3.11 \(\int \frac {(A+B x) (b x+c x^2)}{x^6} \, dx\)

Optimal. Leaf size=33 \[ -\frac {A c+b B}{3 x^3}-\frac {A b}{4 x^4}-\frac {B c}{2 x^2} \]

[Out]

-1/4*A*b/x^4+1/3*(-A*c-B*b)/x^3-1/2*B*c/x^2

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Rubi [A]  time = 0.02, antiderivative size = 33, 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} \[ -\frac {A c+b B}{3 x^3}-\frac {A b}{4 x^4}-\frac {B c}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^6,x]

[Out]

-(A*b)/(4*x^4) - (b*B + A*c)/(3*x^3) - (B*c)/(2*x^2)

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^6} \, dx &=\int \left (\frac {A b}{x^5}+\frac {b B+A c}{x^4}+\frac {B c}{x^3}\right ) \, dx\\ &=-\frac {A b}{4 x^4}-\frac {b B+A c}{3 x^3}-\frac {B c}{2 x^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 0.88 \[ -\frac {3 A b+4 A c x+4 b B x+6 B c x^2}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/x^6,x]

[Out]

-1/12*(3*A*b + 4*b*B*x + 4*A*c*x + 6*B*c*x^2)/x^4

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fricas [A]  time = 0.88, size = 27, normalized size = 0.82 \[ -\frac {6 \, B c x^{2} + 3 \, A b + 4 \, {\left (B b + A c\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^6,x, algorithm="fricas")

[Out]

-1/12*(6*B*c*x^2 + 3*A*b + 4*(B*b + A*c)*x)/x^4

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giac [A]  time = 0.15, size = 27, normalized size = 0.82 \[ -\frac {6 \, B c x^{2} + 4 \, B b x + 4 \, A c x + 3 \, A b}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^6,x, algorithm="giac")

[Out]

-1/12*(6*B*c*x^2 + 4*B*b*x + 4*A*c*x + 3*A*b)/x^4

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maple [A]  time = 0.05, size = 28, normalized size = 0.85 \[ -\frac {B c}{2 x^{2}}-\frac {A b}{4 x^{4}}-\frac {A c +b B}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)/x^6,x)

[Out]

-1/4*A*b/x^4-1/3*(A*c+B*b)/x^3-1/2*B*c/x^2

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maxima [A]  time = 0.90, size = 27, normalized size = 0.82 \[ -\frac {6 \, B c x^{2} + 3 \, A b + 4 \, {\left (B b + A c\right )} x}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^6,x, algorithm="maxima")

[Out]

-1/12*(6*B*c*x^2 + 3*A*b + 4*(B*b + A*c)*x)/x^4

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mupad [B]  time = 0.04, size = 28, normalized size = 0.85 \[ -\frac {\frac {B\,c\,x^2}{2}+\left (\frac {A\,c}{3}+\frac {B\,b}{3}\right )\,x+\frac {A\,b}{4}}{x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)*(A + B*x))/x^6,x)

[Out]

-((A*b)/4 + x*((A*c)/3 + (B*b)/3) + (B*c*x^2)/2)/x^4

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sympy [A]  time = 0.34, size = 31, normalized size = 0.94 \[ \frac {- 3 A b - 6 B c x^{2} + x \left (- 4 A c - 4 B b\right )}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)/x**6,x)

[Out]

(-3*A*b - 6*B*c*x**2 + x*(-4*A*c - 4*B*b))/(12*x**4)

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