### 3.1150 $$\int \frac{(a+b x+c x^2)^3}{(b d+2 c d x)^9} \, dx$$

Optimal. Leaf size=37 $\frac{\left (a+b x+c x^2\right )^4}{4 d^9 \left (b^2-4 a c\right ) (b+2 c x)^8}$

[Out]

(a + b*x + c*x^2)^4/(4*(b^2 - 4*a*c)*d^9*(b + 2*c*x)^8)

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Rubi [A]  time = 0.0133028, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.042, Rules used = {682} $\frac{\left (a+b x+c x^2\right )^4}{4 d^9 \left (b^2-4 a c\right ) (b+2 c x)^8}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^9,x]

[Out]

(a + b*x + c*x^2)^4/(4*(b^2 - 4*a*c)*d^9*(b + 2*c*x)^8)

Rule 682

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(2*c*(d + e*x)^(m +
1)*(a + b*x + c*x^2)^(p + 1))/(e*(p + 1)*(b^2 - 4*a*c)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*
a*c, 0] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx &=\frac{\left (a+b x+c x^2\right )^4}{4 \left (b^2-4 a c\right ) d^9 (b+2 c x)^8}\\ \end{align*}

Mathematica [B]  time = 0.0524216, size = 96, normalized size = 2.59 $\frac{48 a^2 b^2 c^2-64 a^3 c^3+6 \left (b^2-4 a c\right ) (b+2 c x)^4-4 \left (b^2-4 a c\right )^2 (b+2 c x)^2-12 a b^4 c+b^6-4 (b+2 c x)^6}{1024 c^4 d^9 (b+2 c x)^8}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^9,x]

[Out]

(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3 - 4*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 6*(b^2 - 4*a*c)*(b + 2*c*x
)^4 - 4*(b + 2*c*x)^6)/(1024*c^4*d^9*(b + 2*c*x)^8)

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Maple [B]  time = 0.046, size = 121, normalized size = 3.3 \begin{align*}{\frac{1}{{d}^{9}} \left ( -{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{768\,{c}^{4} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{1}{256\,{c}^{4} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{1024\,{c}^{4} \left ( 2\,cx+b \right ) ^{8}}}-{\frac{12\,ac-3\,{b}^{2}}{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)^9,x)

[Out]

1/d^9*(-1/768*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^6-1/256/c^4/(2*c*x+b)^2-1/1024*(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)^8-1/512*(12*a*c-3*b^2)/c^4/(2*c*x+b)^4)

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Maxima [B]  time = 1.43157, size = 359, normalized size = 9.7 \begin{align*} -\frac{256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 96 \,{\left (9 \, b^{2} c^{4} + 4 \, a c^{5}\right )} x^{4} + 64 \,{\left (7 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} x^{3} + 16 \,{\left (7 \, b^{4} c^{2} + 28 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 16 \,{\left (b^{5} c + 4 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{1024 \,{\left (256 \, c^{12} d^{9} x^{8} + 1024 \, b c^{11} d^{9} x^{7} + 1792 \, b^{2} c^{10} d^{9} x^{6} + 1792 \, b^{3} c^{9} d^{9} x^{5} + 1120 \, b^{4} c^{8} d^{9} x^{4} + 448 \, b^{5} c^{7} d^{9} x^{3} + 112 \, b^{6} c^{6} d^{9} x^{2} + 16 \, b^{7} c^{5} d^{9} x + b^{8} c^{4} d^{9}\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)^9,x, algorithm="maxima")

[Out]

-1/1024*(256*c^6*x^6 + 768*b*c^5*x^5 + b^6 + 4*a*b^4*c + 16*a^2*b^2*c^2 + 64*a^3*c^3 + 96*(9*b^2*c^4 + 4*a*c^5
)*x^4 + 64*(7*b^3*c^3 + 12*a*b*c^4)*x^3 + 16*(7*b^4*c^2 + 28*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 16*(b^5*c + 4*a*b^3
*c^2 + 16*a^2*b*c^3)*x)/(256*c^12*d^9*x^8 + 1024*b*c^11*d^9*x^7 + 1792*b^2*c^10*d^9*x^6 + 1792*b^3*c^9*d^9*x^5
+ 1120*b^4*c^8*d^9*x^4 + 448*b^5*c^7*d^9*x^3 + 112*b^6*c^6*d^9*x^2 + 16*b^7*c^5*d^9*x + b^8*c^4*d^9)

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Fricas [B]  time = 1.98935, size = 585, normalized size = 15.81 \begin{align*} -\frac{256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 96 \,{\left (9 \, b^{2} c^{4} + 4 \, a c^{5}\right )} x^{4} + 64 \,{\left (7 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} x^{3} + 16 \,{\left (7 \, b^{4} c^{2} + 28 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 16 \,{\left (b^{5} c + 4 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{1024 \,{\left (256 \, c^{12} d^{9} x^{8} + 1024 \, b c^{11} d^{9} x^{7} + 1792 \, b^{2} c^{10} d^{9} x^{6} + 1792 \, b^{3} c^{9} d^{9} x^{5} + 1120 \, b^{4} c^{8} d^{9} x^{4} + 448 \, b^{5} c^{7} d^{9} x^{3} + 112 \, b^{6} c^{6} d^{9} x^{2} + 16 \, b^{7} c^{5} d^{9} x + b^{8} c^{4} d^{9}\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)^9,x, algorithm="fricas")

