### 3.1152 $$\int \frac{(a+b x+c x^2)^3}{(b d+2 c d x)^{11}} \, dx$$

Optimal. Leaf size=101 $\frac{\left (b^2-4 a c\right )^3}{1280 c^4 d^{11} (b+2 c x)^{10}}-\frac{3 \left (b^2-4 a c\right )^2}{1024 c^4 d^{11} (b+2 c x)^8}+\frac{b^2-4 a c}{256 c^4 d^{11} (b+2 c x)^6}-\frac{1}{512 c^4 d^{11} (b+2 c x)^4}$

[Out]

(b^2 - 4*a*c)^3/(1280*c^4*d^11*(b + 2*c*x)^10) - (3*(b^2 - 4*a*c)^2)/(1024*c^4*d^11*(b + 2*c*x)^8) + (b^2 - 4*
a*c)/(256*c^4*d^11*(b + 2*c*x)^6) - 1/(512*c^4*d^11*(b + 2*c*x)^4)

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Rubi [A]  time = 0.0814115, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {683} $\frac{\left (b^2-4 a c\right )^3}{1280 c^4 d^{11} (b+2 c x)^{10}}-\frac{3 \left (b^2-4 a c\right )^2}{1024 c^4 d^{11} (b+2 c x)^8}+\frac{b^2-4 a c}{256 c^4 d^{11} (b+2 c x)^6}-\frac{1}{512 c^4 d^{11} (b+2 c x)^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^11,x]

[Out]

(b^2 - 4*a*c)^3/(1280*c^4*d^11*(b + 2*c*x)^10) - (3*(b^2 - 4*a*c)^2)/(1024*c^4*d^11*(b + 2*c*x)^8) + (b^2 - 4*
a*c)/(256*c^4*d^11*(b + 2*c*x)^6) - 1/(512*c^4*d^11*(b + 2*c*x)^4)

Rule 683

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*} \int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{11}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 d^{11} (b+2 c x)^{11}}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^{11} (b+2 c x)^9}+\frac{3 \left (-b^2+4 a c\right )}{64 c^3 d^{11} (b+2 c x)^7}+\frac{1}{64 c^3 d^{11} (b+2 c x)^5}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^3}{1280 c^4 d^{11} (b+2 c x)^{10}}-\frac{3 \left (b^2-4 a c\right )^2}{1024 c^4 d^{11} (b+2 c x)^8}+\frac{b^2-4 a c}{256 c^4 d^{11} (b+2 c x)^6}-\frac{1}{512 c^4 d^{11} (b+2 c x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0567742, size = 79, normalized size = 0.78 $\frac{20 \left (b^2-4 a c\right ) (b+2 c x)^4-15 \left (b^2-4 a c\right )^2 (b+2 c x)^2+4 \left (b^2-4 a c\right )^3-10 (b+2 c x)^6}{5120 c^4 d^{11} (b+2 c x)^{10}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^11,x]

[Out]

(4*(b^2 - 4*a*c)^3 - 15*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 20*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 10*(b + 2*c*x)^6)/(51
20*c^4*d^11*(b + 2*c*x)^10)

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Maple [A]  time = 0.046, size = 121, normalized size = 1.2 \begin{align*}{\frac{1}{{d}^{11}} \left ( -{\frac{12\,ac-3\,{b}^{2}}{768\,{c}^{4} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{1280\,{c}^{4} \left ( 2\,cx+b \right ) ^{10}}}-{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{1024\,{c}^{4} \left ( 2\,cx+b \right ) ^{8}}}-{\frac{1}{512\,{c}^{4} \left ( 2\,cx+b \right ) ^{4}}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^11,x)

[Out]

1/d^11*(-1/768*(12*a*c-3*b^2)/c^4/(2*c*x+b)^6-1/1280*(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)^
10-1/1024*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^8-1/512/c^4/(2*c*x+b)^4)

