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/(d*x+c)^2

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \[ -\frac {1}{2 d (c+d x)^2} \]

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*1/(d*(c + d*x)^2)

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fricas [A]  time = 1.10, size = 24, normalized size = 1.71 \[ -\frac {1}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]

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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giac [A]  time = 0.17, size = 12, normalized size = 0.86 \[ -\frac {1}{2 \, {\left (d x + c\right )}^{2} d} \]

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)

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maple [A]  time = 0.04, size = 13, normalized size = 0.93 \[ -\frac {1}{2 \left (d x +c \right )^{2} d} \]

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 = 1.04, size = 24, normalized size = 1.71 \[ -\frac {1}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]

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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mupad [B]  time = 0.55, size = 26, normalized size = 1.86 \[ -\frac {1}{2\,c^2\,d+4\,c\,d^2\,x+2\,d^3\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*c^2*d + 2*d^3*x^2 + 4*c*d^2*x)

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sympy [B]  time = 0.26, size = 26, normalized size = 1.86 \[ - \frac {1}{2 c^{2} d + 4 c d^{2} x + 2 d^{3} x^{2}} \]

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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