3.13 \(\int \frac{(A+B x) (b x+c x^2)}{x^8} \, dx\)

Optimal. Leaf size=33 \[ -\frac{A c+b B}{5 x^5}-\frac{A b}{6 x^6}-\frac{B c}{4 x^4} \]

[Out]

-(A*b)/(6*x^6) - (b*B + A*c)/(5*x^5) - (B*c)/(4*x^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0160021, 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} \[ -\frac{A c+b B}{5 x^5}-\frac{A b}{6 x^6}-\frac{B c}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^8,x]

[Out]

-(A*b)/(6*x^6) - (b*B + A*c)/(5*x^5) - (B*c)/(4*x^4)

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^8} \, dx &=\int \left (\frac{A b}{x^7}+\frac{b B+A c}{x^6}+\frac{B c}{x^5}\right ) \, dx\\ &=-\frac{A b}{6 x^6}-\frac{b B+A c}{5 x^5}-\frac{B c}{4 x^4}\\ \end{align*}

Mathematica [A]  time = 0.008755, size = 31, normalized size = 0.94 \[ -\frac{2 A (5 b+6 c x)+3 B x (4 b+5 c x)}{60 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2))/x^8,x]

[Out]

-(3*B*x*(4*b + 5*c*x) + 2*A*(5*b + 6*c*x))/(60*x^6)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 28, normalized size = 0.9 \begin{align*} -{\frac{Ac+bB}{5\,{x}^{5}}}-{\frac{Ab}{6\,{x}^{6}}}-{\frac{Bc}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)/x^8,x)

[Out]

-1/5*(A*c+B*b)/x^5-1/6*A*b/x^6-1/4*B*c/x^4

________________________________________________________________________________________

Maxima [A]  time = 1.03401, size = 36, normalized size = 1.09 \begin{align*} -\frac{15 \, B c x^{2} + 10 \, A b + 12 \,{\left (B b + A c\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^8,x, algorithm="maxima")

[Out]

-1/60*(15*B*c*x^2 + 10*A*b + 12*(B*b + A*c)*x)/x^6

________________________________________________________________________________________

Fricas [A]  time = 1.68832, size = 70, normalized size = 2.12 \begin{align*} -\frac{15 \, B c x^{2} + 10 \, A b + 12 \,{\left (B b + A c\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^8,x, algorithm="fricas")

[Out]

-1/60*(15*B*c*x^2 + 10*A*b + 12*(B*b + A*c)*x)/x^6

________________________________________________________________________________________

Sympy [A]  time = 0.646628, size = 31, normalized size = 0.94 \begin{align*} - \frac{10 A b + 15 B c x^{2} + x \left (12 A c + 12 B b\right )}{60 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)/x**8,x)

[Out]

-(10*A*b + 15*B*c*x**2 + x*(12*A*c + 12*B*b))/(60*x**6)

________________________________________________________________________________________

Giac [A]  time = 1.14288, size = 36, normalized size = 1.09 \begin{align*} -\frac{15 \, B c x^{2} + 12 \, B b x + 12 \, A c x + 10 \, A b}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^8,x, algorithm="giac")

[Out]

-1/60*(15*B*c*x^2 + 12*B*b*x + 12*A*c*x + 10*A*b)/x^6