\(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^{10}} \, dx\) [1151]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{10}} \, dx=\frac {\left (b^2-4 a c\right )^3}{1152 c^4 d^{10} (b+2 c x)^9}-\frac {3 \left (b^2-4 a c\right )^2}{896 c^4 d^{10} (b+2 c x)^7}+\frac {3 \left (b^2-4 a c\right )}{640 c^4 d^{10} (b+2 c x)^5}-\frac {1}{384 c^4 d^{10} (b+2 c x)^3} \]

[Out]

1/1152*(-4*a*c+b^2)^3/c^4/d^10/(2*c*x+b)^9-3/896*(-4*a*c+b^2)^2/c^4/d^10/(2*c*x+b)^7+3/640*(-4*a*c+b^2)/c^4/d^
10/(2*c*x+b)^5-1/384/c^4/d^10/(2*c*x+b)^3

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 101, 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} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{10}} \, dx=\frac {\left (b^2-4 a c\right )^3}{1152 c^4 d^{10} (b+2 c x)^9}-\frac {3 \left (b^2-4 a c\right )^2}{896 c^4 d^{10} (b+2 c x)^7}+\frac {3 \left (b^2-4 a c\right )}{640 c^4 d^{10} (b+2 c x)^5}-\frac {1}{384 c^4 d^{10} (b+2 c x)^3} \]

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^10,x]

[Out]

(b^2 - 4*a*c)^3/(1152*c^4*d^10*(b + 2*c*x)^9) - (3*(b^2 - 4*a*c)^2)/(896*c^4*d^10*(b + 2*c*x)^7) + (3*(b^2 - 4
*a*c))/(640*c^4*d^10*(b + 2*c*x)^5) - 1/(384*c^4*d^10*(b + 2*c*x)^3)

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^{10} (b+2 c x)^{10}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^{10} (b+2 c x)^8}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^{10} (b+2 c x)^6}+\frac {1}{64 c^3 d^{10} (b+2 c x)^4}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^3}{1152 c^4 d^{10} (b+2 c x)^9}-\frac {3 \left (b^2-4 a c\right )^2}{896 c^4 d^{10} (b+2 c x)^7}+\frac {3 \left (b^2-4 a c\right )}{640 c^4 d^{10} (b+2 c x)^5}-\frac {1}{384 c^4 d^{10} (b+2 c x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{10}} \, dx=\frac {35 \left (b^2-4 a c\right )^3-135 \left (b^2-4 a c\right )^2 (b+2 c x)^2+189 \left (b^2-4 a c\right ) (b+2 c x)^4-105 (b+2 c x)^6}{40320 c^4 d^{10} (b+2 c x)^9} \]

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^10,x]

[Out]

(35*(b^2 - 4*a*c)^3 - 135*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 189*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 105*(b + 2*c*x)^6)
/(40320*c^4*d^10*(b + 2*c*x)^9)

Maple [A] (verified)

