3.3.57 \(\int \frac {(A+B x) (a+c x^2)}{x^3} \, dx\) [257]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {780} \begin {gather*} -\frac {a A}{2 x^2}-\frac {a B}{x}+A c \log (x)+B c x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2))/x^3,x]

[Out]

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

Rule 780

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.05, size = 25, normalized size = 0.96

method	result	size
default	\(-\frac {a A}{2 x^{2}}-\frac {a B}{x}+B c x +A c \ln \left (x \right )\)	\(25\)
risch	\(B c x +\frac {-B a x -\frac {1}{2} A a}{x^{2}}+A c \ln \left (x \right )\)	\(25\)
norman	\(\frac {B c \,x^{3}-\frac {1}{2} A a -B a x}{x^{2}}+A c \ln \left (x \right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 24, normalized size = 0.92 \begin {gather*} B c x + A c \log \left (x\right ) - \frac {2 \, B a x + A a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.96, size = 31, normalized size = 1.19 \begin {gather*} \frac {2 \, B c x^{3} + 2 \, A c x^{2} \log \left (x\right ) - 2 \, B a x - A a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.28, size = 27, normalized size = 1.04 \begin {gather*} A c \log {\left (x \right )} + B c x + \frac {- A a - 2 B a x}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 2.12, size = 25, normalized size = 0.96 \begin {gather*} B c x + A c \log \left ({\left | x \right |}\right ) - \frac {2 \, B a x + A a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.03, size = 24, normalized size = 0.92 \begin {gather*} B\,c\,x-\frac {\frac {A\,a}{2}+B\,a\,x}{x^2}+A\,c\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)*(A + B*x))/x^3,x)

[Out]

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

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