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

Optimal. Leaf size=33 \[ -\frac{a B+A b}{4 x^4}-\frac{a A}{5 x^5}-\frac{b B}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0131032, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {76} \[ -\frac{a B+A b}{4 x^4}-\frac{a A}{5 x^5}-\frac{b B}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x))/x^6,x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0097122, size = 31, normalized size = 0.94 \[ -\frac{3 a (4 A+5 B x)+5 b x (3 A+4 B x)}{60 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 28, normalized size = 0.9 \begin{align*} -{\frac{Bb}{3\,{x}^{3}}}-{\frac{Aa}{5\,{x}^{5}}}-{\frac{Ab+Ba}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01521, size = 36, normalized size = 1.09 \begin{align*} -\frac{20 \, B b x^{2} + 12 \, A a + 15 \,{\left (B a + A b\right )} x}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^6,x, algorithm="maxima")

[Out]

-1/60*(20*B*b*x^2 + 12*A*a + 15*(B*a + A*b)*x)/x^5

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Fricas [A]  time = 1.84178, size = 70, normalized size = 2.12 \begin{align*} -\frac{20 \, B b x^{2} + 12 \, A a + 15 \,{\left (B a + A b\right )} x}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^6,x, algorithm="fricas")

[Out]

-1/60*(20*B*b*x^2 + 12*A*a + 15*(B*a + A*b)*x)/x^5

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Sympy [A]  time = 0.604936, size = 31, normalized size = 0.94 \begin{align*} - \frac{12 A a + 20 B b x^{2} + x \left (15 A b + 15 B a\right )}{60 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x**6,x)

[Out]

-(12*A*a + 20*B*b*x**2 + x*(15*A*b + 15*B*a))/(60*x**5)

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Giac [A]  time = 1.28133, size = 36, normalized size = 1.09 \begin{align*} -\frac{20 \, B b x^{2} + 15 \, B a x + 15 \, A b x + 12 \, A a}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^6,x, algorithm="giac")

[Out]

-1/60*(20*B*b*x^2 + 15*B*a*x + 15*A*b*x + 12*A*a)/x^5