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

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {a B+A b}{2 x^2}-\frac {a A}{3 x^3}-\frac {b B}{x} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.90 \begin {gather*} -\frac {a (2 A+3 B x)+3 b x (A+2 B x)}{6 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (A+B x)}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.27, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B b x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \end {gather*}

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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giac [A]  time = 1.25, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B b x^{2} + 3 \, B a x + 3 \, A b x + 2 \, A a}{6 \, x^{3}} \end {gather*}

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

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maple [A]  time = 0.00, size = 28, normalized size = 0.90 \begin {gather*} -\frac {B b}{x}-\frac {A a}{3 x^{3}}-\frac {A b +B a}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.14, size = 27, normalized size = 0.87 \begin {gather*} -\frac {6 \, B b x^{2} + 2 \, A a + 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \end {gather*}

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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mupad [B]  time = 0.05, size = 27, normalized size = 0.87 \begin {gather*} -\frac {B\,b\,x^2+\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,x+\frac {A\,a}{3}}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x))/x^4,x)

[Out]

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

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sympy [A]  time = 0.32, size = 31, normalized size = 1.00 \begin {gather*} \frac {- 2 A a - 6 B b x^{2} + x \left (- 3 A b - 3 B a\right )}{6 x^{3}} \end {gather*}

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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