\(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^7} \, dx\) [1148]

Optimal result

Integrand size = 24, antiderivative size = 100 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx=\frac {\left (b^2-4 a c\right )^3}{768 c^4 d^7 (b+2 c x)^6}-\frac {3 \left (b^2-4 a c\right )^2}{512 c^4 d^7 (b+2 c x)^4}+\frac {3 \left (b^2-4 a c\right )}{256 c^4 d^7 (b+2 c x)^2}+\frac {\log (b+2 c x)}{128 c^4 d^7} \]

[Out]

1/768*(-4*a*c+b^2)^3/c^4/d^7/(2*c*x+b)^6-3/512*(-4*a*c+b^2)^2/c^4/d^7/(2*c*x+b)^4+3/256*(-4*a*c+b^2)/c^4/d^7/(
2*c*x+b)^2+1/128*ln(2*c*x+b)/c^4/d^7

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx=\frac {\left (b^2-4 a c\right )^3}{768 c^4 d^7 (b+2 c x)^6}-\frac {3 \left (b^2-4 a c\right )^2}{512 c^4 d^7 (b+2 c x)^4}+\frac {3 \left (b^2-4 a c\right )}{256 c^4 d^7 (b+2 c x)^2}+\frac {\log (b+2 c x)}{128 c^4 d^7} \]

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^7,x]

[Out]

(b^2 - 4*a*c)^3/(768*c^4*d^7*(b + 2*c*x)^6) - (3*(b^2 - 4*a*c)^2)/(512*c^4*d^7*(b + 2*c*x)^4) + (3*(b^2 - 4*a*
c))/(256*c^4*d^7*(b + 2*c*x)^2) + Log[b + 2*c*x]/(128*c^4*d^7)

Rule 697

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] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 d^7 (b+2 c x)^7}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^7 (b+2 c x)^5}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^7 (b+2 c x)^3}+\frac {1}{64 c^3 d^7 (b+2 c x)}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^3}{768 c^4 d^7 (b+2 c x)^6}-\frac {3 \left (b^2-4 a c\right )^2}{512 c^4 d^7 (b+2 c x)^4}+\frac {3 \left (b^2-4 a c\right )}{256 c^4 d^7 (b+2 c x)^2}+\frac {\log (b+2 c x)}{128 c^4 d^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx=\frac {\frac {2 \left (b^2-4 a c\right )^3}{(b+2 c x)^6}-\frac {9 \left (b^2-4 a c\right )^2}{(b+2 c x)^4}+\frac {18 \left (b^2-4 a c\right )}{(b+2 c x)^2}+12 \log (b+2 c x)}{1536 c^4 d^7} \]

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^7,x]

[Out]

((2*(b^2 - 4*a*c)^3)/(b + 2*c*x)^6 - (9*(b^2 - 4*a*c)^2)/(b + 2*c*x)^4 + (18*(b^2 - 4*a*c))/(b + 2*c*x)^2 + 12
*Log[b + 2*c*x])/(1536*c^4*d^7)

