*c)^2) 
/(b + 2*c*x)^2 - 12*(b^2 - 4*a*c)*Log[b + 2*c*x])/(512*c^4*d^5)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1107, 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.

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

\(\Big \downarrow \) 1107

\(\displaystyle \int \left (\frac {\left (4 a c-b^2\right )^3}{64 c^3 d^5 (b+2 c x)^5}+\frac {3 \left (4 a c-b^2\right )^2}{64 c^3 d^5 (b+2 c x)^3}+\frac {3 \left (4 a c-b^2\right )}{64 c^3 d^5 (b+2 c x)}+\frac {b}{64 c^3 d^5}+\frac {x}{32 c^2 d^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (b^2-4 a c\right )^3}{512 c^4 d^5 (b+2 c x)^4}-\frac {3 \left (b^2-4 a c\right )^2}{256 c^4 d^5 (b+2 c x)^2}-\frac {3 \left (b^2-4 a c\right ) \log (b+2 c x)}{128 c^4 d^5}+\frac {b x}{64 c^3 d^5}+\frac {x^2}{64 c^2 d^5}\)

Input:

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^5,x]
 

Output:

(b*x)/(64*c^3*d^5) + x^2/(64*c^2*d^5) + (b^2 - 4*a*c)^3/(512*c^4*d^5*(b + 
2*c*x)^4) - (3*(b^2 - 4*a*c)^2)/(256*c^4*d^5*(b + 2*c*x)^2) - (3*(b^2 - 4* 
a*c)*Log[b + 2*c*x])/(128*c^4*d^5)
 

Defintions of rubi rules used

rule 1107

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ 
Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; F 
reeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b*e, 0] && IGtQ[p, 0] &&  !(EqQ[ 
m, 3] && NeQ[p, 1])
 

rule 2009

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

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.13

method	result	size
default	\(\frac {\frac {c \,x^{2}+b x}{64 c^{3}}-\frac {64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{512 c^{4} \left (2 c x +b \right )^{4}}+\frac {\left (12 a c -3 b^{2}\right ) \ln \left (2 c x +b \right )}{128 c^{4}}-\frac {48 a^{2} c^{2}-24 c a \,b^{2}+3 b^{4}}{256 c^{4} \left (2 c x +b \right )^{2}}}{d^{5}}\)	\(121\)
risch	\(\frac {x^{2}}{64 c^{2} d^{5}}+\frac {b x}{64 c^{3} d^{5}}+\frac {-3 c \,d^{5} \left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) x^{2}+\left (-48 a^{2} b \,c^{2} d^{5}+24 a \,b^{3} d^{5} c -3 b^{5} d^{5}\right ) x -\frac {d^{5} \left (64 a^{3} c^{3}+48 a^{2} b^{2} c^{2}-36 a \,b^{4} c +5 b^{6}\right )}{8 c}}{64 d^{10} c^{3} \left (2 c x +b \right )^{4}}+\frac {3 \ln \left (2 c x +b \right ) a}{32 c^{3} d^{5}}-\frac {3 \ln \left (2 c x +b \right ) b^{2}}{128 c^{4} d^{5}}\)	\(175\)
norman	\(\frac {\frac {c^{2} x^{6}}{4 d}-\frac {64 a^{3} c^{3}+48 a^{2} b^{2} c^{2}-36 a \,b^{4} c +33 b^{6}}{512 c^{4} d}+\frac {3 b c \,x^{5}}{4 d}-\frac {3 \left (8 a^{2} c^{2}-4 c a \,b^{2}+13 b^{4}\right ) x^{2}}{32 c^{2} d}-\frac {5 b^{3} x^{3}}{4 d c}-\frac {b \left (24 a^{2} c^{2}-12 c a \,b^{2}+15 b^{4}\right ) x}{32 d \,c^{3}}}{d^{4} \left (2 c x +b \right )^{4}}+\frac {3 \left (4 a c -b^{2}\right ) \ln \left (2 c x +b \right )}{128 c^{4} d^{5}}\)	\(178\)
parallelrisch	\(\frac {-33 b^{6}+192 a \,b^{2} c^{3} x^{2}+1536 \ln \left (\frac {b}{2}+c x \right ) x^{3} a b \,c^{4}+36 a \,b^{4} c +48 \ln \left (\frac {b}{2}+c x \right ) a \,b^{4} c +192 x a \,b^{3} c^{2}-64 a^{3} c^{3}-640 b^{3} c^{3} x^{3}+128 x^{6} c^{6}-288 \ln \left (\frac {b}{2}+c x \right ) x^{2} b^{4} c^{2}-96 \ln \left (\frac {b}{2}+c x \right ) x \,b^{5} c +1152 \ln \left (\frac {b}{2}+c x \right ) x^{2} a \,b^{2} c^{3}+384 \ln \left (\frac {b}{2}+c x \right ) x a \,b^{3} c^{2}+384 x^{5} b \,c^{5}-624 c^{2} x^{2} b^{4}-240 x c \,b^{5}+768 \ln \left (\frac {b}{2}+c x \right ) x^{4} a \,c^{5}-192 \ln \left (\frac {b}{2}+c x \right ) x^{4} b^{2} c^{4}-384 \ln \left (\frac {b}{2}+c x \right ) x^{3} b^{3} c^{3}-384 a^{2} b \,c^{3} x -12 \ln \left (\frac {b}{2}+c x \right ) b^{6}-48 a^{2} b^{2} c^{2}-384 a^{2} c^{4} x^{2}}{512 c^{4} d^{5} \left (2 c x +b \right )^{4}}\)	\(311\)

