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

Optimal. Leaf size=83 \[ -\frac{3 a^2 b}{x}-\frac{a^3}{3 x^3}+\frac{3}{5} c x^5 \left (a c+b^2\right )+\frac{1}{3} b x^3 \left (6 a c+b^2\right )+3 a x \left (a c+b^2\right )+\frac{3}{7} b c^2 x^7+\frac{c^3 x^9}{9} \]

[Out]

-a^3/(3*x^3) - (3*a^2*b)/x + 3*a*(b^2 + a*c)*x + (b*(b^2 + 6*a*c)*x^3)/3 + (3*c*(b^2 + a*c)*x^5)/5 + (3*b*c^2*
x^7)/7 + (c^3*x^9)/9

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Rubi [A]  time = 0.0406804, antiderivative size = 83, 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 = {1108} \[ -\frac{3 a^2 b}{x}-\frac{a^3}{3 x^3}+\frac{3}{5} c x^5 \left (a c+b^2\right )+\frac{1}{3} b x^3 \left (6 a c+b^2\right )+3 a x \left (a c+b^2\right )+\frac{3}{7} b c^2 x^7+\frac{c^3 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x^4,x]

[Out]

-a^3/(3*x^3) - (3*a^2*b)/x + 3*a*(b^2 + a*c)*x + (b*(b^2 + 6*a*c)*x^3)/3 + (3*c*(b^2 + a*c)*x^5)/5 + (3*b*c^2*
x^7)/7 + (c^3*x^9)/9

Rule 1108

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^3}{x^4} \, dx &=\int \left (3 a \left (b^2+a c\right )+\frac{a^3}{x^4}+\frac{3 a^2 b}{x^2}+b \left (b^2+6 a c\right ) x^2+3 c \left (b^2+a c\right ) x^4+3 b c^2 x^6+c^3 x^8\right ) \, dx\\ &=-\frac{a^3}{3 x^3}-\frac{3 a^2 b}{x}+3 a \left (b^2+a c\right ) x+\frac{1}{3} b \left (b^2+6 a c\right ) x^3+\frac{3}{5} c \left (b^2+a c\right ) x^5+\frac{3}{7} b c^2 x^7+\frac{c^3 x^9}{9}\\ \end{align*}

Mathematica [A]  time = 0.0247444, size = 83, normalized size = 1. \[ -\frac{3 a^2 b}{x}-\frac{a^3}{3 x^3}+\frac{3}{5} c x^5 \left (a c+b^2\right )+\frac{1}{3} b x^3 \left (6 a c+b^2\right )+3 a x \left (a c+b^2\right )+\frac{3}{7} b c^2 x^7+\frac{c^3 x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-a^3/(3*x^3) - (3*a^2*b)/x + 3*a*(b^2 + a*c)*x + (b*(b^2 + 6*a*c)*x^3)/3 + (3*c*(b^2 + a*c)*x^5)/5 + (3*b*c^2*
x^7)/7 + (c^3*x^9)/9

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Maple [A]  time = 0.049, size = 84, normalized size = 1. \begin{align*}{\frac{{c}^{3}{x}^{9}}{9}}+{\frac{3\,b{c}^{2}{x}^{7}}{7}}+{\frac{3\,{x}^{5}a{c}^{2}}{5}}+{\frac{3\,{b}^{2}c{x}^{5}}{5}}+2\,{x}^{3}abc+{\frac{{b}^{3}{x}^{3}}{3}}+3\,{a}^{2}cx+3\,{b}^{2}ax-{\frac{{a}^{3}}{3\,{x}^{3}}}-3\,{\frac{b{a}^{2}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^3*x^9+3/7*b*c^2*x^7+3/5*x^5*a*c^2+3/5*b^2*c*x^5+2*x^3*a*b*c+1/3*b^3*x^3+3*a^2*c*x+3*b^2*a*x-1/3*a^3/x^3-
3*a^2*b/x

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Maxima [A]  time = 0.970253, size = 108, normalized size = 1.3 \begin{align*} \frac{1}{9} \, c^{3} x^{9} + \frac{3}{7} \, b c^{2} x^{7} + \frac{3}{5} \,{\left (b^{2} c + a c^{2}\right )} x^{5} + \frac{1}{3} \,{\left (b^{3} + 6 \, a b c\right )} x^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x - \frac{9 \, a^{2} b x^{2} + a^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^3*x^9 + 3/7*b*c^2*x^7 + 3/5*(b^2*c + a*c^2)*x^5 + 1/3*(b^3 + 6*a*b*c)*x^3 + 3*(a*b^2 + a^2*c)*x - 1/3*(9
*a^2*b*x^2 + a^3)/x^3

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Fricas [A]  time = 1.39957, size = 197, normalized size = 2.37 \begin{align*} \frac{35 \, c^{3} x^{12} + 135 \, b c^{2} x^{10} + 189 \,{\left (b^{2} c + a c^{2}\right )} x^{8} + 105 \,{\left (b^{3} + 6 \, a b c\right )} x^{6} - 945 \, a^{2} b x^{2} + 945 \,{\left (a b^{2} + a^{2} c\right )} x^{4} - 105 \, a^{3}}{315 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(35*c^3*x^12 + 135*b*c^2*x^10 + 189*(b^2*c + a*c^2)*x^8 + 105*(b^3 + 6*a*b*c)*x^6 - 945*a^2*b*x^2 + 945*
(a*b^2 + a^2*c)*x^4 - 105*a^3)/x^3

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Sympy [A]  time = 0.405694, size = 88, normalized size = 1.06 \begin{align*} \frac{3 b c^{2} x^{7}}{7} + \frac{c^{3} x^{9}}{9} + x^{5} \left (\frac{3 a c^{2}}{5} + \frac{3 b^{2} c}{5}\right ) + x^{3} \left (2 a b c + \frac{b^{3}}{3}\right ) + x \left (3 a^{2} c + 3 a b^{2}\right ) - \frac{a^{3} + 9 a^{2} b x^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b*c**2*x**7/7 + c**3*x**9/9 + x**5*(3*a*c**2/5 + 3*b**2*c/5) + x**3*(2*a*b*c + b**3/3) + x*(3*a**2*c + 3*a*b
**2) - (a**3 + 9*a**2*b*x**2)/(3*x**3)

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Giac [A]  time = 1.12945, size = 113, normalized size = 1.36 \begin{align*} \frac{1}{9} \, c^{3} x^{9} + \frac{3}{7} \, b c^{2} x^{7} + \frac{3}{5} \, b^{2} c x^{5} + \frac{3}{5} \, a c^{2} x^{5} + \frac{1}{3} \, b^{3} x^{3} + 2 \, a b c x^{3} + 3 \, a b^{2} x + 3 \, a^{2} c x - \frac{9 \, a^{2} b x^{2} + a^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^3*x^9 + 3/7*b*c^2*x^7 + 3/5*b^2*c*x^5 + 3/5*a*c^2*x^5 + 1/3*b^3*x^3 + 2*a*b*c*x^3 + 3*a*b^2*x + 3*a^2*c*
x - 1/3*(9*a^2*b*x^2 + a^3)/x^3