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

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

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Rubi [A]  time = 0.01, antiderivative size = 33, 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}{3 x^3}-\frac {a A}{4 x^4}-\frac {b B}{2 x^2} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 0.88 \begin {gather*} -\frac {3 a A+4 a B x+4 A b x+6 b B x^2}{12 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.20, size = 27, normalized size = 0.82 \begin {gather*} -\frac {6 \, B b x^{2} + 3 \, A a + 4 \, {\left (B a + A b\right )} x}{12 \, x^{4}} \end {gather*}

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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giac [A]  time = 1.25, size = 27, normalized size = 0.82 \begin {gather*} -\frac {6 \, B b x^{2} + 4 \, B a x + 4 \, A b x + 3 \, A a}{12 \, x^{4}} \end {gather*}

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

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maple [A]  time = 0.01, size = 28, normalized size = 0.85 \begin {gather*} -\frac {B b}{2 x^{2}}-\frac {A a}{4 x^{4}}-\frac {A b +B a}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.05, size = 27, normalized size = 0.82 \begin {gather*} -\frac {6 \, B b x^{2} + 3 \, A a + 4 \, {\left (B a + A b\right )} x}{12 \, x^{4}} \end {gather*}

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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mupad [B]  time = 0.04, size = 28, normalized size = 0.85 \begin {gather*} -\frac {\frac {B\,b\,x^2}{2}+\left (\frac {A\,b}{3}+\frac {B\,a}{3}\right )\,x+\frac {A\,a}{4}}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x))/x^5,x)

[Out]

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

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sympy [A]  time = 0.40, size = 31, normalized size = 0.94 \begin {gather*} \frac {- 3 A a - 6 B b x^{2} + x \left (- 4 A b - 4 B a\right )}{12 x^{4}} \end {gather*}

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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