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]

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

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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} \[ -\frac {A c+b B}{5 x^5}-\frac {A b}{6 x^6}-\frac {B c}{4 x^4} \]

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*}

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Mathematica [A]  time = 0.01, 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]

-1/60*(3*B*x*(4*b + 5*c*x) + 2*A*(5*b + 6*c*x))/x^6

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fricas [A]  time = 0.70, size = 27, normalized size = 0.82 \[ -\frac {15 \, B c x^{2} + 10 \, A b + 12 \, {\left (B b + A c\right )} x}{60 \, x^{6}} \]

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

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giac [A]  time = 0.15, size = 27, normalized size = 0.82 \[ -\frac {15 \, B c x^{2} + 12 \, B b x + 12 \, A c x + 10 \, A b}{60 \, x^{6}} \]

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

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maple [A]  time = 0.05, size = 28, normalized size = 0.85 \[ -\frac {B c}{4 x^{4}}-\frac {A b}{6 x^{6}}-\frac {A c +b B}{5 x^{5}} \]

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

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maxima [A]  time = 0.88, size = 27, normalized size = 0.82 \[ -\frac {15 \, B c x^{2} + 10 \, A b + 12 \, {\left (B b + A c\right )} x}{60 \, x^{6}} \]

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

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mupad [B]  time = 0.04, size = 28, normalized size = 0.85 \[ -\frac {\frac {B\,c\,x^2}{4}+\left (\frac {A\,c}{5}+\frac {B\,b}{5}\right )\,x+\frac {A\,b}{6}}{x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)*(A + B*x))/x^8,x)

[Out]

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

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sympy [A]  time = 0.49, size = 31, normalized size = 0.94 \[ \frac {- 10 A b - 15 B c x^{2} + x \left (- 12 A c - 12 B b\right )}{60 x^{6}} \]

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)

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