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

Optimal. Leaf size=86 \[ 3 a^2 b \log (x)-\frac{a^3}{2 x^2}+\frac{1}{2} c x^6 \left (a c+b^2\right )+\frac{1}{4} b x^4 \left (6 a c+b^2\right )+\frac{3}{2} a x^2 \left (a c+b^2\right )+\frac{3}{8} b c^2 x^8+\frac{c^3 x^{10}}{10} \]

[Out]

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

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Rubi [A]  time = 0.0778557, antiderivative size = 86, 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} \[ 3 a^2 b \log (x)-\frac{a^3}{2 x^2}+\frac{1}{2} c x^6 \left (a c+b^2\right )+\frac{1}{4} b x^4 \left (6 a c+b^2\right )+\frac{3}{2} a x^2 \left (a c+b^2\right )+\frac{3}{8} b c^2 x^8+\frac{c^3 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0334795, size = 78, normalized size = 0.91 \[ \frac{1}{40} \left (120 a^2 b \log (x)-\frac{20 a^3}{x^2}+20 c x^6 \left (a c+b^2\right )+10 b x^4 \left (6 a c+b^2\right )+60 a x^2 \left (a c+b^2\right )+15 b c^2 x^8+4 c^3 x^{10}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^3/x^3,x]

[Out]

((-20*a^3)/x^2 + 60*a*(b^2 + a*c)*x^2 + 10*b*(b^2 + 6*a*c)*x^4 + 20*c*(b^2 + a*c)*x^6 + 15*b*c^2*x^8 + 4*c^3*x
^10 + 120*a^2*b*Log[x])/40

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Maple [A]  time = 0.047, size = 87, normalized size = 1. \begin{align*}{\frac{{c}^{3}{x}^{10}}{10}}+{\frac{3\,b{c}^{2}{x}^{8}}{8}}+{\frac{{x}^{6}a{c}^{2}}{2}}+{\frac{{x}^{6}{b}^{2}c}{2}}+{\frac{3\,{x}^{4}abc}{2}}+{\frac{{b}^{3}{x}^{4}}{4}}+{\frac{3\,{x}^{2}{a}^{2}c}{2}}+{\frac{3\,a{b}^{2}{x}^{2}}{2}}+3\,{a}^{2}b\ln \left ( x \right ) -{\frac{{a}^{3}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.96496, size = 111, normalized size = 1.29 \begin{align*} \frac{1}{10} \, c^{3} x^{10} + \frac{3}{8} \, b c^{2} x^{8} + \frac{1}{2} \,{\left (b^{2} c + a c^{2}\right )} x^{6} + \frac{1}{4} \,{\left (b^{3} + 6 \, a b c\right )} x^{4} + \frac{3}{2} \, a^{2} b \log \left (x^{2}\right ) + \frac{3}{2} \,{\left (a b^{2} + a^{2} c\right )} x^{2} - \frac{a^{3}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.40602, size = 197, normalized size = 2.29 \begin{align*} \frac{4 \, c^{3} x^{12} + 15 \, b c^{2} x^{10} + 20 \,{\left (b^{2} c + a c^{2}\right )} x^{8} + 10 \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + 120 \, a^{2} b x^{2} \log \left (x\right ) + 60 \,{\left (a b^{2} + a^{2} c\right )} x^{4} - 20 \, a^{3}}{40 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40*(4*c^3*x^12 + 15*b*c^2*x^10 + 20*(b^2*c + a*c^2)*x^8 + 10*(b^3 + 6*a*b*c)*x^6 + 120*a^2*b*x^2*log(x) + 60
*(a*b^2 + a^2*c)*x^4 - 20*a^3)/x^2

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Sympy [A]  time = 0.393836, size = 92, normalized size = 1.07 \begin{align*} - \frac{a^{3}}{2 x^{2}} + 3 a^{2} b \log{\left (x \right )} + \frac{3 b c^{2} x^{8}}{8} + \frac{c^{3} x^{10}}{10} + x^{6} \left (\frac{a c^{2}}{2} + \frac{b^{2} c}{2}\right ) + x^{4} \left (\frac{3 a b c}{2} + \frac{b^{3}}{4}\right ) + x^{2} \left (\frac{3 a^{2} c}{2} + \frac{3 a b^{2}}{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**3,x)

[Out]

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

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Giac [A]  time = 1.12736, size = 132, normalized size = 1.53 \begin{align*} \frac{1}{10} \, c^{3} x^{10} + \frac{3}{8} \, b c^{2} x^{8} + \frac{1}{2} \, b^{2} c x^{6} + \frac{1}{2} \, a c^{2} x^{6} + \frac{1}{4} \, b^{3} x^{4} + \frac{3}{2} \, a b c x^{4} + \frac{3}{2} \, a b^{2} x^{2} + \frac{3}{2} \, a^{2} c x^{2} + \frac{3}{2} \, a^{2} b \log \left (x^{2}\right ) - \frac{3 \, a^{2} b x^{2} + a^{3}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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