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

Optimal. Leaf size=52 \[ -\frac{a^2 A}{x}+a^2 B \log (x)+2 a A c x+a B c x^2+\frac{1}{3} A c^2 x^3+\frac{1}{4} B c^2 x^4 \]

[Out]

-((a^2*A)/x) + 2*a*A*c*x + a*B*c*x^2 + (A*c^2*x^3)/3 + (B*c^2*x^4)/4 + a^2*B*Log[x]

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Rubi [A]  time = 0.025836, antiderivative size = 52, normalized size of antiderivative = 1., 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 = {766} \[ -\frac{a^2 A}{x}+a^2 B \log (x)+2 a A c x+a B c x^2+\frac{1}{3} A c^2 x^3+\frac{1}{4} B c^2 x^4 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/x^2,x]

[Out]

-((a^2*A)/x) + 2*a*A*c*x + a*B*c*x^2 + (A*c^2*x^3)/3 + (B*c^2*x^4)/4 + a^2*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 )^2}{x^2} \, dx &=\int \left (2 a A c+\frac{a^2 A}{x^2}+\frac{a^2 B}{x}+2 a B c x+A c^2 x^2+B c^2 x^3\right ) \, dx\\ &=-\frac{a^2 A}{x}+2 a A c x+a B c x^2+\frac{1}{3} A c^2 x^3+\frac{1}{4} B c^2 x^4+a^2 B \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0066716, size = 52, normalized size = 1. \[ -\frac{a^2 A}{x}+a^2 B \log (x)+2 a A c x+a B c x^2+\frac{1}{3} A c^2 x^3+\frac{1}{4} B c^2 x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^2*A)/x) + 2*a*A*c*x + a*B*c*x^2 + (A*c^2*x^3)/3 + (B*c^2*x^4)/4 + a^2*B*Log[x]

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Maple [A]  time = 0.006, size = 49, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{2}}{x}}+2\,aAcx+aBc{x}^{2}+{\frac{A{c}^{2}{x}^{3}}{3}}+{\frac{B{c}^{2}{x}^{4}}{4}}+{a}^{2}B\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2*A/x+2*a*A*c*x+a*B*c*x^2+1/3*A*c^2*x^3+1/4*B*c^2*x^4+a^2*B*ln(x)

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Maxima [A]  time = 1.06795, size = 65, normalized size = 1.25 \begin{align*} \frac{1}{4} \, B c^{2} x^{4} + \frac{1}{3} \, A c^{2} x^{3} + B a c x^{2} + 2 \, A a c x + B a^{2} \log \left (x\right ) - \frac{A a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*B*c^2*x^4 + 1/3*A*c^2*x^3 + B*a*c*x^2 + 2*A*a*c*x + B*a^2*log(x) - A*a^2/x

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Fricas [A]  time = 1.59884, size = 131, normalized size = 2.52 \begin{align*} \frac{3 \, B c^{2} x^{5} + 4 \, A c^{2} x^{4} + 12 \, B a c x^{3} + 24 \, A a c x^{2} + 12 \, B a^{2} x \log \left (x\right ) - 12 \, A a^{2}}{12 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*B*c^2*x^5 + 4*A*c^2*x^4 + 12*B*a*c*x^3 + 24*A*a*c*x^2 + 12*B*a^2*x*log(x) - 12*A*a^2)/x

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Sympy [A]  time = 0.316253, size = 51, normalized size = 0.98 \begin{align*} - \frac{A a^{2}}{x} + 2 A a c x + \frac{A c^{2} x^{3}}{3} + B a^{2} \log{\left (x \right )} + B a c x^{2} + \frac{B c^{2} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*a**2/x + 2*A*a*c*x + A*c**2*x**3/3 + B*a**2*log(x) + B*a*c*x**2 + B*c**2*x**4/4

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Giac [A]  time = 1.13244, size = 66, normalized size = 1.27 \begin{align*} \frac{1}{4} \, B c^{2} x^{4} + \frac{1}{3} \, A c^{2} x^{3} + B a c x^{2} + 2 \, A a c x + B a^{2} \log \left ({\left | x \right |}\right ) - \frac{A a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*B*c^2*x^4 + 1/3*A*c^2*x^3 + B*a*c*x^2 + 2*A*a*c*x + B*a^2*log(abs(x)) - A*a^2/x