3.1149 \(\int \frac{(a+b x+c x^2)^3}{(b d+2 c d x)^8} \, dx\)

Optimal. Leaf size=101 \[ \frac{\left (b^2-4 a c\right )^3}{896 c^4 d^8 (b+2 c x)^7}-\frac{3 \left (b^2-4 a c\right )^2}{640 c^4 d^8 (b+2 c x)^5}+\frac{b^2-4 a c}{128 c^4 d^8 (b+2 c x)^3}-\frac{1}{128 c^4 d^8 (b+2 c x)} \]

[Out]

(b^2 - 4*a*c)^3/(896*c^4*d^8*(b + 2*c*x)^7) - (3*(b^2 - 4*a*c)^2)/(640*c^4*d^8*(b + 2*c*x)^5) + (b^2 - 4*a*c)/
(128*c^4*d^8*(b + 2*c*x)^3) - 1/(128*c^4*d^8*(b + 2*c*x))

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Rubi [A]  time = 0.080555, 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} \[ \frac{\left (b^2-4 a c\right )^3}{896 c^4 d^8 (b+2 c x)^7}-\frac{3 \left (b^2-4 a c\right )^2}{640 c^4 d^8 (b+2 c x)^5}+\frac{b^2-4 a c}{128 c^4 d^8 (b+2 c x)^3}-\frac{1}{128 c^4 d^8 (b+2 c x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^8,x]

[Out]

(b^2 - 4*a*c)^3/(896*c^4*d^8*(b + 2*c*x)^7) - (3*(b^2 - 4*a*c)^2)/(640*c^4*d^8*(b + 2*c*x)^5) + (b^2 - 4*a*c)/
(128*c^4*d^8*(b + 2*c*x)^3) - 1/(128*c^4*d^8*(b + 2*c*x))

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)^8} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 d^8 (b+2 c x)^8}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^8 (b+2 c x)^6}+\frac{3 \left (-b^2+4 a c\right )}{64 c^3 d^8 (b+2 c x)^4}+\frac{1}{64 c^3 d^8 (b+2 c x)^2}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^3}{896 c^4 d^8 (b+2 c x)^7}-\frac{3 \left (b^2-4 a c\right )^2}{640 c^4 d^8 (b+2 c x)^5}+\frac{b^2-4 a c}{128 c^4 d^8 (b+2 c x)^3}-\frac{1}{128 c^4 d^8 (b+2 c x)}\\ \end{align*}

Mathematica [A]  time = 0.0485275, size = 79, normalized size = 0.78 \[ \frac{35 \left (b^2-4 a c\right ) (b+2 c x)^4-21 \left (b^2-4 a c\right )^2 (b+2 c x)^2+5 \left (b^2-4 a c\right )^3-35 (b+2 c x)^6}{4480 c^4 d^8 (b+2 c x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^8,x]

[Out]

(5*(b^2 - 4*a*c)^3 - 21*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 35*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 35*(b + 2*c*x)^6)/(44
80*c^4*d^8*(b + 2*c*x)^7)

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Maple [A]  time = 0.046, size = 121, normalized size = 1.2 \begin{align*}{\frac{1}{{d}^{8}} \left ( -{\frac{12\,ac-3\,{b}^{2}}{384\,{c}^{4} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{640\,{c}^{4} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{896\,{c}^{4} \left ( 2\,cx+b \right ) ^{7}}}-{\frac{1}{128\,{c}^{4} \left ( 2\,cx+b \right ) }} \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)^8,x)

[Out]

1/d^8*(-1/384*(12*a*c-3*b^2)/c^4/(2*c*x+b)^3-1/640*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^5-1/896*(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)^7-1/128/c^4/(2*c*x+b))

