3.89 \(\int \frac{(a+b x) (A+B x)}{x^5} \, dx\)

Optimal. Leaf size=33 \[ -\frac{a B+A b}{3 x^3}-\frac{a A}{4 x^4}-\frac{b B}{2 x^2} \]

[Out]

-(a*A)/(4*x^4) - (A*b + a*B)/(3*x^3) - (b*B)/(2*x^2)

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Rubi [A]  time = 0.0129332, 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} \[ -\frac{a B+A b}{3 x^3}-\frac{a A}{4 x^4}-\frac{b B}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x))/x^5,x]

[Out]

-(a*A)/(4*x^4) - (A*b + a*B)/(3*x^3) - (b*B)/(2*x^2)

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^5} \, dx &=\int \left (\frac{a A}{x^5}+\frac{A b+a B}{x^4}+\frac{b B}{x^3}\right ) \, dx\\ &=-\frac{a A}{4 x^4}-\frac{A b+a B}{3 x^3}-\frac{b B}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0077658, size = 29, normalized size = 0.88 \[ -\frac{3 a A+4 a B x+4 A b x+6 b B x^2}{12 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(A + B*x))/x^5,x]

[Out]

-(3*a*A + 4*A*b*x + 4*a*B*x + 6*b*B*x^2)/(12*x^4)

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Maple [A]  time = 0.005, size = 28, normalized size = 0.9 \begin{align*} -{\frac{Ab+Ba}{3\,{x}^{3}}}-{\frac{Aa}{4\,{x}^{4}}}-{\frac{Bb}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(B*x+A)/x^5,x)

[Out]

-1/3*(A*b+B*a)/x^3-1/4*a*A/x^4-1/2*b*B/x^2

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Maxima [A]  time = 1.02378, size = 36, normalized size = 1.09 \begin{align*} -\frac{6 \, B b x^{2} + 3 \, A a + 4 \,{\left (B a + A b\right )} x}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^5,x, algorithm="maxima")

[Out]

-1/12*(6*B*b*x^2 + 3*A*a + 4*(B*a + A*b)*x)/x^4

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Fricas [A]  time = 1.8114, size = 66, normalized size = 2. \begin{align*} -\frac{6 \, B b x^{2} + 3 \, A a + 4 \,{\left (B a + A b\right )} x}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^5,x, algorithm="fricas")

[Out]

-1/12*(6*B*b*x^2 + 3*A*a + 4*(B*a + A*b)*x)/x^4

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Sympy [A]  time = 0.533122, size = 31, normalized size = 0.94 \begin{align*} - \frac{3 A a + 6 B b x^{2} + x \left (4 A b + 4 B a\right )}{12 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x**5,x)

[Out]

-(3*A*a + 6*B*b*x**2 + x*(4*A*b + 4*B*a))/(12*x**4)

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Giac [A]  time = 1.18884, size = 36, normalized size = 1.09 \begin{align*} -\frac{6 \, B b x^{2} + 4 \, B a x + 4 \, A b x + 3 \, A a}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^5,x, algorithm="giac")

[Out]

-1/12*(6*B*b*x^2 + 4*B*a*x + 4*A*b*x + 3*A*a)/x^4