Input:

int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^5,x,method=_RETURNVERBOSE)
 

Output:

1/d^5*(1/64/c^3*(c*x^2+b*x)-1/512/c^4*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4* 
c-b^6)/(2*c*x+b)^4+1/128*(12*a*c-3*b^2)/c^4*ln(2*c*x+b)-1/256*(48*a^2*c^2- 
24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (97) = 194\).

Time = 0.08 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.72 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {128 \, c^{6} x^{6} + 384 \, b c^{5} x^{5} + 448 \, b^{2} c^{4} x^{4} + 256 \, b^{3} c^{3} x^{3} - 5 \, b^{6} + 36 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 48 \, {\left (b^{4} c^{2} + 4 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} x^{2} - 16 \, {\left (b^{5} c - 12 \, a b^{3} c^{2} + 24 \, a^{2} b c^{3}\right )} x - 12 \, {\left (b^{6} - 4 \, a b^{4} c + 16 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 32 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 24 \, {\left (b^{4} c^{2} - 4 \, a b^{2} c^{3}\right )} x^{2} + 8 \, {\left (b^{5} c - 4 \, a b^{3} c^{2}\right )} x\right )} \log \left (2 \, c x + b\right )}{512 \, {\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} \] Input:

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^5,x, algorithm="fricas")
 

Output:

1/512*(128*c^6*x^6 + 384*b*c^5*x^5 + 448*b^2*c^4*x^4 + 256*b^3*c^3*x^3 - 5 
*b^6 + 36*a*b^4*c - 48*a^2*b^2*c^2 - 64*a^3*c^3 + 48*(b^4*c^2 + 4*a*b^2*c^ 
3 - 8*a^2*c^4)*x^2 - 16*(b^5*c - 12*a*b^3*c^2 + 24*a^2*b*c^3)*x - 12*(b^6 
- 4*a*b^4*c + 16*(b^2*c^4 - 4*a*c^5)*x^4 + 32*(b^3*c^3 - 4*a*b*c^4)*x^3 + 
24*(b^4*c^2 - 4*a*b^2*c^3)*x^2 + 8*(b^5*c - 4*a*b^3*c^2)*x)*log(2*c*x + b) 
)/(16*c^8*d^5*x^4 + 32*b*c^7*d^5*x^3 + 24*b^2*c^6*d^5*x^2 + 8*b^3*c^5*d^5* 
x + b^4*c^4*d^5)
 

Sympy [A] (verification not implemented)

Time = 1.53 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.95 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {b x}{64 c^{3} d^{5}} + \frac {- 64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 36 a b^{4} c - 5 b^{6} + x^{2} \left (- 384 a^{2} c^{4} + 192 a b^{2} c^{3} - 24 b^{4} c^{2}\right ) + x \left (- 384 a^{2} b c^{3} + 192 a b^{3} c^{2} - 24 b^{5} c\right )}{512 b^{4} c^{4} d^{5} + 4096 b^{3} c^{5} d^{5} x + 12288 b^{2} c^{6} d^{5} x^{2} + 16384 b c^{7} d^{5} x^{3} + 8192 c^{8} d^{5} x^{4}} + \frac {x^{2}}{64 c^{2} d^{5}} + \frac {3 \cdot \left (4 a c - b^{2}\right ) \log {\left (b + 2 c x \right )}}{128 c^{4} d^{5}} \] Input:

integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**5,x)
 

