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

Optimal. Leaf size=24 \[ x (A c+b B)+A b \log (x)+\frac {1}{2} B c x^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, 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} \begin {gather*} x (A c+b B)+A b \log (x)+\frac {1}{2} B c x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2))/x^2,x]

[Out]

(b*B + A*c)*x + (B*c*x^2)/2 + A*b*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 (b x+c x^2\right )}{x^2} \, dx &=\int \left (b B+A c+\frac {A b}{x}+B c x\right ) \, dx\\ &=(b B+A c) x+\frac {1}{2} B c x^2+A b \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} x (A c+b B)+A b \log (x)+\frac {1}{2} B c x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*B + A*c)*x + (B*c*x^2)/2 + A*b*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 (b x+c x^2\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, B c x^{2} + A b \log \relax (x) + {\left (B b + A c\right )} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.14, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, B c x^{2} + B b x + A c x + A b \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 22, normalized size = 0.92 \begin {gather*} \frac {B c \,x^{2}}{2}+A b \ln \relax (x )+A c x +B b x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.90, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, B c x^{2} + A b \log \relax (x) + {\left (B b + A c\right )} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 22, normalized size = 0.92 \begin {gather*} x\,\left (A\,c+B\,b\right )+\frac {B\,c\,x^2}{2}+A\,b\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b*x + c*x^2)*(A + B*x))/x^2,x)

[Out]

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

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sympy [A]  time = 0.11, size = 22, normalized size = 0.92 \begin {gather*} A b \log {\relax (x )} + \frac {B c x^{2}}{2} + x \left (A c + B b\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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