Time = 2.39 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {-\frac {12 a c -3 b^{2}}{640 c^{4} \left (2 c x +b \right )^{5}}-\frac {1}{384 c^{4} \left (2 c x +b \right )^{3}}-\frac {48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}}{896 c^{4} \left (2 c x +b \right )^{7}}-\frac {64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{1152 c^{4} \left (2 c x +b \right )^{9}}}{d^{10}}\)	\(121\)
risch	\(\frac {-\frac {c^{2} x^{6}}{6}-\frac {b c \,x^{5}}{2}+\left (-\frac {3 a c}{10}-\frac {11 b^{2}}{20}\right ) x^{4}-\frac {b \left (9 a c +4 b^{2}\right ) x^{3}}{15 c}-\frac {\left (15 a^{2} c^{2}+24 a \,b^{2} c +4 b^{4}\right ) x^{2}}{70 c^{2}}-\frac {b \left (30 a^{2} c^{2}+6 a \,b^{2} c +b^{4}\right ) x}{140 c^{3}}-\frac {140 c^{3} a^{3}+30 a^{2} b^{2} c^{2}+6 a \,b^{4} c +b^{6}}{2520 c^{4}}}{d^{10} \left (2 c x +b \right )^{9}}\)	\(152\)
gosper	\(-\frac {420 c^{6} x^{6}+1260 b \,c^{5} x^{5}+756 a \,c^{5} x^{4}+1386 b^{2} c^{4} x^{4}+1512 a b \,c^{4} x^{3}+672 x^{3} b^{3} c^{3}+540 a^{2} c^{4} x^{2}+864 a \,b^{2} c^{3} x^{2}+144 x^{2} b^{4} c^{2}+540 a^{2} b \,c^{3} x +108 x a \,b^{3} c^{2}+18 x \,b^{5} c +140 c^{3} a^{3}+30 a^{2} b^{2} c^{2}+6 a \,b^{4} c +b^{6}}{2520 \left (2 c x +b \right )^{9} d^{10} c^{4}}\)	\(166\)
norman	\(\frac {\frac {a^{3} x}{b d}+\frac {\left (16 c \,a^{3}+3 a^{2} b^{2}\right ) x^{2}}{2 b^{2} d}+\frac {\left (112 a^{3} c^{2}+24 a^{2} b^{2} c +3 a \,b^{4}\right ) x^{3}}{3 b^{3} d}+\frac {\left (448 c^{3} a^{3}+96 a^{2} b^{2} c^{2}+18 a \,b^{4} c +b^{6}\right ) x^{4}}{4 b^{4} d}+\frac {32 c^{4} \left (140 c^{3} a^{3}+30 a^{2} b^{2} c^{2}+6 a \,b^{4} c +b^{6}\right ) x^{8}}{35 b^{8} d}+\frac {64 c^{3} \left (140 c^{3} a^{3}+30 a^{2} b^{2} c^{2}+6 a \,b^{4} c +b^{6}\right ) x^{7}}{35 b^{7} d}+\frac {c \left (2240 c^{3} a^{3}+480 a^{2} b^{2} c^{2}+96 a \,b^{4} c +11 b^{6}\right ) x^{5}}{10 b^{5} d}+\frac {c^{2} \left (8960 c^{3} a^{3}+1920 a^{2} b^{2} c^{2}+384 a \,b^{4} c +59 b^{6}\right ) x^{6}}{30 b^{6} d}+\frac {64 c^{5} \left (140 c^{3} a^{3}+30 a^{2} b^{2} c^{2}+6 a \,b^{4} c +b^{6}\right ) x^{9}}{315 b^{9} d}}{d^{9} \left (2 c x +b \right )^{9}}\)	\(349\)
parallelrisch	\(\frac {35840 x^{9} a^{3} c^{8}+7680 x^{9} a^{2} b^{2} c^{7}+1536 x^{9} a \,b^{4} c^{6}+256 x^{9} b^{6} c^{5}+161280 x^{8} a^{3} b \,c^{7}+34560 x^{8} a^{2} b^{3} c^{6}+6912 x^{8} a \,b^{5} c^{5}+1152 x^{8} b^{7} c^{4}+322560 x^{7} a^{3} b^{2} c^{6}+69120 x^{7} a^{2} b^{4} c^{5}+13824 x^{7} a \,b^{6} c^{4}+2304 x^{7} b^{8} c^{3}+376320 x^{6} a^{3} b^{3} c^{5}+80640 x^{6} a^{2} b^{5} c^{4}+16128 x^{6} a \,b^{7} c^{3}+2478 x^{6} b^{9} c^{2}+282240 x^{5} a^{3} b^{4} c^{4}+60480 x^{5} a^{2} b^{6} c^{3}+12096 x^{5} a \,b^{8} c^{2}+1386 x^{5} b^{10} c +141120 x^{4} a^{3} b^{5} c^{3}+30240 x^{4} a^{2} b^{7} c^{2}+5670 x^{4} a \,b^{9} c +315 x^{4} b^{11}+47040 x^{3} a^{3} b^{6} c^{2}+10080 x^{3} a^{2} b^{8} c +1260 x^{3} a \,b^{10}+10080 x^{2} a^{3} b^{7} c +1890 x^{2} a^{2} b^{9}+1260 a^{3} b^{8} x}{1260 b^{9} d^{10} \left (2 c x +b \right )^{9}}\)	\(379\)