Output:

b*x/(64*c**3*d**5) + (-64*a**3*c**3 - 48*a**2*b**2*c**2 + 36*a*b**4*c - 5* 
b**6 + x**2*(-384*a**2*c**4 + 192*a*b**2*c**3 - 24*b**4*c**2) + x*(-384*a* 
*2*b*c**3 + 192*a*b**3*c**2 - 24*b**5*c))/(512*b**4*c**4*d**5 + 4096*b**3* 
c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c* 
*8*d**5*x**4) + x**2/(64*c**2*d**5) + 3*(4*a*c - b**2)*log(b + 2*c*x)/(128 
*c**4*d**5)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.81 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=-\frac {5 \, b^{6} - 36 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{512 \, {\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} + \frac {c x^{2} + b x}{64 \, c^{3} d^{5}} - \frac {3 \, {\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{128 \, c^{4} d^{5}} \] Input:

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^5,x, algorithm="maxima")
 

Output:

-1/512*(5*b^6 - 36*a*b^4*c + 48*a^2*b^2*c^2 + 64*a^3*c^3 + 24*(b^4*c^2 - 8 
*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 24*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x)/ 
(16*c^8*d^5*x^4 + 32*b*c^7*d^5*x^3 + 24*b^2*c^6*d^5*x^2 + 8*b^3*c^5*d^5*x 
+ b^4*c^4*d^5) + 1/64*(c*x^2 + b*x)/(c^3*d^5) - 3/128*(b^2 - 4*a*c)*log(2* 
c*x + b)/(c^4*d^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (97) = 194\).

Time = 0.12 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.45 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {3 \, {\left (b^{2} - 4 \, a c\right )} \log \left (\frac {1}{4 \, {\left (2 \, c d x + b d\right )}^{2} c^{2} d^{2}}\right )}{256 \, c^{4} d^{5}} - \frac {{\left (2 \, c d x + b d\right )}^{2} {\left (\frac {3 \, b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {12 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )}}{256 \, c^{4} d^{7}} + \frac {\frac {b^{6} c^{8} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac {12 \, a b^{4} c^{9} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} + \frac {48 \, a^{2} b^{2} c^{10} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac {64 \, a^{3} c^{11} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac {6 \, b^{4} c^{8} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {48 \, a b^{2} c^{9} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {96 \, a^{2} c^{10} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}}}{512 \, c^{12} d^{18}} \] Input:

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^5,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

3/256*(b^2 - 4*a*c)*log(1/4/((2*c*d*x + b*d)^2*c^2*d^2))/(c^4*d^5) - 1/256 
*(2*c*d*x + b*d)^2*(3*b^2*d^2/(2*c*d*x + b*d)^2 - 12*a*c*d^2/(2*c*d*x + b* 
d)^2 - 1)/(c^4*d^7) + 1/512*(b^6*c^8*d^17/(2*c*d*x + b*d)^4 - 12*a*b^4*c^9 
*d^17/(2*c*d*x + b*d)^4 + 48*a^2*b^2*c^10*d^17/(2*c*d*x + b*d)^4 - 64*a^3* 
c^11*d^17/(2*c*d*x + b*d)^4 - 6*b^4*c^8*d^15/(2*c*d*x + b*d)^2 + 48*a*b^2* 
c^9*d^15/(2*c*d*x + b*d)^2 - 96*a^2*c^10*d^15/(2*c*d*x + b*d)^2)/(c^12*d^1 
8)
 

Mupad [B] (verification not implemented)

