### 3.1145 $$\int \frac{(a+b x+c x^2)^3}{(b d+2 c d x)^4} \, dx$$

Optimal. Leaf size=103 $\frac{\left (b^2-4 a c\right )^3}{384 c^4 d^4 (b+2 c x)^3}-\frac{3 \left (b^2-4 a c\right )^2}{128 c^4 d^4 (b+2 c x)}-\frac{x \left (b^2-6 a c\right )}{32 c^3 d^4}+\frac{b x^2}{32 c^2 d^4}+\frac{x^3}{48 c d^4}$

[Out]

-((b^2 - 6*a*c)*x)/(32*c^3*d^4) + (b*x^2)/(32*c^2*d^4) + x^3/(48*c*d^4) + (b^2 - 4*a*c)^3/(384*c^4*d^4*(b + 2*
c*x)^3) - (3*(b^2 - 4*a*c)^2)/(128*c^4*d^4*(b + 2*c*x))

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Rubi [A]  time = 0.0941266, antiderivative size = 103, 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} $\frac{\left (b^2-4 a c\right )^3}{384 c^4 d^4 (b+2 c x)^3}-\frac{3 \left (b^2-4 a c\right )^2}{128 c^4 d^4 (b+2 c x)}-\frac{x \left (b^2-6 a c\right )}{32 c^3 d^4}+\frac{b x^2}{32 c^2 d^4}+\frac{x^3}{48 c d^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^4,x]

[Out]

-((b^2 - 6*a*c)*x)/(32*c^3*d^4) + (b*x^2)/(32*c^2*d^4) + x^3/(48*c*d^4) + (b^2 - 4*a*c)^3/(384*c^4*d^4*(b + 2*
c*x)^3) - (3*(b^2 - 4*a*c)^2)/(128*c^4*d^4*(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)^4} \, dx &=\int \left (\frac{-b^2+6 a c}{32 c^3 d^4}+\frac{b x}{16 c^2 d^4}+\frac{x^2}{16 c d^4}+\frac{\left (-b^2+4 a c\right )^3}{64 c^3 d^4 (b+2 c x)^4}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^4 (b+2 c x)^2}\right ) \, dx\\ &=-\frac{\left (b^2-6 a c\right ) x}{32 c^3 d^4}+\frac{b x^2}{32 c^2 d^4}+\frac{x^3}{48 c d^4}+\frac{\left (b^2-4 a c\right )^3}{384 c^4 d^4 (b+2 c x)^3}-\frac{3 \left (b^2-4 a c\right )^2}{128 c^4 d^4 (b+2 c x)}\\ \end{align*}

Mathematica [A]  time = 0.064124, size = 81, normalized size = 0.79 $\frac{\frac{\left (b^2-4 a c\right )^3}{(b+2 c x)^3}-\frac{9 \left (b^2-4 a c\right )^2}{b+2 c x}+12 c x \left (6 a c-b^2\right )+12 b c^2 x^2+8 c^3 x^3}{384 c^4 d^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^4,x]

[Out]

(12*c*(-b^2 + 6*a*c)*x + 12*b*c^2*x^2 + 8*c^3*x^3 + (b^2 - 4*a*c)^3/(b + 2*c*x)^3 - (9*(b^2 - 4*a*c)^2)/(b + 2
*c*x))/(384*c^4*d^4)

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Maple [A]  time = 0.046, size = 116, normalized size = 1.1 \begin{align*}{\frac{1}{{d}^{4}} \left ({\frac{1}{32\,{c}^{3}} \left ({\frac{2\,{x}^{3}{c}^{2}}{3}}+bc{x}^{2}+6\,acx-{b}^{2}x \right ) }-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{384\,{c}^{4} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{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)^4,x)

[Out]

1/d^4*(1/32/c^3*(2/3*x^3*c^2+b*c*x^2+6*a*c*x-b^2*x)-1/384*(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)^3-1/128*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b))

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Maxima [A]  time = 1.1684, size = 236, normalized size = 2.29 \begin{align*} -\frac{2 \, b^{6} - 15 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + 9 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 9 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{96 \,{\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\right )}} + \frac{2 \, c^{2} x^{3} + 3 \, b c x^{2} - 3 \,{\left (b^{2} - 6 \, a c\right )} x}{96 \, c^{3} d^{4}} \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)^4,x, algorithm="maxima")

[Out]

-1/96*(2*b^6 - 15*a*b^4*c + 24*a^2*b^2*c^2 + 16*a^3*c^3 + 9*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 9*(b^5*
c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x)/(8*c^7*d^4*x^3 + 12*b*c^6*d^4*x^2 + 6*b^2*c^5*d^4*x + b^3*c^4*d^4) + 1/96*(
2*c^2*x^3 + 3*b*c*x^2 - 3*(b^2 - 6*a*c)*x)/(c^3*d^4)