[In]

int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^10,x,method=_RETURNVERBOSE)

[Out]

1/d^10*(-1/640*(12*a*c-3*b^2)/c^4/(2*c*x+b)^5-1/384/c^4/(2*c*x+b)^3-1/896*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2
*c*x+b)^7-1/1152*(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)^9)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (93) = 186\).

Time = 0.39 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.77 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{10}} \, dx=-\frac {420 \, c^{6} x^{6} + 1260 \, b c^{5} x^{5} + b^{6} + 6 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} + 140 \, a^{3} c^{3} + 126 \, {\left (11 \, b^{2} c^{4} + 6 \, a c^{5}\right )} x^{4} + 168 \, {\left (4 \, b^{3} c^{3} + 9 \, a b c^{4}\right )} x^{3} + 36 \, {\left (4 \, b^{4} c^{2} + 24 \, a b^{2} c^{3} + 15 \, a^{2} c^{4}\right )} x^{2} + 18 \, {\left (b^{5} c + 6 \, a b^{3} c^{2} + 30 \, a^{2} b c^{3}\right )} x}{2520 \, {\left (512 \, c^{13} d^{10} x^{9} + 2304 \, b c^{12} d^{10} x^{8} + 4608 \, b^{2} c^{11} d^{10} x^{7} + 5376 \, b^{3} c^{10} d^{10} x^{6} + 4032 \, b^{4} c^{9} d^{10} x^{5} + 2016 \, b^{5} c^{8} d^{10} x^{4} + 672 \, b^{6} c^{7} d^{10} x^{3} + 144 \, b^{7} c^{6} d^{10} x^{2} + 18 \, b^{8} c^{5} d^{10} x + b^{9} c^{4} d^{10}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^10,x, algorithm="fricas")

[Out]

-1/2520*(420*c^6*x^6 + 1260*b*c^5*x^5 + b^6 + 6*a*b^4*c + 30*a^2*b^2*c^2 + 140*a^3*c^3 + 126*(11*b^2*c^4 + 6*a
*c^5)*x^4 + 168*(4*b^3*c^3 + 9*a*b*c^4)*x^3 + 36*(4*b^4*c^2 + 24*a*b^2*c^3 + 15*a^2*c^4)*x^2 + 18*(b^5*c + 6*a
*b^3*c^2 + 30*a^2*b*c^3)*x)/(512*c^13*d^10*x^9 + 2304*b*c^12*d^10*x^8 + 4608*b^2*c^11*d^10*x^7 + 5376*b^3*c^10
*d^10*x^6 + 4032*b^4*c^9*d^10*x^5 + 2016*b^5*c^8*d^10*x^4 + 672*b^6*c^7*d^10*x^3 + 144*b^7*c^6*d^10*x^2 + 18*b
^8*c^5*d^10*x + b^9*c^4*d^10)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (102) = 204\).

Time = 5.01 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.95 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{10}} \, dx=\frac {- 140 a^{3} c^{3} - 30 a^{2} b^{2} c^{2} - 6 a b^{4} c - b^{6} - 1260 b c^{5} x^{5} - 420 c^{6} x^{6} + x^{4} \left (- 756 a c^{5} - 1386 b^{2} c^{4}\right ) + x^{3} \left (- 1512 a b c^{4} - 672 b^{3} c^{3}\right ) + x^{2} \left (- 540 a^{2} c^{4} - 864 a b^{2} c^{3} - 144 b^{4} c^{2}\right ) + x \left (- 540 a^{2} b c^{3} - 108 a b^{3} c^{2} - 18 b^{5} c\right )}{2520 b^{9} c^{4} d^{10} + 45360 b^{8} c^{5} d^{10} x + 362880 b^{7} c^{6} d^{10} x^{2} + 1693440 b^{6} c^{7} d^{10} x^{3} + 5080320 b^{5} c^{8} d^{10} x^{4} + 10160640 b^{4} c^{9} d^{10} x^{5} + 13547520 b^{3} c^{10} d^{10} x^{6} + 11612160 b^{2} c^{11} d^{10} x^{7} + 5806080 b c^{12} d^{10} x^{8} + 1290240 c^{13} d^{10} x^{9}} \]

