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

Optimal. Leaf size=47 \[ -\frac{a B+A b}{4 x^4}-\frac{a A}{5 x^5}-\frac{A c+b B}{3 x^3}-\frac{B c}{2 x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0243442, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {765} \[ -\frac{a B+A b}{4 x^4}-\frac{a A}{5 x^5}-\frac{A c+b B}{3 x^3}-\frac{B c}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0152054, size = 44, normalized size = 0.94 \[ -\frac{3 a (4 A+5 B x)+5 x \left (3 A b+4 A c x+4 b B x+6 B c x^2\right )}{60 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 40, normalized size = 0.9 \begin{align*} -{\frac{Ac+bB}{3\,{x}^{3}}}-{\frac{Bc}{2\,{x}^{2}}}-{\frac{aA}{5\,{x}^{5}}}-{\frac{Ab+aB}{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+a)/x^6,x)

[Out]

-1/3*(A*c+B*b)/x^3-1/2*B*c/x^2-1/5*a*A/x^5-1/4*(A*b+B*a)/x^4

________________________________________________________________________________________

Maxima [A]  time = 0.99919, size = 53, normalized size = 1.13 \begin{align*} -\frac{30 \, B c x^{3} + 20 \,{\left (B b + A c\right )} x^{2} + 12 \, A a + 15 \,{\left (B a + A b\right )} x}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(30*B*c*x^3 + 20*(B*b + A*c)*x^2 + 12*A*a + 15*(B*a + A*b)*x)/x^5

________________________________________________________________________________________

Fricas [A]  time = 1.13698, size = 99, normalized size = 2.11 \begin{align*} -\frac{30 \, B c x^{3} + 20 \,{\left (B b + A c\right )} x^{2} + 12 \, A a + 15 \,{\left (B a + A b\right )} x}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(30*B*c*x^3 + 20*(B*b + A*c)*x^2 + 12*A*a + 15*(B*a + A*b)*x)/x^5

________________________________________________________________________________________

Sympy [A]  time = 3.4854, size = 44, normalized size = 0.94 \begin{align*} - \frac{12 A a + 30 B c x^{3} + x^{2} \left (20 A c + 20 B b\right ) + x \left (15 A b + 15 B a\right )}{60 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(12*A*a + 30*B*c*x**3 + x**2*(20*A*c + 20*B*b) + x*(15*A*b + 15*B*a))/(60*x**5)

________________________________________________________________________________________

Giac [A]  time = 1.37833, size = 55, normalized size = 1.17 \begin{align*} -\frac{30 \, B c x^{3} + 20 \, B b x^{2} + 20 \, A c x^{2} + 15 \, B a x + 15 \, A b x + 12 \, A a}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(30*B*c*x^3 + 20*B*b*x^2 + 20*A*c*x^2 + 15*B*a*x + 15*A*b*x + 12*A*a)/x^5