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {-\frac {48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}}{512 c^{4} \left (2 c x +b \right )^{4}}+\frac {\ln \left (2 c x +b \right )}{128 c^{4}}-\frac {64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{768 c^{4} \left (2 c x +b \right )^{6}}-\frac {12 a c -3 b^{2}}{256 c^{4} \left (2 c x +b \right )^{2}}}{d^{7}}\)	\(120\)
risch	\(\frac {\left (-\frac {3 a c}{4}+\frac {3 b^{2}}{16}\right ) x^{4}-\frac {3 b \left (4 a c -b^{2}\right ) x^{3}}{8 c}-\frac {3 \left (16 a^{2} c^{2}+40 a \,b^{2} c -11 b^{4}\right ) x^{2}}{128 c^{2}}-\frac {3 b \left (16 a^{2} c^{2}+8 a \,b^{2} c -3 b^{4}\right ) x}{128 c^{3}}-\frac {128 c^{3} a^{3}+48 a^{2} b^{2} c^{2}+24 a \,b^{4} c -11 b^{6}}{1536 c^{4}}}{d^{7} \left (2 c x +b \right )^{6}}+\frac {\ln \left (2 c x +b \right )}{128 c^{4} d^{7}}\)	\(157\)
norman	\(\frac {-\frac {128 a^{3} c^{5}+48 b^{2} c^{4} a^{2}+24 a \,b^{4} c^{3}-11 c^{2} b^{6}}{1536 c^{6} d}-\frac {3 \left (4 c^{3} a -b^{2} c^{2}\right ) x^{4}}{16 c^{2} d}-\frac {3 \left (16 a^{2} c^{4}+40 b^{2} c^{3} a -11 b^{4} c^{2}\right ) x^{2}}{128 c^{4} d}-\frac {b \left (12 c^{3} a -3 b^{2} c^{2}\right ) x^{3}}{8 c^{3} d}-\frac {3 b \left (16 a^{2} c^{4}+8 b^{2} c^{3} a -3 b^{4} c^{2}\right ) x}{128 c^{5} d}}{d^{6} \left (2 c x +b \right )^{6}}+\frac {\ln \left (2 c x +b \right )}{128 c^{4} d^{7}}\)	\(201\)
parallelrisch	\(\frac {11 c^{2} b^{6}+396 x^{2} b^{4} c^{4}-288 x a \,b^{3} c^{4}-48 b^{2} c^{4} a^{2}-128 a^{3} c^{5}-24 a \,b^{4} c^{3}+576 x^{3} b^{3} c^{5}-1152 x^{4} a \,c^{7}+288 x^{4} b^{2} c^{6}+768 \ln \left (\frac {b}{2}+c x \right ) x^{6} c^{8}+12 \ln \left (\frac {b}{2}+c x \right ) b^{6} c^{2}+108 x \,b^{5} c^{3}-576 x^{2} a^{2} c^{6}-576 x \,a^{2} b \,c^{5}-1440 x^{2} a \,b^{2} c^{5}-2304 x^{3} a b \,c^{6}+2304 \ln \left (\frac {b}{2}+c x \right ) x^{5} b \,c^{7}+2880 \ln \left (\frac {b}{2}+c x \right ) x^{4} b^{2} c^{6}+1920 \ln \left (\frac {b}{2}+c x \right ) x^{3} b^{3} c^{5}+720 \ln \left (\frac {b}{2}+c x \right ) x^{2} b^{4} c^{4}+144 \ln \left (\frac {b}{2}+c x \right ) x \,b^{5} c^{3}}{1536 c^{6} d^{7} \left (2 c x +b \right )^{6}}\)	\(281\)

[In]

int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^7,x,method=_RETURNVERBOSE)

[Out]

1/d^7*(-1/512*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^4+1/128/c^4*ln(2*c*x+b)-1/768*(64*a^3*c^3-48*a^2*b^2
*c^2+12*a*b^4*c-b^6)/c^4/(2*c*x+b)^6-1/256*(12*a*c-3*b^2)/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. 292 vs. \(2 (92) = 184\).

Time = 0.26 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.92 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx=\frac {11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \, {\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x + 12 \, {\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \log \left (2 \, c x + b\right )}{1536 \, {\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^7,x, algorithm="fricas")

[Out]

1/1536*(11*b^6 - 24*a*b^4*c - 48*a^2*b^2*c^2 - 128*a^3*c^3 + 288*(b^2*c^4 - 4*a*c^5)*x^4 + 576*(b^3*c^3 - 4*a*
b*c^4)*x^3 + 36*(11*b^4*c^2 - 40*a*b^2*c^3 - 16*a^2*c^4)*x^2 + 36*(3*b^5*c - 8*a*b^3*c^2 - 16*a^2*b*c^3)*x + 1
2*(64*c^6*x^6 + 192*b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^6)*log(2*c
*x + b))/(64*c^10*d^7*x^6 + 192*b*c^9*d^7*x^5 + 240*b^2*c^8*d^7*x^4 + 160*b^3*c^7*d^7*x^3 + 60*b^4*c^6*d^7*x^2
+ 12*b^5*c^5*d^7*x + b^6*c^4*d^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (100) = 200\).

Time = 4.52 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.45 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx=\frac {- 128 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} - 24 a b^{4} c + 11 b^{6} + x^{4} \left (- 1152 a c^{5} + 288 b^{2} c^{4}\right ) + x^{3} \left (- 2304 a b c^{4} + 576 b^{3} c^{3}\right ) + x^{2} \left (- 576 a^{2} c^{4} - 1440 a b^{2} c^{3} + 396 b^{4} c^{2}\right ) + x \left (- 576 a^{2} b c^{3} - 288 a b^{3} c^{2} + 108 b^{5} c\right )}{1536 b^{6} c^{4} d^{7} + 18432 b^{5} c^{5} d^{7} x + 92160 b^{4} c^{6} d^{7} x^{2} + 245760 b^{3} c^{7} d^{7} x^{3} + 368640 b^{2} c^{8} d^{7} x^{4} + 294912 b c^{9} d^{7} x^{5} + 98304 c^{10} d^{7} x^{6}} + \frac {\log {\left (b + 2 c x \right )}}{128 c^{4} d^{7}} \]

[In]

integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**7,x)

[Out]

(-128*a**3*c**3 - 48*a**2*b**2*c**2 - 24*a*b**4*c + 11*b**6 + x**4*(-1152*a*c**5 + 288*b**2*c**4) + x**3*(-230
4*a*b*c**4 + 576*b**3*c**3) + x**2*(-576*a**2*c**4 - 1440*a*b**2*c**3 + 396*b**4*c**2) + x*(-576*a**2*b*c**3 -
288*a*b**3*c**2 + 108*b**5*c))/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 24
5760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + log
(b + 2*c*x)/(128*c**4*d**7)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (92) = 184\).

