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

Optimal. Leaf size=27 \[ -\frac{a B+A b}{x}-\frac{a A}{2 x^2}+b B \log (x) \]

[Out]

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

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Rubi [A]  time = 0.013042, antiderivative size = 27, 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 = {76} \[ -\frac{a B+A b}{x}-\frac{a A}{2 x^2}+b B \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x))/x^3,x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0105576, size = 28, normalized size = 1.04 \[ \frac{-a B-A b}{x}-\frac{a A}{2 x^2}+b B \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 28, normalized size = 1. \begin{align*} bB\ln \left ( x \right ) -{\frac{Aa}{2\,{x}^{2}}}-{\frac{Ab}{x}}-{\frac{Ba}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15508, size = 34, normalized size = 1.26 \begin{align*} B b \log \left (x\right ) - \frac{A a + 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.9138, size = 70, normalized size = 2.59 \begin{align*} \frac{2 \, B b x^{2} \log \left (x\right ) - A a - 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.42327, size = 26, normalized size = 0.96 \begin{align*} B b \log{\left (x \right )} - \frac{A a + x \left (2 A b + 2 B a\right )}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.2282, size = 35, normalized size = 1.3 \begin{align*} B b \log \left ({\left | x \right |}\right ) - \frac{A a + 2 \,{\left (B a + A b\right )} x}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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