3.1.9 \(\int \frac {(A+B x) (b x+c x^2)}{x^4} \, dx\)

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

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Rubi [A]  time = 0.02, antiderivative size = 27, 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*} -\frac {A c+b B}{x}-\frac {A b}{2 x^2}+B c \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^4,x]

[Out]

-(A*b)/(2*x^2) - (b*B + A*c)/x + B*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^4} \, dx &=\int \left (\frac {A b}{x^3}+\frac {b B+A c}{x^2}+\frac {B c}{x}\right ) \, dx\\ &=-\frac {A b}{2 x^2}-\frac {b B+A c}{x}+B c \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.04 \begin {gather*} \frac {-A c-b B}{x}-\frac {A b}{2 x^2}+B c \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/x^4,x]

[Out]

-1/2*(A*b)/x^2 + (-(b*B) - A*c)/x + B*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^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2))/x^4,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2))/x^4, x]

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fricas [A]  time = 0.40, size = 29, normalized size = 1.07 \begin {gather*} \frac {2 \, B c x^{2} \log \relax (x) - A b - 2 \, {\left (B b + A c\right )} x}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^4,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 26, normalized size = 0.96 \begin {gather*} B c \log \left ({\left | x \right |}\right ) - \frac {A b + 2 \, {\left (B b + A c\right )} x}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^4,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 28, normalized size = 1.04 \begin {gather*} B c \ln \relax (x )-\frac {A c}{x}-\frac {B b}{x}-\frac {A b}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)/x^4,x)

[Out]

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

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maxima [A]  time = 0.84, size = 25, normalized size = 0.93 \begin {gather*} B c \log \relax (x) - \frac {A b + 2 \, {\left (B b + A c\right )} x}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^4,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.02, size = 25, normalized size = 0.93 \begin {gather*} B\,c\,\ln \relax (x)-\frac {\frac {A\,b}{2}+x\,\left (A\,c+B\,b\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)*(A + B*x))/x^4,x)

[Out]

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

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sympy [A]  time = 0.22, size = 27, normalized size = 1.00 \begin {gather*} B c \log {\relax (x )} + \frac {- A b + x \left (- 2 A c - 2 B b\right )}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)/x**4,x)

[Out]

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

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