3.8 \(\int \frac{(A+B x) (b x+c x^2)}{x^3} \, dx\)

Optimal. Leaf size=22 \[ \log (x) (A c+b B)-\frac{A b}{x}+B c x \]

[Out]

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

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Rubi [A]  time = 0.0158835, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {765} \[ \log (x) (A c+b B)-\frac{A b}{x}+B c x \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^3,x]

[Out]

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

Mathematica [A]  time = 0.008849, size = 22, normalized size = 1. \[ \log (x) (A c+b B)-\frac{A b}{x}+B c x \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/x^3,x]

[Out]

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

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Maple [A]  time = 0.02, size = 23, normalized size = 1.1 \begin{align*} Bcx+Ac\ln \left ( x \right ) +B\ln \left ( x \right ) b-{\frac{Ab}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)/x^3,x)

[Out]

B*c*x+A*c*ln(x)+B*ln(x)*b-A*b/x

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Maxima [A]  time = 0.965725, size = 30, normalized size = 1.36 \begin{align*} B c x +{\left (B b + A c\right )} \log \left (x\right ) - \frac{A b}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^3,x, algorithm="maxima")

[Out]

B*c*x + (B*b + A*c)*log(x) - A*b/x

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Fricas [A]  time = 1.77541, size = 57, normalized size = 2.59 \begin{align*} \frac{B c x^{2} +{\left (B b + A c\right )} x \log \left (x\right ) - A b}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.31423, size = 19, normalized size = 0.86 \begin{align*} - \frac{A b}{x} + B c x + \left (A c + B b\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)/x**3,x)

[Out]

-A*b/x + B*c*x + (A*c + B*b)*log(x)

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Giac [A]  time = 1.18852, size = 31, normalized size = 1.41 \begin{align*} B c x +{\left (B b + A c\right )} \log \left ({\left | x \right |}\right ) - \frac{A b}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^3,x, algorithm="giac")

[Out]

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