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Maxima [B]  time = 1.23609, size = 335, normalized size = 3.32 \begin{align*} -\frac{140 \, c^{6} x^{6} + 420 \, b c^{5} x^{5} + b^{6} + 2 \, a b^{4} c + 6 \, a^{2} b^{2} c^{2} + 20 \, a^{3} c^{3} + 70 \,{\left (7 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 280 \,{\left (b^{3} c^{3} + a b c^{4}\right )} x^{3} + 84 \,{\left (b^{4} c^{2} + 2 \, a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2} + 14 \,{\left (b^{5} c + 2 \, a b^{3} c^{2} + 6 \, a^{2} b c^{3}\right )} x}{280 \,{\left (128 \, c^{11} d^{8} x^{7} + 448 \, b c^{10} d^{8} x^{6} + 672 \, b^{2} c^{9} d^{8} x^{5} + 560 \, b^{3} c^{8} d^{8} x^{4} + 280 \, b^{4} c^{7} d^{8} x^{3} + 84 \, b^{5} c^{6} d^{8} x^{2} + 14 \, b^{6} c^{5} d^{8} x + b^{7} c^{4} d^{8}\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)^8,x, algorithm="maxima")

[Out]

-1/280*(140*c^6*x^6 + 420*b*c^5*x^5 + b^6 + 2*a*b^4*c + 6*a^2*b^2*c^2 + 20*a^3*c^3 + 70*(7*b^2*c^4 + 2*a*c^5)*
x^4 + 280*(b^3*c^3 + a*b*c^4)*x^3 + 84*(b^4*c^2 + 2*a*b^2*c^3 + a^2*c^4)*x^2 + 14*(b^5*c + 2*a*b^3*c^2 + 6*a^2
*b*c^3)*x)/(128*c^11*d^8*x^7 + 448*b*c^10*d^8*x^6 + 672*b^2*c^9*d^8*x^5 + 560*b^3*c^8*d^8*x^4 + 280*b^4*c^7*d^
8*x^3 + 84*b^5*c^6*d^8*x^2 + 14*b^6*c^5*d^8*x + b^7*c^4*d^8)

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Fricas [B]  time = 1.95219, size = 529, normalized size = 5.24 \begin{align*} -\frac{140 \, c^{6} x^{6} + 420 \, b c^{5} x^{5} + b^{6} + 2 \, a b^{4} c + 6 \, a^{2} b^{2} c^{2} + 20 \, a^{3} c^{3} + 70 \,{\left (7 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 280 \,{\left (b^{3} c^{3} + a b c^{4}\right )} x^{3} + 84 \,{\left (b^{4} c^{2} + 2 \, a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2} + 14 \,{\left (b^{5} c + 2 \, a b^{3} c^{2} + 6 \, a^{2} b c^{3}\right )} x}{280 \,{\left (128 \, c^{11} d^{8} x^{7} + 448 \, b c^{10} d^{8} x^{6} + 672 \, b^{2} c^{9} d^{8} x^{5} + 560 \, b^{3} c^{8} d^{8} x^{4} + 280 \, b^{4} c^{7} d^{8} x^{3} + 84 \, b^{5} c^{6} d^{8} x^{2} + 14 \, b^{6} c^{5} d^{8} x + b^{7} c^{4} d^{8}\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)^8,x, algorithm="fricas")

[Out]

-1/280*(140*c^6*x^6 + 420*b*c^5*x^5 + b^6 + 2*a*b^4*c + 6*a^2*b^2*c^2 + 20*a^3*c^3 + 70*(7*b^2*c^4 + 2*a*c^5)*
x^4 + 280*(b^3*c^3 + a*b*c^4)*x^3 + 84*(b^4*c^2 + 2*a*b^2*c^3 + a^2*c^4)*x^2 + 14*(b^5*c + 2*a*b^3*c^2 + 6*a^2
*b*c^3)*x)/(128*c^11*d^8*x^7 + 448*b*c^10*d^8*x^6 + 672*b^2*c^9*d^8*x^5 + 560*b^3*c^8*d^8*x^4 + 280*b^4*c^7*d^
8*x^3 + 84*b^5*c^6*d^8*x^2 + 14*b^6*c^5*d^8*x + b^7*c^4*d^8)