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Maxima [B]  time = 2.35701, size = 397, normalized size = 3.93 \begin{align*} -\frac{640 \, c^{6} x^{6} + 1920 \, b c^{5} x^{5} + b^{6} + 8 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3} + 160 \,{\left (13 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} + 320 \,{\left (3 \, b^{3} c^{3} + 8 \, a b c^{4}\right )} x^{3} + 60 \,{\left (3 \, b^{4} c^{2} + 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 20 \,{\left (b^{5} c + 8 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x}{5120 \,{\left (1024 \, c^{14} d^{11} x^{10} + 5120 \, b c^{13} d^{11} x^{9} + 11520 \, b^{2} c^{12} d^{11} x^{8} + 15360 \, b^{3} c^{11} d^{11} x^{7} + 13440 \, b^{4} c^{10} d^{11} x^{6} + 8064 \, b^{5} c^{9} d^{11} x^{5} + 3360 \, b^{6} c^{8} d^{11} x^{4} + 960 \, b^{7} c^{7} d^{11} x^{3} + 180 \, b^{8} c^{6} d^{11} x^{2} + 20 \, b^{9} c^{5} d^{11} x + b^{10} c^{4} d^{11}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^11,x, algorithm="maxima")

[Out]

-1/5120*(640*c^6*x^6 + 1920*b*c^5*x^5 + b^6 + 8*a*b^4*c + 48*a^2*b^2*c^2 + 256*a^3*c^3 + 160*(13*b^2*c^4 + 8*a
*c^5)*x^4 + 320*(3*b^3*c^3 + 8*a*b*c^4)*x^3 + 60*(3*b^4*c^2 + 24*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 20*(b^5*c + 8*a
*b^3*c^2 + 48*a^2*b*c^3)*x)/(1024*c^14*d^11*x^10 + 5120*b*c^13*d^11*x^9 + 11520*b^2*c^12*d^11*x^8 + 15360*b^3*
c^11*d^11*x^7 + 13440*b^4*c^10*d^11*x^6 + 8064*b^5*c^9*d^11*x^5 + 3360*b^6*c^8*d^11*x^4 + 960*b^7*c^7*d^11*x^3
+ 180*b^8*c^6*d^11*x^2 + 20*b^9*c^5*d^11*x + b^10*c^4*d^11)

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Fricas [B]  time = 2.05677, size = 678, normalized size = 6.71 \begin{align*} -\frac{640 \, c^{6} x^{6} + 1920 \, b c^{5} x^{5} + b^{6} + 8 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3} + 160 \,{\left (13 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} + 320 \,{\left (3 \, b^{3} c^{3} + 8 \, a b c^{4}\right )} x^{3} + 60 \,{\left (3 \, b^{4} c^{2} + 24 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 20 \,{\left (b^{5} c + 8 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x}{5120 \,{\left (1024 \, c^{14} d^{11} x^{10} + 5120 \, b c^{13} d^{11} x^{9} + 11520 \, b^{2} c^{12} d^{11} x^{8} + 15360 \, b^{3} c^{11} d^{11} x^{7} + 13440 \, b^{4} c^{10} d^{11} x^{6} + 8064 \, b^{5} c^{9} d^{11} x^{5} + 3360 \, b^{6} c^{8} d^{11} x^{4} + 960 \, b^{7} c^{7} d^{11} x^{3} + 180 \, b^{8} c^{6} d^{11} x^{2} + 20 \, b^{9} c^{5} d^{11} x + b^{10} c^{4} d^{11}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^11,x, algorithm="fricas")

[Out]

-1/5120*(640*c^6*x^6 + 1920*b*c^5*x^5 + b^6 + 8*a*b^4*c + 48*a^2*b^2*c^2 + 256*a^3*c^3 + 160*(13*b^2*c^4 + 8*a
*c^5)*x^4 + 320*(3*b^3*c^3 + 8*a*b*c^4)*x^3 + 60*(3*b^4*c^2 + 24*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 20*(b^5*c + 8*a
*b^3*c^2 + 48*a^2*b*c^3)*x)/(1024*c^14*d^11*x^10 + 5120*b*c^13*d^11*x^9 + 11520*b^2*c^12*d^11*x^8 + 15360*b^3*
c^11*d^11*x^7 + 13440*b^4*c^10*d^11*x^6 + 8064*b^5*c^9*d^11*x^5 + 3360*b^6*c^8*d^11*x^4 + 960*b^7*c^7*d^11*x^3
+ 180*b^8*c^6*d^11*x^2 + 20*b^9*c^5*d^11*x + b^10*c^4*d^11)

