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

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

[Out]

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

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Rubi [A]  time = 0.0166305, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {765} \[ -\frac{A c+b B}{2 x^2}-\frac{A b}{3 x^3}-\frac{B c}{x} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^5,x]

[Out]

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

Mathematica [A]  time = 0.008783, size = 28, normalized size = 0.9 \[ -\frac{A (2 b+3 c x)+3 B x (b+2 c x)}{6 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)/x^5,x)

[Out]

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

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Maxima [A]  time = 1.02574, size = 36, normalized size = 1.16 \begin{align*} -\frac{6 \, B c x^{2} + 2 \, A b + 3 \,{\left (B b + A c\right )} x}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.6412, size = 65, normalized size = 2.1 \begin{align*} -\frac{6 \, B c x^{2} + 2 \, A b + 3 \,{\left (B b + A c\right )} x}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.435433, size = 31, normalized size = 1. \begin{align*} - \frac{2 A b + 6 B c x^{2} + x \left (3 A c + 3 B b\right )}{6 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)/x**5,x)

[Out]

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

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Giac [A]  time = 1.14418, size = 36, normalized size = 1.16 \begin{align*} -\frac{6 \, B c x^{2} + 3 \, B b x + 3 \, A c x + 2 \, A b}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)/x^5,x, algorithm="giac")

[Out]

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