3.844 \(\int \frac{(a+b x^2+c x^4)^3}{x} \, dx\)

Optimal. Leaf size=85 \[ \frac{3}{2} a^2 b x^2+a^3 \log (x)+\frac{3}{8} c x^8 \left (a c+b^2\right )+\frac{1}{6} b x^6 \left (6 a c+b^2\right )+\frac{3}{4} a x^4 \left (a c+b^2\right )+\frac{3}{10} b c^2 x^{10}+\frac{c^3 x^{12}}{12} \]

[Out]

(3*a^2*b*x^2)/2 + (3*a*(b^2 + a*c)*x^4)/4 + (b*(b^2 + 6*a*c)*x^6)/6 + (3*c*(b^2 + a*c)*x^8)/8 + (3*b*c^2*x^10)
/10 + (c^3*x^12)/12 + a^3*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0742707, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1114, 698} \[ \frac{3}{2} a^2 b x^2+a^3 \log (x)+\frac{3}{8} c x^8 \left (a c+b^2\right )+\frac{1}{6} b x^6 \left (6 a c+b^2\right )+\frac{3}{4} a x^4 \left (a c+b^2\right )+\frac{3}{10} b c^2 x^{10}+\frac{c^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x,x]

[Out]

(3*a^2*b*x^2)/2 + (3*a*(b^2 + a*c)*x^4)/4 + (b*(b^2 + 6*a*c)*x^6)/6 + (3*c*(b^2 + a*c)*x^8)/8 + (3*b*c^2*x^10)
/10 + (c^3*x^12)/12 + a^3*Log[x]

Rule 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^3}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^3}{x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (3 a^2 b+\frac{a^3}{x}+3 a \left (b^2+a c\right ) x+b \left (b^2+6 a c\right ) x^2+3 c \left (b^2+a c\right ) x^3+3 b c^2 x^4+c^3 x^5\right ) \, dx,x,x^2\right )\\ &=\frac{3}{2} a^2 b x^2+\frac{3}{4} a \left (b^2+a c\right ) x^4+\frac{1}{6} b \left (b^2+6 a c\right ) x^6+\frac{3}{8} c \left (b^2+a c\right ) x^8+\frac{3}{10} b c^2 x^{10}+\frac{c^3 x^{12}}{12}+a^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0211895, size = 85, normalized size = 1. \[ \frac{3}{2} a^2 b x^2+a^3 \log (x)+\frac{3}{8} c x^8 \left (a c+b^2\right )+\frac{1}{6} b x^6 \left (6 a c+b^2\right )+\frac{3}{4} a x^4 \left (a c+b^2\right )+\frac{3}{10} b c^2 x^{10}+\frac{c^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^3/x,x]

[Out]

(3*a^2*b*x^2)/2 + (3*a*(b^2 + a*c)*x^4)/4 + (b*(b^2 + 6*a*c)*x^6)/6 + (3*c*(b^2 + a*c)*x^8)/8 + (3*b*c^2*x^10)
/10 + (c^3*x^12)/12 + a^3*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.045, size = 85, normalized size = 1. \begin{align*}{\frac{{c}^{3}{x}^{12}}{12}}+{\frac{3\,b{c}^{2}{x}^{10}}{10}}+{\frac{3\,{x}^{8}a{c}^{2}}{8}}+{\frac{3\,{x}^{8}{b}^{2}c}{8}}+{x}^{6}abc+{\frac{{b}^{3}{x}^{6}}{6}}+{\frac{3\,{a}^{2}c{x}^{4}}{4}}+{\frac{3\,a{x}^{4}{b}^{2}}{4}}+{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{a}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^3/x,x)

[Out]

1/12*c^3*x^12+3/10*b*c^2*x^10+3/8*x^8*a*c^2+3/8*x^8*b^2*c+x^6*a*b*c+1/6*b^3*x^6+3/4*a^2*c*x^4+3/4*a*x^4*b^2+3/
2*a^2*b*x^2+a^3*ln(x)

________________________________________________________________________________________

Maxima [A]  time = 0.978995, size = 111, normalized size = 1.31 \begin{align*} \frac{1}{12} \, c^{3} x^{12} + \frac{3}{10} \, b c^{2} x^{10} + \frac{3}{8} \,{\left (b^{2} c + a c^{2}\right )} x^{8} + \frac{1}{6} \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + \frac{3}{2} \, a^{2} b x^{2} + \frac{3}{4} \,{\left (a b^{2} + a^{2} c\right )} x^{4} + \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x,x, algorithm="maxima")

[Out]

1/12*c^3*x^12 + 3/10*b*c^2*x^10 + 3/8*(b^2*c + a*c^2)*x^8 + 1/6*(b^3 + 6*a*b*c)*x^6 + 3/2*a^2*b*x^2 + 3/4*(a*b
^2 + a^2*c)*x^4 + 1/2*a^3*log(x^2)

________________________________________________________________________________________

Fricas [A]  time = 1.75492, size = 189, normalized size = 2.22 \begin{align*} \frac{1}{12} \, c^{3} x^{12} + \frac{3}{10} \, b c^{2} x^{10} + \frac{3}{8} \,{\left (b^{2} c + a c^{2}\right )} x^{8} + \frac{1}{6} \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + \frac{3}{2} \, a^{2} b x^{2} + \frac{3}{4} \,{\left (a b^{2} + a^{2} c\right )} x^{4} + a^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x,x, algorithm="fricas")

[Out]

1/12*c^3*x^12 + 3/10*b*c^2*x^10 + 3/8*(b^2*c + a*c^2)*x^8 + 1/6*(b^3 + 6*a*b*c)*x^6 + 3/2*a^2*b*x^2 + 3/4*(a*b
^2 + a^2*c)*x^4 + a^3*log(x)

________________________________________________________________________________________

Sympy [A]  time = 0.367357, size = 92, normalized size = 1.08 \begin{align*} a^{3} \log{\left (x \right )} + \frac{3 a^{2} b x^{2}}{2} + \frac{3 b c^{2} x^{10}}{10} + \frac{c^{3} x^{12}}{12} + x^{8} \left (\frac{3 a c^{2}}{8} + \frac{3 b^{2} c}{8}\right ) + x^{6} \left (a b c + \frac{b^{3}}{6}\right ) + x^{4} \left (\frac{3 a^{2} c}{4} + \frac{3 a b^{2}}{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**3/x,x)

[Out]

a**3*log(x) + 3*a**2*b*x**2/2 + 3*b*c**2*x**10/10 + c**3*x**12/12 + x**8*(3*a*c**2/8 + 3*b**2*c/8) + x**6*(a*b
*c + b**3/6) + x**4*(3*a**2*c/4 + 3*a*b**2/4)

________________________________________________________________________________________

Giac [A]  time = 1.14849, size = 117, normalized size = 1.38 \begin{align*} \frac{1}{12} \, c^{3} x^{12} + \frac{3}{10} \, b c^{2} x^{10} + \frac{3}{8} \, b^{2} c x^{8} + \frac{3}{8} \, a c^{2} x^{8} + \frac{1}{6} \, b^{3} x^{6} + a b c x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{4} \, a^{2} c x^{4} + \frac{3}{2} \, a^{2} b x^{2} + \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^3/x,x, algorithm="giac")

[Out]

1/12*c^3*x^12 + 3/10*b*c^2*x^10 + 3/8*b^2*c*x^8 + 3/8*a*c^2*x^8 + 1/6*b^3*x^6 + a*b*c*x^6 + 3/4*a*b^2*x^4 + 3/
4*a^2*c*x^4 + 3/2*a^2*b*x^2 + 1/2*a^3*log(x^2)