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Sympy [B]  time = 42.6112, size = 308, normalized size = 3.05 \begin{align*} - \frac{256 a^{3} c^{3} + 48 a^{2} b^{2} c^{2} + 8 a b^{4} c + b^{6} + 1920 b c^{5} x^{5} + 640 c^{6} x^{6} + x^{4} \left (1280 a c^{5} + 2080 b^{2} c^{4}\right ) + x^{3} \left (2560 a b c^{4} + 960 b^{3} c^{3}\right ) + x^{2} \left (960 a^{2} c^{4} + 1440 a b^{2} c^{3} + 180 b^{4} c^{2}\right ) + x \left (960 a^{2} b c^{3} + 160 a b^{3} c^{2} + 20 b^{5} c\right )}{5120 b^{10} c^{4} d^{11} + 102400 b^{9} c^{5} d^{11} x + 921600 b^{8} c^{6} d^{11} x^{2} + 4915200 b^{7} c^{7} d^{11} x^{3} + 17203200 b^{6} c^{8} d^{11} x^{4} + 41287680 b^{5} c^{9} d^{11} x^{5} + 68812800 b^{4} c^{10} d^{11} x^{6} + 78643200 b^{3} c^{11} d^{11} x^{7} + 58982400 b^{2} c^{12} d^{11} x^{8} + 26214400 b c^{13} d^{11} x^{9} + 5242880 c^{14} d^{11} x^{10}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**11,x)

[Out]

-(256*a**3*c**3 + 48*a**2*b**2*c**2 + 8*a*b**4*c + b**6 + 1920*b*c**5*x**5 + 640*c**6*x**6 + x**4*(1280*a*c**5
+ 2080*b**2*c**4) + x**3*(2560*a*b*c**4 + 960*b**3*c**3) + x**2*(960*a**2*c**4 + 1440*a*b**2*c**3 + 180*b**4*
c**2) + x*(960*a**2*b*c**3 + 160*a*b**3*c**2 + 20*b**5*c))/(5120*b**10*c**4*d**11 + 102400*b**9*c**5*d**11*x +
921600*b**8*c**6*d**11*x**2 + 4915200*b**7*c**7*d**11*x**3 + 17203200*b**6*c**8*d**11*x**4 + 41287680*b**5*c*
*9*d**11*x**5 + 68812800*b**4*c**10*d**11*x**6 + 78643200*b**3*c**11*d**11*x**7 + 58982400*b**2*c**12*d**11*x*
*8 + 26214400*b*c**13*d**11*x**9 + 5242880*c**14*d**11*x**10)

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Giac [A]  time = 1.17969, size = 223, normalized size = 2.21 \begin{align*} -\frac{640 \, c^{6} x^{6} + 1920 \, b c^{5} x^{5} + 2080 \, b^{2} c^{4} x^{4} + 1280 \, a c^{5} x^{4} + 960 \, b^{3} c^{3} x^{3} + 2560 \, a b c^{4} x^{3} + 180 \, b^{4} c^{2} x^{2} + 1440 \, a b^{2} c^{3} x^{2} + 960 \, a^{2} c^{4} x^{2} + 20 \, b^{5} c x + 160 \, a b^{3} c^{2} x + 960 \, a^{2} b c^{3} x + b^{6} + 8 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 256 \, a^{3} c^{3}}{5120 \,{\left (2 \, c x + b\right )}^{10} c^{4} d^{11}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^11,x, algorithm="giac")

[Out]

-1/5120*(640*c^6*x^6 + 1920*b*c^5*x^5 + 2080*b^2*c^4*x^4 + 1280*a*c^5*x^4 + 960*b^3*c^3*x^3 + 2560*a*b*c^4*x^3
+ 180*b^4*c^2*x^2 + 1440*a*b^2*c^3*x^2 + 960*a^2*c^4*x^2 + 20*b^5*c*x + 160*a*b^3*c^2*x + 960*a^2*b*c^3*x + b
^6 + 8*a*b^4*c + 48*a^2*b^2*c^2 + 256*a^3*c^3)/((2*c*x + b)^10*c^4*d^11)