3.8.74 \(\int \frac {(A+B x) (a+b x+c x^2)}{x^3} \, dx\)

Optimal. Leaf size=36 \[ -\frac {a B+A b}{x}-\frac {a A}{2 x^2}+\log (x) (A c+b B)+B c x \]

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Rubi [A]  time = 0.02, antiderivative size = 36, 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}{x}-\frac {a A}{2 x^2}+\log (x) (A c+b B)+B c x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.02, size = 37, normalized size = 1.03 \begin {gather*} \frac {-a B-A b}{x}-\frac {a A}{2 x^2}+\log (x) (A c+b B)+B c x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a*A)/x^2 + (-(A*b) - a*B)/x + B*c*x + (b*B + A*c)*Log[x]

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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^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.40, size = 41, normalized size = 1.14 \begin {gather*} \frac {2 \, B c x^{3} + 2 \, {\left (B b + A c\right )} x^{2} \log \relax (x) - A a - 2 \, {\left (B a + A b\right )} x}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 35, normalized size = 0.97 \begin {gather*} B c x + {\left (B b + A c\right )} \log \left ({\left | x \right |}\right ) - \frac {A a + 2 \, {\left (B a + A b\right )} x}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 37, normalized size = 1.03 \begin {gather*} A c \ln \relax (x )+B b \ln \relax (x )+B c x -\frac {A b}{x}-\frac {B a}{x}-\frac {A a}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x+a)/x^3,x)

[Out]

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

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maxima [A]  time = 0.50, size = 34, normalized size = 0.94 \begin {gather*} B c x + {\left (B b + A c\right )} \log \relax (x) - \frac {A a + 2 \, {\left (B a + A b\right )} x}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x+a)/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.17, size = 34, normalized size = 0.94 \begin {gather*} \ln \relax (x)\,\left (A\,c+B\,b\right )-\frac {\frac {A\,a}{2}+x\,\left (A\,b+B\,a\right )}{x^2}+B\,c\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2))/x^3,x)

[Out]

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

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sympy [A]  time = 0.31, size = 36, normalized size = 1.00 \begin {gather*} B c x + \left (A c + B b\right ) \log {\relax (x )} + \frac {- A a + x \left (- 2 A b - 2 B a\right )}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x+a)/x**3,x)

[Out]

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

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