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

Optimal. Leaf size=53 \[ a^2 A \log (x)+a^2 B x+a A c x^2+\frac{2}{3} a B c x^3+\frac{1}{4} A c^2 x^4+\frac{1}{5} B c^2 x^5 \]

[Out]

a^2*B*x + a*A*c*x^2 + (2*a*B*c*x^3)/3 + (A*c^2*x^4)/4 + (B*c^2*x^5)/5 + a^2*A*Log[x]

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Rubi [A]  time = 0.0209783, antiderivative size = 53, 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} \[ a^2 A \log (x)+a^2 B x+a A c x^2+\frac{2}{3} a B c x^3+\frac{1}{4} A c^2 x^4+\frac{1}{5} B c^2 x^5 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/x,x]

[Out]

a^2*B*x + a*A*c*x^2 + (2*a*B*c*x^3)/3 + (A*c^2*x^4)/4 + (B*c^2*x^5)/5 + a^2*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 )^2}{x} \, dx &=\int \left (a^2 B+\frac{a^2 A}{x}+2 a A c x+2 a B c x^2+A c^2 x^3+B c^2 x^4\right ) \, dx\\ &=a^2 B x+a A c x^2+\frac{2}{3} a B c x^3+\frac{1}{4} A c^2 x^4+\frac{1}{5} B c^2 x^5+a^2 A \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0063978, size = 53, normalized size = 1. \[ a^2 A \log (x)+a^2 B x+a A c x^2+\frac{2}{3} a B c x^3+\frac{1}{4} A c^2 x^4+\frac{1}{5} B c^2 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*B*x + a*A*c*x^2 + (2*a*B*c*x^3)/3 + (A*c^2*x^4)/4 + (B*c^2*x^5)/5 + a^2*A*Log[x]

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Maple [A]  time = 0.003, size = 48, normalized size = 0.9 \begin{align*}{a}^{2}Bx+aAc{x}^{2}+{\frac{2\,aBc{x}^{3}}{3}}+{\frac{A{c}^{2}{x}^{4}}{4}}+{\frac{B{c}^{2}{x}^{5}}{5}}+{a}^{2}A\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,x)

[Out]

a^2*B*x+a*A*c*x^2+2/3*a*B*c*x^3+1/4*A*c^2*x^4+1/5*B*c^2*x^5+a^2*A*ln(x)

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Maxima [A]  time = 1.03605, size = 63, normalized size = 1.19 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \, A c^{2} x^{4} + \frac{2}{3} \, B a c x^{3} + A a c x^{2} + B a^{2} x + A a^{2} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.48841, size = 113, normalized size = 2.13 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \, A c^{2} x^{4} + \frac{2}{3} \, B a c x^{3} + A a c x^{2} + B a^{2} x + A a^{2} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.32886, size = 54, normalized size = 1.02 \begin{align*} A a^{2} \log{\left (x \right )} + A a c x^{2} + \frac{A c^{2} x^{4}}{4} + B a^{2} x + \frac{2 B a c x^{3}}{3} + \frac{B c^{2} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**2*log(x) + A*a*c*x**2 + A*c**2*x**4/4 + B*a**2*x + 2*B*a*c*x**3/3 + B*c**2*x**5/5

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Giac [A]  time = 1.14094, size = 65, normalized size = 1.23 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \, A c^{2} x^{4} + \frac{2}{3} \, B a c x^{3} + A a c x^{2} + B a^{2} x + A a^{2} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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