3.1.10 \(\int \frac {(A+B x) (b x+c x^2)}{x^5} \, dx\)

Optimal. Leaf size=31 \[ -\frac {A c+b B}{2 x^2}-\frac {A b}{3 x^3}-\frac {B c}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 31, 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}{2 x^2}-\frac {A b}{3 x^3}-\frac {B c}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^5,x]

[Out]

-(A*b)/(3*x^3) - (b*B + A*c)/(2*x^2) - (B*c)/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^5} \, dx &=\int \left (\frac {A b}{x^4}+\frac {b B+A c}{x^3}+\frac {B c}{x^2}\right ) \, dx\\ &=-\frac {A b}{3 x^3}-\frac {b B+A c}{2 x^2}-\frac {B c}{x}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.90 \begin {gather*} -\frac {A (2 b+3 c x)+3 B x (b+2 c x)}{6 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/x^5,x]

[Out]

-1/6*(3*B*x*(b + 2*c*x) + A*(2*b + 3*c*x))/x^3

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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^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2))/x^5,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2))/x^5, x]

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fricas [A]  time = 0.39, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B c x^{2} + 2 \, A b + 3 \, {\left (B b + A c\right )} x}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^5,x, algorithm="fricas")

[Out]

-1/6*(6*B*c*x^2 + 2*A*b + 3*(B*b + A*c)*x)/x^3

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giac [A]  time = 0.15, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B c x^{2} + 3 \, B b x + 3 \, A c x + 2 \, A b}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^5,x, algorithm="giac")

[Out]

-1/6*(6*B*c*x^2 + 3*B*b*x + 3*A*c*x + 2*A*b)/x^3

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maple [A]  time = 0.05, size = 28, normalized size = 0.90 \begin {gather*} -\frac {B c}{x}-\frac {A b}{3 x^{3}}-\frac {A c +b 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^5,x)

[Out]

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

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maxima [A]  time = 0.88, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B c x^{2} + 2 \, A b + 3 \, {\left (B b + A c\right )} x}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^5,x, algorithm="maxima")

[Out]

-1/6*(6*B*c*x^2 + 2*A*b + 3*(B*b + A*c)*x)/x^3

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mupad [B]  time = 0.04, size = 27, normalized size = 0.87 \begin {gather*} -\frac {B\,c\,x^2+\left (\frac {A\,c}{2}+\frac {B\,b}{2}\right )\,x+\frac {A\,b}{3}}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)*(A + B*x))/x^5,x)

[Out]

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

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sympy [A]  time = 0.28, size = 31, normalized size = 1.00 \begin {gather*} \frac {- 2 A b - 6 B c x^{2} + x \left (- 3 A c - 3 B b\right )}{6 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)/x**5,x)

[Out]

(-2*A*b - 6*B*c*x**2 + x*(-3*A*c - 3*B*b))/(6*x**3)

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