[Out]

-1/1024*(256*c^6*x^6 + 768*b*c^5*x^5 + b^6 + 4*a*b^4*c + 16*a^2*b^2*c^2 + 64*a^3*c^3 + 96*(9*b^2*c^4 + 4*a*c^5
)*x^4 + 64*(7*b^3*c^3 + 12*a*b*c^4)*x^3 + 16*(7*b^4*c^2 + 28*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 16*(b^5*c + 4*a*b^3
*c^2 + 16*a^2*b*c^3)*x)/(256*c^12*d^9*x^8 + 1024*b*c^11*d^9*x^7 + 1792*b^2*c^10*d^9*x^6 + 1792*b^3*c^9*d^9*x^5
+ 1120*b^4*c^8*d^9*x^4 + 448*b^5*c^7*d^9*x^3 + 112*b^6*c^6*d^9*x^2 + 16*b^7*c^5*d^9*x + b^8*c^4*d^9)

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Sympy [B]  time = 22.5226, size = 277, normalized size = 7.49 \begin{align*} - \frac{64 a^{3} c^{3} + 16 a^{2} b^{2} c^{2} + 4 a b^{4} c + b^{6} + 768 b c^{5} x^{5} + 256 c^{6} x^{6} + x^{4} \left (384 a c^{5} + 864 b^{2} c^{4}\right ) + x^{3} \left (768 a b c^{4} + 448 b^{3} c^{3}\right ) + x^{2} \left (256 a^{2} c^{4} + 448 a b^{2} c^{3} + 112 b^{4} c^{2}\right ) + x \left (256 a^{2} b c^{3} + 64 a b^{3} c^{2} + 16 b^{5} c\right )}{1024 b^{8} c^{4} d^{9} + 16384 b^{7} c^{5} d^{9} x + 114688 b^{6} c^{6} d^{9} x^{2} + 458752 b^{5} c^{7} d^{9} x^{3} + 1146880 b^{4} c^{8} d^{9} x^{4} + 1835008 b^{3} c^{9} d^{9} x^{5} + 1835008 b^{2} c^{10} d^{9} x^{6} + 1048576 b c^{11} d^{9} x^{7} + 262144 c^{12} d^{9} x^{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)**9,x)

[Out]

-(64*a**3*c**3 + 16*a**2*b**2*c**2 + 4*a*b**4*c + b**6 + 768*b*c**5*x**5 + 256*c**6*x**6 + x**4*(384*a*c**5 +
864*b**2*c**4) + x**3*(768*a*b*c**4 + 448*b**3*c**3) + x**2*(256*a**2*c**4 + 448*a*b**2*c**3 + 112*b**4*c**2)
+ x*(256*a**2*b*c**3 + 64*a*b**3*c**2 + 16*b**5*c))/(1024*b**8*c**4*d**9 + 16384*b**7*c**5*d**9*x + 114688*b**
6*c**6*d**9*x**2 + 458752*b**5*c**7*d**9*x**3 + 1146880*b**4*c**8*d**9*x**4 + 1835008*b**3*c**9*d**9*x**5 + 18
35008*b**2*c**10*d**9*x**6 + 1048576*b*c**11*d**9*x**7 + 262144*c**12*d**9*x**8)

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Giac [B]  time = 1.17934, size = 223, normalized size = 6.03 \begin{align*} -\frac{256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + 864 \, b^{2} c^{4} x^{4} + 384 \, a c^{5} x^{4} + 448 \, b^{3} c^{3} x^{3} + 768 \, a b c^{4} x^{3} + 112 \, b^{4} c^{2} x^{2} + 448 \, a b^{2} c^{3} x^{2} + 256 \, a^{2} c^{4} x^{2} + 16 \, b^{5} c x + 64 \, a b^{3} c^{2} x + 256 \, a^{2} b c^{3} x + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3}}{1024 \,{\left (2 \, c x + b\right )}^{8} c^{4} d^{9}} \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)^9,x, algorithm="giac")

[Out]

-1/1024*(256*c^6*x^6 + 768*b*c^5*x^5 + 864*b^2*c^4*x^4 + 384*a*c^5*x^4 + 448*b^3*c^3*x^3 + 768*a*b*c^4*x^3 + 1
12*b^4*c^2*x^2 + 448*a*b^2*c^3*x^2 + 256*a^2*c^4*x^2 + 16*b^5*c*x + 64*a*b^3*c^2*x + 256*a^2*b*c^3*x + b^6 + 4
*a*b^4*c + 16*a^2*b^2*c^2 + 64*a^3*c^3)/((2*c*x + b)^8*c^4*d^9)