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Sympy [B]  time = 9.30292, size = 262, normalized size = 2.59 \begin{align*} - \frac{20 a^{3} c^{3} + 6 a^{2} b^{2} c^{2} + 2 a b^{4} c + b^{6} + 420 b c^{5} x^{5} + 140 c^{6} x^{6} + x^{4} \left (140 a c^{5} + 490 b^{2} c^{4}\right ) + x^{3} \left (280 a b c^{4} + 280 b^{3} c^{3}\right ) + x^{2} \left (84 a^{2} c^{4} + 168 a b^{2} c^{3} + 84 b^{4} c^{2}\right ) + x \left (84 a^{2} b c^{3} + 28 a b^{3} c^{2} + 14 b^{5} c\right )}{280 b^{7} c^{4} d^{8} + 3920 b^{6} c^{5} d^{8} x + 23520 b^{5} c^{6} d^{8} x^{2} + 78400 b^{4} c^{7} d^{8} x^{3} + 156800 b^{3} c^{8} d^{8} x^{4} + 188160 b^{2} c^{9} d^{8} x^{5} + 125440 b c^{10} d^{8} x^{6} + 35840 c^{11} d^{8} x^{7}} \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)**8,x)

[Out]

-(20*a**3*c**3 + 6*a**2*b**2*c**2 + 2*a*b**4*c + b**6 + 420*b*c**5*x**5 + 140*c**6*x**6 + x**4*(140*a*c**5 + 4
90*b**2*c**4) + x**3*(280*a*b*c**4 + 280*b**3*c**3) + x**2*(84*a**2*c**4 + 168*a*b**2*c**3 + 84*b**4*c**2) + x
*(84*a**2*b*c**3 + 28*a*b**3*c**2 + 14*b**5*c))/(280*b**7*c**4*d**8 + 3920*b**6*c**5*d**8*x + 23520*b**5*c**6*
d**8*x**2 + 78400*b**4*c**7*d**8*x**3 + 156800*b**3*c**8*d**8*x**4 + 188160*b**2*c**9*d**8*x**5 + 125440*b*c**
10*d**8*x**6 + 35840*c**11*d**8*x**7)

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Giac [A]  time = 1.18763, size = 223, normalized size = 2.21 \begin{align*} -\frac{140 \, c^{6} x^{6} + 420 \, b c^{5} x^{5} + 490 \, b^{2} c^{4} x^{4} + 140 \, a c^{5} x^{4} + 280 \, b^{3} c^{3} x^{3} + 280 \, a b c^{4} x^{3} + 84 \, b^{4} c^{2} x^{2} + 168 \, a b^{2} c^{3} x^{2} + 84 \, a^{2} c^{4} x^{2} + 14 \, b^{5} c x + 28 \, a b^{3} c^{2} x + 84 \, a^{2} b c^{3} x + b^{6} + 2 \, a b^{4} c + 6 \, a^{2} b^{2} c^{2} + 20 \, a^{3} c^{3}}{280 \,{\left (2 \, c x + b\right )}^{7} c^{4} d^{8}} \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)^8,x, algorithm="giac")

[Out]

-1/280*(140*c^6*x^6 + 420*b*c^5*x^5 + 490*b^2*c^4*x^4 + 140*a*c^5*x^4 + 280*b^3*c^3*x^3 + 280*a*b*c^4*x^3 + 84
*b^4*c^2*x^2 + 168*a*b^2*c^3*x^2 + 84*a^2*c^4*x^2 + 14*b^5*c*x + 28*a*b^3*c^2*x + 84*a^2*b*c^3*x + b^6 + 2*a*b
^4*c + 6*a^2*b^2*c^2 + 20*a^3*c^3)/((2*c*x + b)^7*c^4*d^8)