3.850 \(\int \frac{(A+B x) (a+b x+c x^2)}{x^4} \, dx\)

Optimal. Leaf size=41 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{3 x^3}-\frac{A c+b B}{x}+B c \log (x) \]

[Out]

-(a*A)/(3*x^3) - (A*b + a*B)/(2*x^2) - (b*B + A*c)/x + B*c*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0253187, antiderivative size = 41, 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} \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{3 x^3}-\frac{A c+b B}{x}+B c \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2))/x^4,x]

[Out]

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

Mathematica [A]  time = 0.0272206, size = 41, normalized size = 1. \[ B c \log (x)-\frac{a (2 A+3 B x)+3 x (A b+2 A c x+2 b B x)}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2))/x^4,x]

[Out]

-(a*(2*A + 3*B*x) + 3*x*(A*b + 2*b*B*x + 2*A*c*x))/(6*x^3) + B*c*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 42, normalized size = 1. \begin{align*} Bc\ln \left ( x \right ) -{\frac{aA}{3\,{x}^{3}}}-{\frac{Ab}{2\,{x}^{2}}}-{\frac{aB}{2\,{x}^{2}}}-{\frac{Ac}{x}}-{\frac{bB}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)/x^4,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02508, size = 51, normalized size = 1.24 \begin{align*} B c \log \left (x\right ) - \frac{6 \,{\left (B b + A c\right )} x^{2} + 2 \, A a + 3 \,{\left (B a + A b\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+a)/x^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.25606, size = 100, normalized size = 2.44 \begin{align*} \frac{6 \, B c x^{3} \log \left (x\right ) - 6 \,{\left (B b + A c\right )} x^{2} - 2 \, A a - 3 \,{\left (B a + A b\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+a)/x^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 1.42573, size = 41, normalized size = 1. \begin{align*} B c \log{\left (x \right )} - \frac{2 A a + x^{2} \left (6 A c + 6 B b\right ) + x \left (3 A b + 3 B a\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+a)/x**4,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.2402, size = 53, normalized size = 1.29 \begin{align*} B c \log \left ({\left | x \right |}\right ) - \frac{6 \,{\left (B b + A c\right )} x^{2} + 2 \, A a + 3 \,{\left (B a + A b\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+a)/x^4,x, algorithm="giac")

[Out]

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