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

Optimal. Leaf size=47 \[ -\frac {a B+A b}{6 x^6}-\frac {a A}{7 x^7}-\frac {A c+b B}{5 x^5}-\frac {B c}{4 x^4} \]

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Rubi [A]  time = 0.03, 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}{6 x^6}-\frac {a A}{7 x^7}-\frac {A c+b B}{5 x^5}-\frac {B c}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*A)/(7*x^7) - (A*b + 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 (a+b x+c x^2\right )}{x^8} \, dx &=\int \left (\frac {a A}{x^8}+\frac {A b+a B}{x^7}+\frac {b B+A c}{x^6}+\frac {B c}{x^5}\right ) \, dx\\ &=-\frac {a A}{7 x^7}-\frac {A b+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 = 0.02, size = 46, normalized size = 0.98 \begin {gather*} -\frac {10 a (6 A+7 B x)+7 x (2 A (5 b+6 c x)+3 B x (4 b+5 c x))}{420 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/420*(10*a*(6*A + 7*B*x) + 7*x*(3*B*x*(4*b + 5*c*x) + 2*A*(5*b + 6*c*x)))/x^7

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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^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.38, size = 39, normalized size = 0.83 \begin {gather*} -\frac {105 \, B c x^{3} + 84 \, {\left (B b + A c\right )} x^{2} + 60 \, A a + 70 \, {\left (B a + A b\right )} x}{420 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/420*(105*B*c*x^3 + 84*(B*b + A*c)*x^2 + 60*A*a + 70*(B*a + A*b)*x)/x^7

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giac [A]  time = 0.17, size = 41, normalized size = 0.87 \begin {gather*} -\frac {105 \, B c x^{3} + 84 \, B b x^{2} + 84 \, A c x^{2} + 70 \, B a x + 70 \, A b x + 60 \, A a}{420 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/420*(105*B*c*x^3 + 84*B*b*x^2 + 84*A*c*x^2 + 70*B*a*x + 70*A*b*x + 60*A*a)/x^7

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maple [A]  time = 0.06, size = 40, normalized size = 0.85 \begin {gather*} -\frac {B c}{4 x^{4}}-\frac {A a}{7 x^{7}}-\frac {A c +b B}{5 x^{5}}-\frac {A b +B a}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(A*c+B*b)/x^5-1/4*B*c/x^4-1/7*a*A/x^7-1/6*(A*b+B*a)/x^6

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maxima [A]  time = 0.49, size = 39, normalized size = 0.83 \begin {gather*} -\frac {105 \, B c x^{3} + 84 \, {\left (B b + A c\right )} x^{2} + 60 \, A a + 70 \, {\left (B a + A b\right )} x}{420 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/420*(105*B*c*x^3 + 84*(B*b + A*c)*x^2 + 60*A*a + 70*(B*a + A*b)*x)/x^7

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mupad [B]  time = 0.03, size = 41, normalized size = 0.87 \begin {gather*} -\frac {\frac {B\,c\,x^3}{4}+\left (\frac {A\,c}{5}+\frac {B\,b}{5}\right )\,x^2+\left (\frac {A\,b}{6}+\frac {B\,a}{6}\right )\,x+\frac {A\,a}{7}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((A*a)/7 + x*((A*b)/6 + (B*a)/6) + x^2*((A*c)/5 + (B*b)/5) + (B*c*x^3)/4)/x^7

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sympy [A]  time = 3.87, size = 46, normalized size = 0.98 \begin {gather*} \frac {- 60 A a - 105 B c x^{3} + x^{2} \left (- 84 A c - 84 B b\right ) + x \left (- 70 A b - 70 B a\right )}{420 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-60*A*a - 105*B*c*x**3 + x**2*(-84*A*c - 84*B*b) + x*(-70*A*b - 70*B*a))/(420*x**7)

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