3.853 \(\int \frac{(A+B x) (a+b x+c x^2)}{x^7} \, dx\)

Optimal. Leaf size=47 \[ -\frac{a B+A b}{5 x^5}-\frac{a A}{6 x^6}-\frac{A c+b B}{4 x^4}-\frac{B c}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0234613, 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} \[ -\frac{a B+A b}{5 x^5}-\frac{a A}{6 x^6}-\frac{A c+b B}{4 x^4}-\frac{B c}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2))/x^7,x]

[Out]

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

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^7} \, dx &=\int \left (\frac{a A}{x^7}+\frac{A b+a B}{x^6}+\frac{b B+A c}{x^5}+\frac{B c}{x^4}\right ) \, dx\\ &=-\frac{a A}{6 x^6}-\frac{A b+a B}{5 x^5}-\frac{b B+A c}{4 x^4}-\frac{B c}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0153881, size = 45, normalized size = 0.96 \[ -\frac{2 a (5 A+6 B x)+x (3 A (4 b+5 c x)+5 B x (3 b+4 c x))}{60 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2))/x^7,x]

[Out]

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

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Maple [A]  time = 0.004, size = 40, normalized size = 0.9 \begin{align*} -{\frac{Bc}{3\,{x}^{3}}}-{\frac{Ab+aB}{5\,{x}^{5}}}-{\frac{Ac+bB}{4\,{x}^{4}}}-{\frac{aA}{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^7,x)

[Out]

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

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Maxima [A]  time = 1.08778, size = 53, normalized size = 1.13 \begin{align*} -\frac{20 \, B c x^{3} + 15 \,{\left (B b + A c\right )} x^{2} + 10 \, A a + 12 \,{\left (B a + A b\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/x^7,x, algorithm="maxima")

[Out]

-1/60*(20*B*c*x^3 + 15*(B*b + A*c)*x^2 + 10*A*a + 12*(B*a + A*b)*x)/x^6

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Fricas [A]  time = 1.21965, size = 99, normalized size = 2.11 \begin{align*} -\frac{20 \, B c x^{3} + 15 \,{\left (B b + A c\right )} x^{2} + 10 \, A a + 12 \,{\left (B a + A b\right )} x}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/x^7,x, algorithm="fricas")

[Out]

-1/60*(20*B*c*x^3 + 15*(B*b + A*c)*x^2 + 10*A*a + 12*(B*a + A*b)*x)/x^6

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Sympy [A]  time = 3.91403, size = 44, normalized size = 0.94 \begin{align*} - \frac{10 A a + 20 B c x^{3} + x^{2} \left (15 A c + 15 B b\right ) + x \left (12 A b + 12 B a\right )}{60 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)/x**7,x)

[Out]

-(10*A*a + 20*B*c*x**3 + x**2*(15*A*c + 15*B*b) + x*(12*A*b + 12*B*a))/(60*x**6)

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Giac [A]  time = 1.30237, size = 55, normalized size = 1.17 \begin{align*} -\frac{20 \, B c x^{3} + 15 \, B b x^{2} + 15 \, A c x^{2} + 12 \, B a x + 12 \, A b x + 10 \, A a}{60 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/x^7,x, algorithm="giac")

[Out]

-1/60*(20*B*c*x^3 + 15*B*b*x^2 + 15*A*c*x^2 + 12*B*a*x + 12*A*b*x + 10*A*a)/x^6