[In]

integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**10,x)

[Out]

(-140*a**3*c**3 - 30*a**2*b**2*c**2 - 6*a*b**4*c - b**6 - 1260*b*c**5*x**5 - 420*c**6*x**6 + x**4*(-756*a*c**5
- 1386*b**2*c**4) + x**3*(-1512*a*b*c**4 - 672*b**3*c**3) + x**2*(-540*a**2*c**4 - 864*a*b**2*c**3 - 144*b**4
*c**2) + x*(-540*a**2*b*c**3 - 108*a*b**3*c**2 - 18*b**5*c))/(2520*b**9*c**4*d**10 + 45360*b**8*c**5*d**10*x +
362880*b**7*c**6*d**10*x**2 + 1693440*b**6*c**7*d**10*x**3 + 5080320*b**5*c**8*d**10*x**4 + 10160640*b**4*c**
9*d**10*x**5 + 13547520*b**3*c**10*d**10*x**6 + 11612160*b**2*c**11*d**10*x**7 + 5806080*b*c**12*d**10*x**8 +
1290240*c**13*d**10*x**9)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (93) = 186\).

Time = 0.21 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.77 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{10}} \, dx=-\frac {420 \, c^{6} x^{6} + 1260 \, b c^{5} x^{5} + b^{6} + 6 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} + 140 \, a^{3} c^{3} + 126 \, {\left (11 \, b^{2} c^{4} + 6 \, a c^{5}\right )} x^{4} + 168 \, {\left (4 \, b^{3} c^{3} + 9 \, a b c^{4}\right )} x^{3} + 36 \, {\left (4 \, b^{4} c^{2} + 24 \, a b^{2} c^{3} + 15 \, a^{2} c^{4}\right )} x^{2} + 18 \, {\left (b^{5} c + 6 \, a b^{3} c^{2} + 30 \, a^{2} b c^{3}\right )} x}{2520 \, {\left (512 \, c^{13} d^{10} x^{9} + 2304 \, b c^{12} d^{10} x^{8} + 4608 \, b^{2} c^{11} d^{10} x^{7} + 5376 \, b^{3} c^{10} d^{10} x^{6} + 4032 \, b^{4} c^{9} d^{10} x^{5} + 2016 \, b^{5} c^{8} d^{10} x^{4} + 672 \, b^{6} c^{7} d^{10} x^{3} + 144 \, b^{7} c^{6} d^{10} x^{2} + 18 \, b^{8} c^{5} d^{10} x + b^{9} c^{4} d^{10}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^10,x, algorithm="maxima")

[Out]

-1/2520*(420*c^6*x^6 + 1260*b*c^5*x^5 + b^6 + 6*a*b^4*c + 30*a^2*b^2*c^2 + 140*a^3*c^3 + 126*(11*b^2*c^4 + 6*a
*c^5)*x^4 + 168*(4*b^3*c^3 + 9*a*b*c^4)*x^3 + 36*(4*b^4*c^2 + 24*a*b^2*c^3 + 15*a^2*c^4)*x^2 + 18*(b^5*c + 6*a
*b^3*c^2 + 30*a^2*b*c^3)*x)/(512*c^13*d^10*x^9 + 2304*b*c^12*d^10*x^8 + 4608*b^2*c^11*d^10*x^7 + 5376*b^3*c^10
*d^10*x^6 + 4032*b^4*c^9*d^10*x^5 + 2016*b^5*c^8*d^10*x^4 + 672*b^6*c^7*d^10*x^3 + 144*b^7*c^6*d^10*x^2 + 18*b
^8*c^5*d^10*x + b^9*c^4*d^10)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{10}} \, dx=-\frac {420 \, c^{6} x^{6} + 1260 \, b c^{5} x^{5} + 1386 \, b^{2} c^{4} x^{4} + 756 \, a c^{5} x^{4} + 672 \, b^{3} c^{3} x^{3} + 1512 \, a b c^{4} x^{3} + 144 \, b^{4} c^{2} x^{2} + 864 \, a b^{2} c^{3} x^{2} + 540 \, a^{2} c^{4} x^{2} + 18 \, b^{5} c x + 108 \, a b^{3} c^{2} x + 540 \, a^{2} b c^{3} x + b^{6} + 6 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} + 140 \, a^{3} c^{3}}{2520 \, {\left (2 \, c x + b\right )}^{9} c^{4} d^{10}} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^10,x, algorithm="giac")

