### 3.1823 $$\int \frac{(a+b x)^3}{(a c+(b c+a d) x+b d x^2)^3} \, dx$$

Optimal. Leaf size=14 $-\frac{1}{2 d (c+d x)^2}$

[Out]

-1/(2*d*(c + d*x)^2)

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Rubi [A]  time = 0.0081064, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.069, Rules used = {626, 32} $-\frac{1}{2 d (c+d x)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(2*d*(c + d*x)^2)

Rule 626

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{(a+b x)^3}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=\int \frac{1}{(c+d x)^3} \, dx\\ &=-\frac{1}{2 d (c+d x)^2}\\ \end{align*}

Mathematica [A]  time = 0.002949, size = 14, normalized size = 1. $-\frac{1}{2 d (c+d x)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-1/(2*d*(c + d*x)^2)

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Maple [A]  time = 0.039, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,d \left ( dx+c \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/d/(d*x+c)^2

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Maxima [A]  time = 1.11179, size = 32, normalized size = 2.29 \begin{align*} -\frac{1}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(d^3*x^2 + 2*c*d^2*x + c^2*d)

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Fricas [A]  time = 1.59199, size = 49, normalized size = 3.5 \begin{align*} -\frac{1}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(d^3*x^2 + 2*c*d^2*x + c^2*d)

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Sympy [B]  time = 0.45007, size = 26, normalized size = 1.86 \begin{align*} - \frac{1}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*c**2*d + 4*c*d**2*x + 2*d**3*x**2)

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Giac [A]  time = 1.27555, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2 \,{\left (d x + c\right )}^{2} d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/((d*x + c)^2*d)