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

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

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \begin {gather*} \log (x) (A c+b B)-\frac {A b}{x}+B c x \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.00 \begin {gather*} \log (x) (A c+b B)-\frac {A b}{x}+B c x \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (b x+c x^2\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 26, normalized size = 1.18 \begin {gather*} \frac {B c x^{2} + {\left (B b + A c\right )} x \log \relax (x) - A b}{x} \end {gather*}

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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giac [A]  time = 0.15, size = 23, normalized size = 1.05 \begin {gather*} B c x + {\left (B b + A c\right )} \log \left ({\left | x \right |}\right ) - \frac {A b}{x} \end {gather*}

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

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maple [A]  time = 0.05, size = 23, normalized size = 1.05 \begin {gather*} A c \ln \relax (x )+B b \ln \relax (x )+B c x -\frac {A b}{x} \end {gather*}

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

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maxima [A]  time = 0.87, size = 22, normalized size = 1.00 \begin {gather*} B c x + {\left (B b + A c\right )} \log \relax (x) - \frac {A b}{x} \end {gather*}

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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mupad [B]  time = 0.04, size = 22, normalized size = 1.00 \begin {gather*} \ln \relax (x)\,\left (A\,c+B\,b\right )+B\,c\,x-\frac {A\,b}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)*(A + B*x))/x^3,x)

[Out]

log(x)*(A*c + B*b) + B*c*x - (A*b)/x

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sympy [A]  time = 0.15, size = 19, normalized size = 0.86 \begin {gather*} - \frac {A b}{x} + B c x + \left (A c + B b\right ) \log {\relax (x )} \end {gather*}

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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