[Out]

-1/2520*(420*c^6*x^6 + 1260*b*c^5*x^5 + 1386*b^2*c^4*x^4 + 756*a*c^5*x^4 + 672*b^3*c^3*x^3 + 1512*a*b*c^4*x^3
+ 144*b^4*c^2*x^2 + 864*a*b^2*c^3*x^2 + 540*a^2*c^4*x^2 + 18*b^5*c*x + 108*a*b^3*c^2*x + 540*a^2*b*c^3*x + b^6
+ 6*a*b^4*c + 30*a^2*b^2*c^2 + 140*a^3*c^3)/((2*c*x + b)^9*c^4*d^10)

Mupad [B] (verification not implemented)

Time = 9.77 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.65 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{10}} \, dx=-\frac {\frac {140\,a^3\,c^3+30\,a^2\,b^2\,c^2+6\,a\,b^4\,c+b^6}{2520\,c^4}+x^4\,\left (\frac {11\,b^2}{20}+\frac {3\,a\,c}{10}\right )+\frac {c^2\,x^6}{6}+\frac {x^2\,\left (15\,a^2\,c^2+24\,a\,b^2\,c+4\,b^4\right )}{70\,c^2}+\frac {b\,c\,x^5}{2}+\frac {x^3\,\left (4\,b^3+9\,a\,c\,b\right )}{15\,c}+\frac {b\,x\,\left (30\,a^2\,c^2+6\,a\,b^2\,c+b^4\right )}{140\,c^3}}{b^9\,d^{10}+18\,b^8\,c\,d^{10}\,x+144\,b^7\,c^2\,d^{10}\,x^2+672\,b^6\,c^3\,d^{10}\,x^3+2016\,b^5\,c^4\,d^{10}\,x^4+4032\,b^4\,c^5\,d^{10}\,x^5+5376\,b^3\,c^6\,d^{10}\,x^6+4608\,b^2\,c^7\,d^{10}\,x^7+2304\,b\,c^8\,d^{10}\,x^8+512\,c^9\,d^{10}\,x^9} \]

[In]

int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^10,x)

[Out]

-((b^6 + 140*a^3*c^3 + 30*a^2*b^2*c^2 + 6*a*b^4*c)/(2520*c^4) + x^4*((3*a*c)/10 + (11*b^2)/20) + (c^2*x^6)/6 +
(x^2*(4*b^4 + 15*a^2*c^2 + 24*a*b^2*c))/(70*c^2) + (b*c*x^5)/2 + (x^3*(4*b^3 + 9*a*b*c))/(15*c) + (b*x*(b^4 +
30*a^2*c^2 + 6*a*b^2*c))/(140*c^3))/(b^9*d^10 + 512*c^9*d^10*x^9 + 2304*b*c^8*d^10*x^8 + 144*b^7*c^2*d^10*x^2
+ 672*b^6*c^3*d^10*x^3 + 2016*b^5*c^4*d^10*x^4 + 4032*b^4*c^5*d^10*x^5 + 5376*b^3*c^6*d^10*x^6 + 4608*b^2*c^7
*d^10*x^7 + 18*b^8*c*d^10*x)