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

Optimal. Leaf size=80 \[ 3 a^2 b x-\frac{a^3}{x}+\frac{3}{7} c x^7 \left (a c+b^2\right )+\frac{1}{5} b x^5 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11} \]

[Out]

-(a^3/x) + 3*a^2*b*x + a*(b^2 + a*c)*x^3 + (b*(b^2 + 6*a*c)*x^5)/5 + (3*c*(b^2 + a*c)*x^7)/7 + (b*c^2*x^9)/3 +
(c^3*x^11)/11

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Rubi [A]  time = 0.039367, antiderivative size = 80, 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} \[ 3 a^2 b x-\frac{a^3}{x}+\frac{3}{7} c x^7 \left (a c+b^2\right )+\frac{1}{5} b x^5 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^3/x^2,x]

[Out]

-(a^3/x) + 3*a^2*b*x + a*(b^2 + a*c)*x^3 + (b*(b^2 + 6*a*c)*x^5)/5 + (3*c*(b^2 + a*c)*x^7)/7 + (b*c^2*x^9)/3 +
(c^3*x^11)/11

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^2} \, dx &=\int \left (3 a^2 b+\frac{a^3}{x^2}+3 a \left (b^2+a c\right ) x^2+b \left (b^2+6 a c\right ) x^4+3 c \left (b^2+a c\right ) x^6+3 b c^2 x^8+c^3 x^{10}\right ) \, dx\\ &=-\frac{a^3}{x}+3 a^2 b x+a \left (b^2+a c\right ) x^3+\frac{1}{5} b \left (b^2+6 a c\right ) x^5+\frac{3}{7} c \left (b^2+a c\right ) x^7+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0231187, size = 80, normalized size = 1. \[ 3 a^2 b x-\frac{a^3}{x}+\frac{3}{7} c x^7 \left (a c+b^2\right )+\frac{1}{5} b x^5 \left (6 a c+b^2\right )+a x^3 \left (a c+b^2\right )+\frac{1}{3} b c^2 x^9+\frac{c^3 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3/x) + 3*a^2*b*x + a*(b^2 + a*c)*x^3 + (b*(b^2 + 6*a*c)*x^5)/5 + (3*c*(b^2 + a*c)*x^7)/7 + (b*c^2*x^9)/3 +
(c^3*x^11)/11

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Maple [A]  time = 0.046, size = 84, normalized size = 1.1 \begin{align*}{\frac{{c}^{3}{x}^{11}}{11}}+{\frac{b{c}^{2}{x}^{9}}{3}}+{\frac{3\,{x}^{7}a{c}^{2}}{7}}+{\frac{3\,{b}^{2}c{x}^{7}}{7}}+{\frac{6\,{x}^{5}abc}{5}}+{\frac{{b}^{3}{x}^{5}}{5}}+{x}^{3}{a}^{2}c+a{x}^{3}{b}^{2}+3\,{a}^{2}bx-{\frac{{a}^{3}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^3*x^11+1/3*b*c^2*x^9+3/7*x^7*a*c^2+3/7*b^2*c*x^7+6/5*x^5*a*b*c+1/5*b^3*x^5+x^3*a^2*c+a*x^3*b^2+3*a^2*b*
x-a^3/x

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Maxima [A]  time = 0.955623, size = 105, normalized size = 1.31 \begin{align*} \frac{1}{11} \, c^{3} x^{11} + \frac{1}{3} \, b c^{2} x^{9} + \frac{3}{7} \,{\left (b^{2} c + a c^{2}\right )} x^{7} + \frac{1}{5} \,{\left (b^{3} + 6 \, a b c\right )} x^{5} + 3 \, a^{2} b x +{\left (a b^{2} + a^{2} c\right )} x^{3} - \frac{a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^3*x^11 + 1/3*b*c^2*x^9 + 3/7*(b^2*c + a*c^2)*x^7 + 1/5*(b^3 + 6*a*b*c)*x^5 + 3*a^2*b*x + (a*b^2 + a^2*c
)*x^3 - a^3/x

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Fricas [A]  time = 1.46594, size = 201, normalized size = 2.51 \begin{align*} \frac{105 \, c^{3} x^{12} + 385 \, b c^{2} x^{10} + 495 \,{\left (b^{2} c + a c^{2}\right )} x^{8} + 231 \,{\left (b^{3} + 6 \, a b c\right )} x^{6} + 3465 \, a^{2} b x^{2} + 1155 \,{\left (a b^{2} + a^{2} c\right )} x^{4} - 1155 \, a^{3}}{1155 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1155*(105*c^3*x^12 + 385*b*c^2*x^10 + 495*(b^2*c + a*c^2)*x^8 + 231*(b^3 + 6*a*b*c)*x^6 + 3465*a^2*b*x^2 + 1
155*(a*b^2 + a^2*c)*x^4 - 1155*a^3)/x

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Sympy [A]  time = 0.359609, size = 82, normalized size = 1.02 \begin{align*} - \frac{a^{3}}{x} + 3 a^{2} b x + \frac{b c^{2} x^{9}}{3} + \frac{c^{3} x^{11}}{11} + x^{7} \left (\frac{3 a c^{2}}{7} + \frac{3 b^{2} c}{7}\right ) + x^{5} \left (\frac{6 a b c}{5} + \frac{b^{3}}{5}\right ) + x^{3} \left (a^{2} c + a b^{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**2,x)

[Out]

-a**3/x + 3*a**2*b*x + b*c**2*x**9/3 + c**3*x**11/11 + x**7*(3*a*c**2/7 + 3*b**2*c/7) + x**5*(6*a*b*c/5 + b**3
/5) + x**3*(a**2*c + a*b**2)

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Giac [A]  time = 1.16089, size = 112, normalized size = 1.4 \begin{align*} \frac{1}{11} \, c^{3} x^{11} + \frac{1}{3} \, b c^{2} x^{9} + \frac{3}{7} \, b^{2} c x^{7} + \frac{3}{7} \, a c^{2} x^{7} + \frac{1}{5} \, b^{3} x^{5} + \frac{6}{5} \, a b c x^{5} + a b^{2} x^{3} + a^{2} c x^{3} + 3 \, a^{2} b x - \frac{a^{3}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^3*x^11 + 1/3*b*c^2*x^9 + 3/7*b^2*c*x^7 + 3/7*a*c^2*x^7 + 1/5*b^3*x^5 + 6/5*a*b*c*x^5 + a*b^2*x^3 + a^2*
c*x^3 + 3*a^2*b*x - a^3/x