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

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

[Out]

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

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Rubi [A]  time = 0.0235477, antiderivative size = 45, normalized size of antiderivative = 1., 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} \[ -\frac{a B+A b}{3 x^3}-\frac{a A}{4 x^4}-\frac{A c+b B}{2 x^2}-\frac{B c}{x} \]

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*}

Mathematica [A]  time = 0.0155992, size = 42, normalized size = 0.93 \[ -\frac{a (3 A+4 B x)+2 x (A (2 b+3 c x)+3 B x (b+2 c x))}{12 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 40, normalized size = 0.9 \begin{align*} -{\frac{Ab+aB}{3\,{x}^{3}}}-{\frac{Ac+bB}{2\,{x}^{2}}}-{\frac{Bc}{x}}-{\frac{aA}{4\,{x}^{4}}} \end{align*}

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]

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

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Maxima [A]  time = 1.04347, size = 53, normalized size = 1.18 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.29218, size = 95, normalized size = 2.11 \begin{align*} -\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{align*}

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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Sympy [A]  time = 2.69182, size = 44, normalized size = 0.98 \begin{align*} - \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{align*}

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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Giac [A]  time = 1.30148, size = 55, normalized size = 1.22 \begin{align*} -\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{align*}

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