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Fricas [B]  time = 1.92167, size = 417, normalized size = 4.05 \begin{align*} \frac{16 \, c^{6} x^{6} + 48 \, b c^{5} x^{5} - 2 \, b^{6} + 15 \, a b^{4} c - 24 \, a^{2} b^{2} c^{2} - 16 \, a^{3} c^{3} + 24 \,{\left (b^{2} c^{4} + 6 \, a c^{5}\right )} x^{4} - 8 \,{\left (2 \, b^{3} c^{3} - 27 \, a b c^{4}\right )} x^{3} - 12 \,{\left (2 \, b^{4} c^{2} - 15 \, a b^{2} c^{3} + 12 \, a^{2} c^{4}\right )} x^{2} - 6 \,{\left (2 \, b^{5} c - 15 \, a b^{3} c^{2} + 24 \, a^{2} b c^{3}\right )} x}{96 \,{\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\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)^4,x, algorithm="fricas")

[Out]

1/96*(16*c^6*x^6 + 48*b*c^5*x^5 - 2*b^6 + 15*a*b^4*c - 24*a^2*b^2*c^2 - 16*a^3*c^3 + 24*(b^2*c^4 + 6*a*c^5)*x^
4 - 8*(2*b^3*c^3 - 27*a*b*c^4)*x^3 - 12*(2*b^4*c^2 - 15*a*b^2*c^3 + 12*a^2*c^4)*x^2 - 6*(2*b^5*c - 15*a*b^3*c^
2 + 24*a^2*b*c^3)*x)/(8*c^7*d^4*x^3 + 12*b*c^6*d^4*x^2 + 6*b^2*c^5*d^4*x + b^3*c^4*d^4)

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Sympy [A]  time = 2.40521, size = 185, normalized size = 1.8 \begin{align*} \frac{b x^{2}}{32 c^{2} d^{4}} - \frac{16 a^{3} c^{3} + 24 a^{2} b^{2} c^{2} - 15 a b^{4} c + 2 b^{6} + x^{2} \left (144 a^{2} c^{4} - 72 a b^{2} c^{3} + 9 b^{4} c^{2}\right ) + x \left (144 a^{2} b c^{3} - 72 a b^{3} c^{2} + 9 b^{5} c\right )}{96 b^{3} c^{4} d^{4} + 576 b^{2} c^{5} d^{4} x + 1152 b c^{6} d^{4} x^{2} + 768 c^{7} d^{4} x^{3}} + \frac{x^{3}}{48 c d^{4}} + \frac{x \left (6 a c - b^{2}\right )}{32 c^{3} d^{4}} \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)**4,x)

[Out]

b*x**2/(32*c**2*d**4) - (16*a**3*c**3 + 24*a**2*b**2*c**2 - 15*a*b**4*c + 2*b**6 + x**2*(144*a**2*c**4 - 72*a*
b**2*c**3 + 9*b**4*c**2) + x*(144*a**2*b*c**3 - 72*a*b**3*c**2 + 9*b**5*c))/(96*b**3*c**4*d**4 + 576*b**2*c**5
*d**4*x + 1152*b*c**6*d**4*x**2 + 768*c**7*d**4*x**3) + x**3/(48*c*d**4) + x*(6*a*c - b**2)/(32*c**3*d**4)

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Giac [A]  time = 1.201, size = 221, normalized size = 2.15 \begin{align*} -\frac{9 \, b^{4} c^{2} x^{2} - 72 \, a b^{2} c^{3} x^{2} + 144 \, a^{2} c^{4} x^{2} + 9 \, b^{5} c x - 72 \, a b^{3} c^{2} x + 144 \, a^{2} b c^{3} x + 2 \, b^{6} - 15 \, a b^{4} c + 24 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}}{96 \,{\left (2 \, c x + b\right )}^{3} c^{4} d^{4}} + \frac{2 \, c^{11} d^{8} x^{3} + 3 \, b c^{10} d^{8} x^{2} - 3 \, b^{2} c^{9} d^{8} x + 18 \, a c^{10} d^{8} x}{96 \, c^{12} d^{12}} \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)^4,x, algorithm="giac")

[Out]

-1/96*(9*b^4*c^2*x^2 - 72*a*b^2*c^3*x^2 + 144*a^2*c^4*x^2 + 9*b^5*c*x - 72*a*b^3*c^2*x + 144*a^2*b*c^3*x + 2*b
^6 - 15*a*b^4*c + 24*a^2*b^2*c^2 + 16*a^3*c^3)/((2*c*x + b)^3*c^4*d^4) + 1/96*(2*c^11*d^8*x^3 + 3*b*c^10*d^8*x
^2 - 3*b^2*c^9*d^8*x + 18*a*c^10*d^8*x)/(c^12*d^12)