### 3.1822 $$\int \frac{(a+b x)^4}{(a c+(b c+a d) x+b d x^2)^3} \, dx$$

Optimal. Leaf size=28 $\frac{(a+b x)^2}{2 (c+d x)^2 (b c-a d)}$

[Out]

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

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Rubi [A]  time = 0.0093834, antiderivative size = 28, 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, 37} $\frac{(a+b x)^2}{2 (c+d x)^2 (b c-a d)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a + b*x)^2/(2*(b*c - a*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 37

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

Rubi steps

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

Mathematica [A]  time = 0.0087586, size = 26, normalized size = 0.93 $-\frac{a d+b (c+2 d x)}{2 d^2 (c+d x)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 35, normalized size = 1.3 \begin{align*} -{\frac{ad-bc}{2\,{d}^{2} \left ( dx+c \right ) ^{2}}}-{\frac{b}{{d}^{2} \left ( dx+c \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x)

[Out]

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

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Maxima [A]  time = 1.08441, size = 51, normalized size = 1.82 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.62033, size = 81, normalized size = 2.89 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.567457, size = 39, normalized size = 1.39 \begin{align*} - \frac{a d + b c + 2 b d x}{2 c^{2} d^{2} + 4 c d^{3} x + 2 d^{4} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4/(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

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

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Giac [A]  time = 1.17273, size = 32, normalized size = 1.14 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (d x + c\right )}^{2} d^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/(a*c+(a*d+b*c)*x+b*d*x^2)^3,x, algorithm="giac")

[Out]

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