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]

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

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Rubi [A]  time = 0.016918, antiderivative size = 33, 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} \[ -\frac{A c+b B}{3 x^3}-\frac{A b}{4 x^4}-\frac{B c}{2 x^2} \]

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*}

Mathematica [A]  time = 0.0074544, 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]

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

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Maple [A]  time = 0.005, size = 28, normalized size = 0.9 \begin{align*} -{\frac{Ab}{4\,{x}^{4}}}-{\frac{Bc}{2\,{x}^{2}}}-{\frac{Ac+bB}{3\,{x}^{3}}} \end{align*}

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

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Maxima [A]  time = 0.987884, size = 36, normalized size = 1.09 \begin{align*} -\frac{6 \, B c x^{2} + 3 \, A b + 4 \,{\left (B b + A c\right )} x}{12 \, x^{4}} \end{align*}

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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Fricas [A]  time = 1.83952, size = 66, normalized size = 2. \begin{align*} -\frac{6 \, B c x^{2} + 3 \, A b + 4 \,{\left (B b + A c\right )} x}{12 \, x^{4}} \end{align*}

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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Sympy [A]  time = 0.555558, size = 31, normalized size = 0.94 \begin{align*} - \frac{3 A b + 6 B c x^{2} + x \left (4 A c + 4 B b\right )}{12 x^{4}} \end{align*}

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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Giac [A]  time = 1.12281, size = 36, normalized size = 1.09 \begin{align*} -\frac{6 \, B c x^{2} + 4 \, B b x + 4 \, A c x + 3 \, A b}{12 \, x^{4}} \end{align*}

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