### 3.1766 $$\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^4} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0109377, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.074, Rules used = {24, 37} $-\frac{(c+d x)^2}{2 (a+b x)^2 (b c-a d)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 24

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((A_.) + (B_.)*(v_) + (C_.)*(v_)^2), x_Symbol] :> Dist[1/b^2, Int[u*(a + b*
v)^(m + 1)*Simp[b*B - a*C + b*C*v, x], x], x] /; FreeQ[{a, b, A, B, C}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0] &&
LeQ[m, -1]

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

Mathematica [A]  time = 0.009448, size = 26, normalized size = 0.93 $-\frac{a d+b (c+2 d x)}{2 b^2 (a+b x)^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 35, normalized size = 1.3 \begin{align*} -{\frac{-ad+bc}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{d}{{b}^{2} \left ( bx+a \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06197, size = 51, normalized size = 1.82 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.6709, size = 81, normalized size = 2.89 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.524705, size = 39, normalized size = 1.39 \begin{align*} - \frac{a d + b c + 2 b d x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17529, size = 32, normalized size = 1.14 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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