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

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

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Rubi [A]  time = 0.03, antiderivative size = 41, 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}{2 x^2}-\frac {a A}{3 x^3}-\frac {A c+b B}{x}+B c \log (x) \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 41, normalized size = 1.00 \begin {gather*} 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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(a*(2*A + 3*B*x) + 3*x*(A*b + 2*b*B*x + 2*A*c*x))/x^3 + B*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^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 41, normalized size = 1.00 \begin {gather*} \frac {6 \, B c x^{3} \log \relax (x) - 6 \, {\left (B b + A c\right )} x^{2} - 2 \, A a - 3 \, {\left (B a + A b\right )} x}{6 \, x^{3}} \end {gather*}

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

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giac [A]  time = 0.17, size = 39, normalized size = 0.95 \begin {gather*} 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 {gather*}

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

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maple [A]  time = 0.05, size = 42, normalized size = 1.02 \begin {gather*} B c \ln \relax (x )-\frac {A c}{x}-\frac {B b}{x}-\frac {A b}{2 x^{2}}-\frac {B a}{2 x^{2}}-\frac {A 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^4,x)

[Out]

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

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maxima [A]  time = 0.62, size = 38, normalized size = 0.93 \begin {gather*} B c \log \relax (x) - \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 {gather*}

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

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mupad [B]  time = 1.18, size = 38, normalized size = 0.93 \begin {gather*} B\,c\,\ln \relax (x)-\frac {\left (A\,c+B\,b\right )\,x^2+\left (\frac {A\,b}{2}+\frac {B\,a}{2}\right )\,x+\frac {A\,a}{3}}{x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.58, size = 44, normalized size = 1.07 \begin {gather*} B c \log {\relax (x )} + \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 {gather*}

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)

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