### 3.1764 $$\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^2} \, dx$$

Optimal. Leaf size=25 $\frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0207689, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.074, Rules used = {24, 43} $\frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d*x)/b + ((b*c - a*d)*Log[a + b*x])/b^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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0071398, size = 25, normalized size = 1. $\frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 32, normalized size = 1.3 \begin{align*}{\frac{dx}{b}}-{\frac{\ln \left ( bx+a \right ) ad}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) c}{b}} \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)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.0755, size = 34, normalized size = 1.36 \begin{align*} \frac{d x}{b} + \frac{{\left (b c - a d\right )} \log \left (b x + a\right )}{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)^2,x, algorithm="maxima")

[Out]

d*x/b + (b*c - a*d)*log(b*x + a)/b^2

________________________________________________________________________________________

Fricas [A]  time = 1.71667, size = 54, normalized size = 2.16 \begin{align*} \frac{b d x +{\left (b c - a d\right )} \log \left (b x + a\right )}{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)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.578064, size = 20, normalized size = 0.8 \begin{align*} \frac{d x}{b} - \frac{\left (a d - b c\right ) \log{\left (a + b x \right )}}{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)**2,x)

[Out]

d*x/b - (a*d - b*c)*log(a + b*x)/b**2

________________________________________________________________________________________

Giac [B]  time = 1.1685, size = 158, normalized size = 6.32 \begin{align*} b d{\left (\frac{2 \, a \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} + \frac{b x + a}{b^{3}} - \frac{a^{2}}{{\left (b x + a\right )} b^{3}}\right )} - \frac{{\left (b c + a d\right )}{\left (\frac{\log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x + a\right )} b}\right )}}{b} - \frac{a c}{{\left (b x + a\right )} b} \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)^2,x, algorithm="giac")

[Out]

b*d*(2*a*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^3 + (b*x + a)/b^3 - a^2/((b*x + a)*b^3)) - (b*c + a*d)*(log(
abs(b*x + a)/((b*x + a)^2*abs(b)))/b - a/((b*x + a)*b))/b - a*c/((b*x + a)*b)