3.848 \(\int \frac{(A+B x) (a+b x+c x^2)}{x^2} \, dx\)

Optimal. Leaf size=36 \[ \log (x) (a B+A b)-\frac{a A}{x}+x (A c+b B)+\frac{1}{2} B c x^2 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0252918, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {765} \[ \log (x) (a B+A b)-\frac{a A}{x}+x (A c+b B)+\frac{1}{2} B c x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )}{x^2} \, dx &=\int \left (b B \left (1+\frac{A c}{b B}\right )+\frac{a A}{x^2}+\frac{A b+a B}{x}+B c x\right ) \, dx\\ &=-\frac{a A}{x}+(b B+A c) x+\frac{1}{2} B c x^2+(A b+a B) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.016727, size = 36, normalized size = 1. \[ \log (x) (a B+A b)-\frac{a A}{x}+x (A c+b B)+\frac{1}{2} B c x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 34, normalized size = 0.9 \begin{align*}{\frac{Bc{x}^{2}}{2}}+Acx+bBx+Ab\ln \left ( x \right ) +aB\ln \left ( x \right ) -{\frac{aA}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)/x^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.07078, size = 46, normalized size = 1.28 \begin{align*} \frac{1}{2} \, B c x^{2} +{\left (B b + A c\right )} x +{\left (B a + A b\right )} \log \left (x\right ) - \frac{A a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.24768, size = 95, normalized size = 2.64 \begin{align*} \frac{B c x^{3} + 2 \,{\left (B b + A c\right )} x^{2} + 2 \,{\left (B a + A b\right )} x \log \left (x\right ) - 2 \, A a}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.376721, size = 31, normalized size = 0.86 \begin{align*} - \frac{A a}{x} + \frac{B c x^{2}}{2} + x \left (A c + B b\right ) + \left (A b + B a\right ) \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)/x**2,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17388, size = 46, normalized size = 1.28 \begin{align*} \frac{1}{2} \, B c x^{2} + B b x + A c x +{\left (B a + A b\right )} \log \left ({\left | x \right |}\right ) - \frac{A a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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