3.270 \(\int \frac{(A+B x) (a+c x^2)^3}{x^2} \, dx\)

Optimal. Leaf size=80 \[ 3 a^2 A c x-\frac{a^3 A}{x}+\frac{3}{2} a^2 B c x^2+a^3 B \log (x)+a A c^2 x^3+\frac{3}{4} a B c^2 x^4+\frac{1}{5} A c^3 x^5+\frac{1}{6} B c^3 x^6 \]

[Out]

-((a^3*A)/x) + 3*a^2*A*c*x + (3*a^2*B*c*x^2)/2 + a*A*c^2*x^3 + (3*a*B*c^2*x^4)/4 + (A*c^3*x^5)/5 + (B*c^3*x^6)
/6 + a^3*B*Log[x]

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Rubi [A]  time = 0.0413895, antiderivative size = 80, 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} \[ 3 a^2 A c x-\frac{a^3 A}{x}+\frac{3}{2} a^2 B c x^2+a^3 B \log (x)+a A c^2 x^3+\frac{3}{4} a B c^2 x^4+\frac{1}{5} A c^3 x^5+\frac{1}{6} B c^3 x^6 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/x^2,x]

[Out]

-((a^3*A)/x) + 3*a^2*A*c*x + (3*a^2*B*c*x^2)/2 + a*A*c^2*x^3 + (3*a*B*c^2*x^4)/4 + (A*c^3*x^5)/5 + (B*c^3*x^6)
/6 + a^3*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 )^3}{x^2} \, dx &=\int \left (3 a^2 A c+\frac{a^3 A}{x^2}+\frac{a^3 B}{x}+3 a^2 B c x+3 a A c^2 x^2+3 a B c^2 x^3+A c^3 x^4+B c^3 x^5\right ) \, dx\\ &=-\frac{a^3 A}{x}+3 a^2 A c x+\frac{3}{2} a^2 B c x^2+a A c^2 x^3+\frac{3}{4} a B c^2 x^4+\frac{1}{5} A c^3 x^5+\frac{1}{6} B c^3 x^6+a^3 B \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0071213, size = 80, normalized size = 1. \[ 3 a^2 A c x-\frac{a^3 A}{x}+\frac{3}{2} a^2 B c x^2+a^3 B \log (x)+a A c^2 x^3+\frac{3}{4} a B c^2 x^4+\frac{1}{5} A c^3 x^5+\frac{1}{6} B c^3 x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^3*A)/x) + 3*a^2*A*c*x + (3*a^2*B*c*x^2)/2 + a*A*c^2*x^3 + (3*a*B*c^2*x^4)/4 + (A*c^3*x^5)/5 + (B*c^3*x^6)
/6 + a^3*B*Log[x]

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Maple [A]  time = 0.005, size = 73, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{3}}{x}}+3\,{a}^{2}Acx+{\frac{3\,{a}^{2}Bc{x}^{2}}{2}}+aA{c}^{2}{x}^{3}+{\frac{3\,aB{c}^{2}{x}^{4}}{4}}+{\frac{A{c}^{3}{x}^{5}}{5}}+{\frac{B{c}^{3}{x}^{6}}{6}}+{a}^{3}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)^3/x^2,x)

[Out]

-a^3*A/x+3*a^2*A*c*x+3/2*a^2*B*c*x^2+a*A*c^2*x^3+3/4*a*B*c^2*x^4+1/5*A*c^3*x^5+1/6*B*c^3*x^6+a^3*B*ln(x)

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Maxima [A]  time = 1.03623, size = 97, normalized size = 1.21 \begin{align*} \frac{1}{6} \, B c^{3} x^{6} + \frac{1}{5} \, A c^{3} x^{5} + \frac{3}{4} \, B a c^{2} x^{4} + A a c^{2} x^{3} + \frac{3}{2} \, B a^{2} c x^{2} + 3 \, A a^{2} c x + B a^{3} \log \left (x\right ) - \frac{A a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*c^3*x^6 + 1/5*A*c^3*x^5 + 3/4*B*a*c^2*x^4 + A*a*c^2*x^3 + 3/2*B*a^2*c*x^2 + 3*A*a^2*c*x + B*a^3*log(x) -
A*a^3/x

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Fricas [A]  time = 1.56812, size = 186, normalized size = 2.32 \begin{align*} \frac{10 \, B c^{3} x^{7} + 12 \, A c^{3} x^{6} + 45 \, B a c^{2} x^{5} + 60 \, A a c^{2} x^{4} + 90 \, B a^{2} c x^{3} + 180 \, A a^{2} c x^{2} + 60 \, B a^{3} x \log \left (x\right ) - 60 \, A a^{3}}{60 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*B*c^3*x^7 + 12*A*c^3*x^6 + 45*B*a*c^2*x^5 + 60*A*a*c^2*x^4 + 90*B*a^2*c*x^3 + 180*A*a^2*c*x^2 + 60*B*
a^3*x*log(x) - 60*A*a^3)/x

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Sympy [A]  time = 0.388647, size = 82, normalized size = 1.02 \begin{align*} - \frac{A a^{3}}{x} + 3 A a^{2} c x + A a c^{2} x^{3} + \frac{A c^{3} x^{5}}{5} + B a^{3} \log{\left (x \right )} + \frac{3 B a^{2} c x^{2}}{2} + \frac{3 B a c^{2} x^{4}}{4} + \frac{B c^{3} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*a**3/x + 3*A*a**2*c*x + A*a*c**2*x**3 + A*c**3*x**5/5 + B*a**3*log(x) + 3*B*a**2*c*x**2/2 + 3*B*a*c**2*x**4
/4 + B*c**3*x**6/6

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Giac [A]  time = 1.12299, size = 99, normalized size = 1.24 \begin{align*} \frac{1}{6} \, B c^{3} x^{6} + \frac{1}{5} \, A c^{3} x^{5} + \frac{3}{4} \, B a c^{2} x^{4} + A a c^{2} x^{3} + \frac{3}{2} \, B a^{2} c x^{2} + 3 \, A a^{2} c x + B a^{3} \log \left ({\left | x \right |}\right ) - \frac{A a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*B*c^3*x^6 + 1/5*A*c^3*x^5 + 3/4*B*a*c^2*x^4 + A*a*c^2*x^3 + 3/2*B*a^2*c*x^2 + 3*A*a^2*c*x + B*a^3*log(abs(
x)) - A*a^3/x