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

Optimal. Leaf size=81 \[ 3 a^2 A c \log (x)-\frac{a^3 A}{2 x^2}+3 a^2 B c x-\frac{a^3 B}{x}+\frac{3}{2} a A c^2 x^2+a B c^2 x^3+\frac{1}{4} A c^3 x^4+\frac{1}{5} B c^3 x^5 \]

[Out]

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

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Rubi [A]  time = 0.0397865, antiderivative size = 81, 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} \[ 3 a^2 A c \log (x)-\frac{a^3 A}{2 x^2}+3 a^2 B c x-\frac{a^3 B}{x}+\frac{3}{2} a A c^2 x^2+a B c^2 x^3+\frac{1}{4} A c^3 x^4+\frac{1}{5} B c^3 x^5 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0072039, size = 81, normalized size = 1. \[ 3 a^2 A c \log (x)-\frac{a^3 A}{2 x^2}+3 a^2 B c x-\frac{a^3 B}{x}+\frac{3}{2} a A c^2 x^2+a B c^2 x^3+\frac{1}{4} A c^3 x^4+\frac{1}{5} B c^3 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 74, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{3}}{2\,{x}^{2}}}-{\frac{B{a}^{3}}{x}}+3\,{a}^{2}Bcx+{\frac{3\,aA{c}^{2}{x}^{2}}{2}}+aB{c}^{2}{x}^{3}+{\frac{A{c}^{3}{x}^{4}}{4}}+{\frac{B{c}^{3}{x}^{5}}{5}}+3\,{a}^{2}Ac\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^3,x)

[Out]

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

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Maxima [A]  time = 1.04403, size = 99, normalized size = 1.22 \begin{align*} \frac{1}{5} \, B c^{3} x^{5} + \frac{1}{4} \, A c^{3} x^{4} + B a c^{2} x^{3} + \frac{3}{2} \, A a c^{2} x^{2} + 3 \, B a^{2} c x + 3 \, A a^{2} c \log \left (x\right ) - \frac{2 \, B a^{3} x + A a^{3}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.49984, size = 185, normalized size = 2.28 \begin{align*} \frac{4 \, B c^{3} x^{7} + 5 \, A c^{3} x^{6} + 20 \, B a c^{2} x^{5} + 30 \, A a c^{2} x^{4} + 60 \, B a^{2} c x^{3} + 60 \, A a^{2} c x^{2} \log \left (x\right ) - 20 \, B a^{3} x - 10 \, A a^{3}}{20 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(4*B*c^3*x^7 + 5*A*c^3*x^6 + 20*B*a*c^2*x^5 + 30*A*a*c^2*x^4 + 60*B*a^2*c*x^3 + 60*A*a^2*c*x^2*log(x) - 2
0*B*a^3*x - 10*A*a^3)/x^2

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Sympy [A]  time = 0.489394, size = 83, normalized size = 1.02 \begin{align*} 3 A a^{2} c \log{\left (x \right )} + \frac{3 A a c^{2} x^{2}}{2} + \frac{A c^{3} x^{4}}{4} + 3 B a^{2} c x + B a c^{2} x^{3} + \frac{B c^{3} x^{5}}{5} - \frac{A a^{3} + 2 B a^{3} x}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08061, size = 100, normalized size = 1.23 \begin{align*} \frac{1}{5} \, B c^{3} x^{5} + \frac{1}{4} \, A c^{3} x^{4} + B a c^{2} x^{3} + \frac{3}{2} \, A a c^{2} x^{2} + 3 \, B a^{2} c x + 3 \, A a^{2} c \log \left ({\left | x \right |}\right ) - \frac{2 \, B a^{3} x + A a^{3}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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