Time = 5.21 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {x^2}{64\,c^2\,d^5}-\frac {\frac {64\,a^3\,c^3+48\,a^2\,b^2\,c^2-36\,a\,b^4\,c+5\,b^6}{8\,c}+x^2\,\left (48\,a^2\,c^3-24\,a\,b^2\,c^2+3\,b^4\,c\right )+x\,\left (48\,a^2\,b\,c^2-24\,a\,b^3\,c+3\,b^5\right )}{64\,b^4\,c^3\,d^5+512\,b^3\,c^4\,d^5\,x+1536\,b^2\,c^5\,d^5\,x^2+2048\,b\,c^6\,d^5\,x^3+1024\,c^7\,d^5\,x^4}+\frac {b\,x}{64\,c^3\,d^5}+\frac {\ln \left (b+2\,c\,x\right )\,\left (12\,a\,c-3\,b^2\right )}{128\,c^4\,d^5} \] Input:

int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^5,x)
 

Output:

x^2/(64*c^2*d^5) - ((5*b^6 + 64*a^3*c^3 + 48*a^2*b^2*c^2 - 36*a*b^4*c)/(8* 
c) + x^2*(3*b^4*c + 48*a^2*c^3 - 24*a*b^2*c^2) + x*(3*b^5 + 48*a^2*b*c^2 - 
 24*a*b^3*c))/(64*b^4*c^3*d^5 + 1024*c^7*d^5*x^4 + 512*b^3*c^4*d^5*x + 204 
8*b*c^6*d^5*x^3 + 1536*b^2*c^5*d^5*x^2) + (b*x)/(64*c^3*d^5) + (log(b + 2* 
c*x)*(12*a*c - 3*b^2))/(128*c^4*d^5)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.11 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^5} \, dx=\frac {48 \,\mathrm {log}\left (2 c x +b \right ) a \,b^{4} c +384 \,\mathrm {log}\left (2 c x +b \right ) a \,b^{3} c^{2} x +1152 \,\mathrm {log}\left (2 c x +b \right ) a \,b^{2} c^{3} x^{2}+1536 \,\mathrm {log}\left (2 c x +b \right ) a b \,c^{4} x^{3}+768 \,\mathrm {log}\left (2 c x +b \right ) a \,c^{5} x^{4}-12 \,\mathrm {log}\left (2 c x +b \right ) b^{6}-96 \,\mathrm {log}\left (2 c x +b \right ) b^{5} c x -288 \,\mathrm {log}\left (2 c x +b \right ) b^{4} c^{2} x^{2}-384 \,\mathrm {log}\left (2 c x +b \right ) b^{3} c^{3} x^{3}-192 \,\mathrm {log}\left (2 c x +b \right ) b^{2} c^{4} x^{4}-64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}-384 a^{2} b \,c^{3} x -384 a^{2} c^{4} x^{2}+36 a \,b^{4} c +192 a \,b^{3} c^{2} x +192 a \,b^{2} c^{3} x^{2}-13 b^{6}-80 b^{5} c x -144 b^{4} c^{2} x^{2}+320 b^{2} c^{4} x^{4}+384 b \,c^{5} x^{5}+128 c^{6} x^{6}}{512 c^{4} d^{5} \left (16 c^{4} x^{4}+32 b \,c^{3} x^{3}+24 b^{2} c^{2} x^{2}+8 b^{3} c x +b^{4}\right )} \] Input:

int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^5,x)
 

Output:

(48*log(b + 2*c*x)*a*b**4*c + 384*log(b + 2*c*x)*a*b**3*c**2*x + 1152*log( 
b + 2*c*x)*a*b**2*c**3*x**2 + 1536*log(b + 2*c*x)*a*b*c**4*x**3 + 768*log( 
b + 2*c*x)*a*c**5*x**4 - 12*log(b + 2*c*x)*b**6 - 96*log(b + 2*c*x)*b**5*c 
*x - 288*log(b + 2*c*x)*b**4*c**2*x**2 - 384*log(b + 2*c*x)*b**3*c**3*x**3 
 - 192*log(b + 2*c*x)*b**2*c**4*x**4 - 64*a**3*c**3 - 48*a**2*b**2*c**2 - 
384*a**2*b*c**3*x - 384*a**2*c**4*x**2 + 36*a*b**4*c + 192*a*b**3*c**2*x + 
 192*a*b**2*c**3*x**2 - 13*b**6 - 80*b**5*c*x - 144*b**4*c**2*x**2 + 320*b 
**2*c**4*x**4 + 384*b*c**5*x**5 + 128*c**6*x**6)/(512*c**4*d**5*(b**4 + 8* 
b**3*c*x + 24*b**2*c**2*x**2 + 32*b*c**3*x**3 + 16*c**4*x**4))