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

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

[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)

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Rubi [A]  time = 0.0254179, 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} \[ -\frac{a B+A b}{6 x^6}-\frac{a A}{7 x^7}-\frac{A c+b B}{5 x^5}-\frac{B c}{4 x^4} \]

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*}

Mathematica [A]  time = 0.017483, size = 46, normalized size = 0.98 \[ -\frac{10 a (6 A+7 B x)+7 x (2 A (5 b+6 c x)+3 B x (4 b+5 c x))}{420 x^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 40, normalized size = 0.9 \begin{align*} -{\frac{aA}{7\,{x}^{7}}}-{\frac{Ac+bB}{5\,{x}^{5}}}-{\frac{Bc}{4\,{x}^{4}}}-{\frac{Ab+aB}{6\,{x}^{6}}} \end{align*}

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]

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

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Maxima [A]  time = 1.07298, size = 53, normalized size = 1.13 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.17057, size = 101, normalized size = 2.15 \begin{align*} -\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{align*}

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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Sympy [A]  time = 5.51216, size = 44, normalized size = 0.94 \begin{align*} - \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{align*}

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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Giac [A]  time = 1.24719, size = 55, normalized size = 1.17 \begin{align*} -\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{align*}

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