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

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

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Rubi [A]  time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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*}

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Mathematica [A]  time = 0.02, size = 44, normalized size = 0.94 \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/60*(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))/x^5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 39, normalized size = 0.83 \begin {gather*} -\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 {gather*}

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

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giac [A]  time = 0.15, size = 41, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.05, size = 40, normalized size = 0.85 \begin {gather*} -\frac {B c}{2 x^{2}}-\frac {A a}{5 x^{5}}-\frac {A c +b B}{3 x^{3}}-\frac {A b +B a}{4 x^{4}} \end {gather*}

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]

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

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maxima [A]  time = 0.67, size = 39, normalized size = 0.83 \begin {gather*} -\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 {gather*}

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

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mupad [B]  time = 0.03, size = 41, normalized size = 0.87 \begin {gather*} -\frac {\frac {B\,c\,x^3}{2}+\left (\frac {A\,c}{3}+\frac {B\,b}{3}\right )\,x^2+\left (\frac {A\,b}{4}+\frac {B\,a}{4}\right )\,x+\frac {A\,a}{5}}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.70, size = 46, normalized size = 0.98 \begin {gather*} \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 {gather*}

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)

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