### 3.1765 $$\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^3} \, dx$$

Optimal. Leaf size=32 $\frac{d \log (a+b x)}{b^2}-\frac{b c-a d}{b^2 (a+b x)}$

[Out]

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

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Rubi [A]  time = 0.024534, antiderivative size = 32, 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} $\frac{d \log (a+b x)}{b^2}-\frac{b c-a d}{b^2 (a+b x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0101296, size = 31, normalized size = 0.97 $\frac{a d-b c}{b^2 (a+b x)}+\frac{d \log (a+b x)}{b^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.045, size = 39, normalized size = 1.2 \begin{align*}{\frac{d\ln \left ( bx+a \right ) }{{b}^{2}}}+{\frac{ad}{{b}^{2} \left ( bx+a \right ) }}-{\frac{c}{b \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)^3,x)

[Out]

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

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Maxima [A]  time = 1.0242, size = 47, normalized size = 1.47 \begin{align*} -\frac{b c - a d}{b^{3} x + a b^{2}} + \frac{d \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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.82422, size = 80, normalized size = 2.5 \begin{align*} -\frac{b c - a d -{\left (b d x + a d\right )} \log \left (b x + a\right )}{b^{3} x + a 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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.510869, size = 27, normalized size = 0.84 \begin{align*} \frac{a d - b c}{a b^{2} + b^{3} x} + \frac{d \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)**3,x)

[Out]

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

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Giac [A]  time = 1.23979, size = 45, normalized size = 1.41 \begin{align*} \frac{d \log \left ({\left | b x + a \right |}\right )}{b^{2}} - \frac{b c - a d}{{\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)^3,x, algorithm="giac")

[Out]

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