3.8.76 \(\int \frac {(A+B x) (a+b x+c x^2)}{x^5} \, dx\)

Optimal. Leaf size=45 \[ -\frac {a B+A b}{3 x^3}-\frac {a A}{4 x^4}-\frac {A c+b B}{2 x^2}-\frac {B c}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {765} \begin {gather*} -\frac {a B+A b}{3 x^3}-\frac {a A}{4 x^4}-\frac {A c+b B}{2 x^2}-\frac {B c}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2))/x^5,x]

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 42, normalized size = 0.93 \begin {gather*} -\frac {a (3 A+4 B x)+2 x (A (2 b+3 c x)+3 B x (b+2 c x))}{12 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2))/x^5,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )}{x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2))/x^5,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2))/x^5, x]

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fricas [A]  time = 0.39, size = 39, normalized size = 0.87 \begin {gather*} -\frac {12 \, B c x^{3} + 6 \, {\left (B b + A c\right )} 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)*(c*x^2+b*x+a)/x^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 41, normalized size = 0.91 \begin {gather*} -\frac {12 \, B c x^{3} + 6 \, B b x^{2} + 6 \, A c 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)*(c*x^2+b*x+a)/x^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 40, normalized size = 0.89 \begin {gather*} -\frac {B c}{x}-\frac {A a}{4 x^{4}}-\frac {A c +b B}{2 x^{2}}-\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)*(c*x^2+b*x+a)/x^5,x)

[Out]

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

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maxima [A]  time = 0.65, size = 39, normalized size = 0.87 \begin {gather*} -\frac {12 \, B c x^{3} + 6 \, {\left (B b + A c\right )} 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)*(c*x^2+b*x+a)/x^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.03, size = 40, normalized size = 0.89 \begin {gather*} -\frac {B\,c\,x^3+\left (\frac {A\,c}{2}+\frac {B\,b}{2}\right )\,x^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 + c*x^2))/x^5,x)

[Out]

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

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sympy [A]  time = 1.04, size = 46, normalized size = 1.02 \begin {gather*} \frac {- 3 A a - 12 B c x^{3} + x^{2} \left (- 6 A c - 6 B b\right ) + 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)*(c*x**2+b*x+a)/x**5,x)

[Out]

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

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