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

Optimal. Leaf size=24 \[ x (A c+b B)+A b \log (x)+\frac{1}{2} B c x^2 \]

[Out]

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

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Rubi [A]  time = 0.0127155, antiderivative size = 24, 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} \[ x (A c+b B)+A b \log (x)+\frac{1}{2} B c x^2 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^2,x]

[Out]

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

Mathematica [A]  time = 0.005294, size = 24, normalized size = 1. \[ x (A c+b B)+A b \log (x)+\frac{1}{2} B c x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 22, normalized size = 0.9 \begin{align*}{\frac{Bc{x}^{2}}{2}}+Acx+bBx+Ab\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.967713, size = 30, normalized size = 1.25 \begin{align*} \frac{1}{2} \, B c x^{2} + A b \log \left (x\right ) +{\left (B b + A c\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.75176, size = 57, normalized size = 2.38 \begin{align*} \frac{1}{2} \, B c x^{2} + A b \log \left (x\right ) +{\left (B b + A c\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.29136, size = 22, normalized size = 0.92 \begin{align*} A b \log{\left (x \right )} + \frac{B c x^{2}}{2} + x \left (A c + B b\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14413, size = 30, normalized size = 1.25 \begin{align*} \frac{1}{2} \, B c x^{2} + B b x + A c x + A b \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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