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

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, size = 83, normalized size = 1.00 \[ -\frac {a^3}{3 x^3}-\frac {3 a^2 b}{x}+\frac {3}{5} c x^5 \left (a c+b^2\right )+\frac {1}{3} b x^3 \left (6 a c+b^2\right )+3 a x \left (a c+b^2\right )+\frac {3}{7} b c^2 x^7+\frac {c^3 x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*a^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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fricas [A]  time = 0.88, size = 83, normalized size = 1.00 \[ \frac {35 \, c^{3} x^{12} + 135 \, b c^{2} x^{10} + 189 \, {\left (b^{2} c + a c^{2}\right )} x^{8} + 105 \, {\left (b^{3} + 6 \, a b c\right )} x^{6} - 945 \, a^{2} b x^{2} + 945 \, {\left (a b^{2} + a^{2} c\right )} x^{4} - 105 \, a^{3}}{315 \, x^{3}} \]

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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giac [A]  time = 0.18, size = 84, normalized size = 1.01 \[ \frac {1}{9} \, c^{3} x^{9} + \frac {3}{7} \, b c^{2} x^{7} + \frac {3}{5} \, b^{2} c x^{5} + \frac {3}{5} \, a c^{2} x^{5} + \frac {1}{3} \, b^{3} x^{3} + 2 \, a b c x^{3} + 3 \, a b^{2} x + 3 \, a^{2} c x - \frac {9 \, a^{2} b x^{2} + a^{3}}{3 \, x^{3}} \]

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

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maple [A]  time = 0.01, size = 84, normalized size = 1.01 \[ \frac {c^{3} x^{9}}{9}+\frac {3 b \,c^{2} x^{7}}{7}+\frac {3 a \,c^{2} x^{5}}{5}+\frac {3 b^{2} c \,x^{5}}{5}+2 a b c \,x^{3}+\frac {b^{3} x^{3}}{3}+3 a^{2} c x +3 a \,b^{2} x -\frac {3 a^{2} b}{x}-\frac {a^{3}}{3 x^{3}} \]

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*a*b^2*x-3*a^2*b/x-1/
3*a^3/x^3

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maxima [A]  time = 1.35, size = 80, normalized size = 0.96 \[ \frac {1}{9} \, c^{3} x^{9} + \frac {3}{7} \, b c^{2} x^{7} + \frac {3}{5} \, {\left (b^{2} c + a c^{2}\right )} x^{5} + \frac {1}{3} \, {\left (b^{3} + 6 \, a b c\right )} x^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x - \frac {9 \, a^{2} b x^{2} + a^{3}}{3 \, x^{3}} \]

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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mupad [B]  time = 0.03, size = 77, normalized size = 0.93 \[ x^3\,\left (\frac {b^3}{3}+2\,a\,c\,b\right )-\frac {\frac {a^3}{3}+3\,b\,a^2\,x^2}{x^3}+\frac {c^3\,x^9}{9}+\frac {3\,b\,c^2\,x^7}{7}+3\,a\,x\,\left (b^2+a\,c\right )+\frac {3\,c\,x^5\,\left (b^2+a\,c\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2 + c*x^4)^3/x^4,x)

[Out]

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

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sympy [A]  time = 0.24, size = 90, normalized size = 1.08 \[ \frac {3 b c^{2} x^{7}}{7} + \frac {c^{3} x^{9}}{9} + x^{5} \left (\frac {3 a c^{2}}{5} + \frac {3 b^{2} c}{5}\right ) + x^{3} \left (2 a b c + \frac {b^{3}}{3}\right ) + x \left (3 a^{2} c + 3 a b^{2}\right ) + \frac {- a^{3} - 9 a^{2} b x^{2}}{3 x^{3}} \]

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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