3.88 \(\int \frac{(a+b x) (A+B x)}{x^4} \, dx\)

Optimal. Leaf size=31 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{3 x^3}-\frac{b B}{x} \]

[Out]

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

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Rubi [A]  time = 0.0133023, antiderivative size = 31, 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} \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{3 x^3}-\frac{b B}{x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(A + B*x))/x^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0089094, size = 28, normalized size = 0.9 \[ -\frac{a (2 A+3 B x)+3 b x (A+2 B x)}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(A + B*x))/x^4,x]

[Out]

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

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Maple [A]  time = 0.004, size = 28, normalized size = 0.9 \begin{align*} -{\frac{Aa}{3\,{x}^{3}}}-{\frac{Ab+Ba}{2\,{x}^{2}}}-{\frac{Bb}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(B*x+A)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.15249, size = 36, normalized size = 1.16 \begin{align*} -\frac{6 \, B b x^{2} + 2 \, A a + 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.81821, size = 65, normalized size = 2.1 \begin{align*} -\frac{6 \, B b x^{2} + 2 \, A a + 3 \,{\left (B a + A b\right )} x}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.465642, size = 31, normalized size = 1. \begin{align*} - \frac{2 A a + 6 B b x^{2} + x \left (3 A b + 3 B a\right )}{6 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x**4,x)

[Out]

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

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Giac [A]  time = 1.17973, size = 36, normalized size = 1.16 \begin{align*} -\frac{6 \, B b x^{2} + 3 \, B a x + 3 \, A b x + 2 \, A a}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(B*x+A)/x^4,x, algorithm="giac")

[Out]

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