Time = 0.21 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.38 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx=\frac {11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \, {\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{1536 \, {\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\right )}} + \frac {\log \left (2 \, c x + b\right )}{128 \, c^{4} d^{7}} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^7,x, algorithm="maxima")

[Out]

1/1536*(11*b^6 - 24*a*b^4*c - 48*a^2*b^2*c^2 - 128*a^3*c^3 + 288*(b^2*c^4 - 4*a*c^5)*x^4 + 576*(b^3*c^3 - 4*a*
b*c^4)*x^3 + 36*(11*b^4*c^2 - 40*a*b^2*c^3 - 16*a^2*c^4)*x^2 + 36*(3*b^5*c - 8*a*b^3*c^2 - 16*a^2*b*c^3)*x)/(6
4*c^10*d^7*x^6 + 192*b*c^9*d^7*x^5 + 240*b^2*c^8*d^7*x^4 + 160*b^3*c^7*d^7*x^3 + 60*b^4*c^6*d^7*x^2 + 12*b^5*c
^5*d^7*x + b^6*c^4*d^7) + 1/128*log(2*c*x + b)/(c^4*d^7)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx=\frac {\log \left ({\left | 2 \, c x + b \right |}\right )}{128 \, c^{4} d^{7}} + \frac {11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \, {\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{1536 \, {\left (2 \, c x + b\right )}^{6} c^{4} d^{7}} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^7,x, algorithm="giac")

[Out]

1/128*log(abs(2*c*x + b))/(c^4*d^7) + 1/1536*(11*b^6 - 24*a*b^4*c - 48*a^2*b^2*c^2 - 128*a^3*c^3 + 288*(b^2*c^
4 - 4*a*c^5)*x^4 + 576*(b^3*c^3 - 4*a*b*c^4)*x^3 + 36*(11*b^4*c^2 - 40*a*b^2*c^3 - 16*a^2*c^4)*x^2 + 36*(3*b^5
*c - 8*a*b^3*c^2 - 16*a^2*b*c^3)*x)/((2*c*x + b)^6*c^4*d^7)

Mupad [B] (verification not implemented)

Time = 9.67 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.29 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx=\frac {\ln \left (b+2\,c\,x\right )}{128\,c^4\,d^7}-\frac {\frac {128\,a^3\,c^3+48\,a^2\,b^2\,c^2+24\,a\,b^4\,c-11\,b^6}{1536\,c^4}+x^4\,\left (\frac {3\,a\,c}{4}-\frac {3\,b^2}{16}\right )+\frac {3\,x^2\,\left (16\,a^2\,c^2+40\,a\,b^2\,c-11\,b^4\right )}{128\,c^2}+\frac {3\,x\,\left (16\,a^2\,b\,c^2+8\,a\,b^3\,c-3\,b^5\right )}{128\,c^3}-\frac {3\,x^3\,\left (b^3-4\,a\,b\,c\right )}{8\,c}}{b^6\,d^7+12\,b^5\,c\,d^7\,x+60\,b^4\,c^2\,d^7\,x^2+160\,b^3\,c^3\,d^7\,x^3+240\,b^2\,c^4\,d^7\,x^4+192\,b\,c^5\,d^7\,x^5+64\,c^6\,d^7\,x^6} \]

[In]

int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^7,x)

[Out]

log(b + 2*c*x)/(128*c^4*d^7) - ((128*a^3*c^3 - 11*b^6 + 48*a^2*b^2*c^2 + 24*a*b^4*c)/(1536*c^4) + x^4*((3*a*c)
/4 - (3*b^2)/16) + (3*x^2*(16*a^2*c^2 - 11*b^4 + 40*a*b^2*c))/(128*c^2) + (3*x*(16*a^2*b*c^2 - 3*b^5 + 8*a*b^3
*c))/(128*c^3) - (3*x^3*(b^3 - 4*a*b*c))/(8*c))/(b^6*d^7 + 64*c^6*d^7*x^6 + 192*b*c^5*d^7*x^5 + 60*b^4*c^2*d^7
*x^2 + 160*b^3*c^3*d^7*x^3 + 240*b^2*c^4*d^7*x^4 + 12*b^5*c*d^7*x)