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

Optimal. Leaf size=26 \[ -\frac {a A}{x}+a B \log (x)+A c x+\frac {1}{2} B c x^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {766} \begin {gather*} -\frac {a A}{x}+a B \log (x)+A c x+\frac {1}{2} B c x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2))/x^2,x]

[Out]

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

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 26, normalized size = 1.00 \begin {gather*} -\frac {a A}{x}+a B \log (x)+A c x+\frac {1}{2} B c x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.38, size = 30, normalized size = 1.15 \begin {gather*} \frac {B c x^{3} + 2 \, A c x^{2} + 2 \, B a x \log \relax (x) - 2 \, A a}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.16, size = 25, normalized size = 0.96 \begin {gather*} \frac {1}{2} \, B c x^{2} + A c x + B a \log \left ({\left | x \right |}\right ) - \frac {A a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 25, normalized size = 0.96 \begin {gather*} \frac {B c \,x^{2}}{2}+A c x +B a \ln \relax (x )-\frac {A a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.60, size = 24, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, B c x^{2} + A c x + B a \log \relax (x) - \frac {A a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 24, normalized size = 0.92 \begin {gather*} A\,c\,x-\frac {A\,a}{x}+\frac {B\,c\,x^2}{2}+B\,a\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)*(A + B*x))/x^2,x)

[Out]

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

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sympy [A]  time = 0.14, size = 24, normalized size = 0.92 \begin {gather*} - \frac {A a}{x} + A c x + B a \log {\relax (x )} + \frac {B c x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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