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

Optimal. Leaf size=28 \[ a A \log (x)+a B x+\frac {1}{2} A c x^2+\frac {1}{3} B c x^3 \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 28, 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*} a A \log (x)+a B x+\frac {1}{2} A c x^2+\frac {1}{3} B c x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2))/x,x]

[Out]

a*B*x + (A*c*x^2)/2 + (B*c*x^3)/3 + a*A*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} \, dx &=\int \left (a B+\frac {a A}{x}+A c x+B c x^2\right ) \, dx\\ &=a B x+\frac {1}{2} A c x^2+\frac {1}{3} B c x^3+a A \log (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} a A \log (x)+a B x+\frac {1}{2} A c x^2+\frac {1}{3} B c x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.39, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, B c x^{3} + \frac {1}{2} \, A c x^{2} + B a x + A a \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.17, size = 25, normalized size = 0.89 \begin {gather*} \frac {1}{3} \, B c x^{3} + \frac {1}{2} \, A c x^{2} + B a x + A a \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+a)/x,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.04, size = 25, normalized size = 0.89 \begin {gather*} \frac {B c \,x^{3}}{3}+\frac {A c \,x^{2}}{2}+A a \ln \relax (x )+B a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.57, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{3} \, B c x^{3} + \frac {1}{2} \, A c x^{2} + B a x + A a \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 24, normalized size = 0.86 \begin {gather*} B\,a\,x+\frac {A\,c\,x^2}{2}+\frac {B\,c\,x^3}{3}+A\,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,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 27, normalized size = 0.96 \begin {gather*} A a \log {\relax (x )} + \frac {A c x^{2}}{2} + B a x + \frac {B c x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________