3.196 \(\int \frac{x (A+B x)}{(a+b x)^3} \, dx\)

Optimal. Leaf size=55 \[ -\frac{A b-2 a B}{b^3 (a+b x)}+\frac{a (A b-a B)}{2 b^3 (a+b x)^2}+\frac{B \log (a+b x)}{b^3} \]

[Out]

(a*(A*b - a*B))/(2*b^3*(a + b*x)^2) - (A*b - 2*a*B)/(b^3*(a + b*x)) + (B*Log[a + b*x])/b^3

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Rubi [A]  time = 0.0377139, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {77} \[ -\frac{A b-2 a B}{b^3 (a+b x)}+\frac{a (A b-a B)}{2 b^3 (a+b x)^2}+\frac{B \log (a+b x)}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[(x*(A + B*x))/(a + b*x)^3,x]

[Out]

(a*(A*b - a*B))/(2*b^3*(a + b*x)^2) - (A*b - 2*a*B)/(b^3*(a + b*x)) + (B*Log[a + b*x])/b^3

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0163888, size = 54, normalized size = 0.98 \[ \frac{3 a^2 B-a b (A-4 B x)+2 B (a+b x)^2 \log (a+b x)-2 A b^2 x}{2 b^3 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(A + B*x))/(a + b*x)^3,x]

[Out]

(3*a^2*B - 2*A*b^2*x - a*b*(A - 4*B*x) + 2*B*(a + b*x)^2*Log[a + b*x])/(2*b^3*(a + b*x)^2)

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Maple [A]  time = 0.004, size = 70, normalized size = 1.3 \begin{align*} -{\frac{A}{{b}^{2} \left ( bx+a \right ) }}+2\,{\frac{Ba}{{b}^{3} \left ( bx+a \right ) }}+{\frac{Aa}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{B{a}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{B\ln \left ( bx+a \right ) }{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(B*x+A)/(b*x+a)^3,x)

[Out]

-1/b^2/(b*x+a)*A+2/b^3/(b*x+a)*B*a+1/2*a/b^2/(b*x+a)^2*A-1/2*a^2/b^3/(b*x+a)^2*B+B*ln(b*x+a)/b^3

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Maxima [A]  time = 1.04585, size = 88, normalized size = 1.6 \begin{align*} \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac{B \log \left (b x + a\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)/(b*x+a)^3,x, algorithm="maxima")

[Out]

1/2*(3*B*a^2 - A*a*b + 2*(2*B*a*b - A*b^2)*x)/(b^5*x^2 + 2*a*b^4*x + a^2*b^3) + B*log(b*x + a)/b^3

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Fricas [A]  time = 1.94273, size = 173, normalized size = 3.15 \begin{align*} \frac{3 \, B a^{2} - A a b + 2 \,{\left (2 \, B a b - A b^{2}\right )} x + 2 \,{\left (B b^{2} x^{2} + 2 \, B a b x + B a^{2}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/2*(3*B*a^2 - A*a*b + 2*(2*B*a*b - A*b^2)*x + 2*(B*b^2*x^2 + 2*B*a*b*x + B*a^2)*log(b*x + a))/(b^5*x^2 + 2*a*
b^4*x + a^2*b^3)

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Sympy [A]  time = 0.628659, size = 63, normalized size = 1.15 \begin{align*} \frac{B \log{\left (a + b x \right )}}{b^{3}} + \frac{- A a b + 3 B a^{2} + x \left (- 2 A b^{2} + 4 B a b\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)/(b*x+a)**3,x)

[Out]

B*log(a + b*x)/b**3 + (-A*a*b + 3*B*a**2 + x*(-2*A*b**2 + 4*B*a*b))/(2*a**2*b**3 + 4*a*b**4*x + 2*b**5*x**2)

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Giac [A]  time = 1.18719, size = 73, normalized size = 1.33 \begin{align*} \frac{B \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac{2 \,{\left (2 \, B a - A b\right )} x + \frac{3 \, B a^{2} - A a b}{b}}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)/(b*x+a)^3,x, algorithm="giac")

[Out]

B*log(abs(b*x + a))/b^3 + 1/2*(2*(2*B*a - A*b)*x + (3*B*a^2 - A*a*b)/b)/((b*x + a)^2*b^2)