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

Optimal. Leaf size=72 \[ \frac{15}{7} a^2 b^4 x^7+4 a^3 b^3 x^5+5 a^4 b^2 x^3+6 a^5 b x-\frac{a^6}{x}+\frac{2}{3} a b^5 x^9+\frac{b^6 x^{11}}{11} \]

[Out]

-(a^6/x) + 6*a^5*b*x + 5*a^4*b^2*x^3 + 4*a^3*b^3*x^5 + (15*a^2*b^4*x^7)/7 + (2*a*b^5*x^9)/3 + (b^6*x^11)/11

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Rubi [A]  time = 0.0392213, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {28, 270} \[ \frac{15}{7} a^2 b^4 x^7+4 a^3 b^3 x^5+5 a^4 b^2 x^3+6 a^5 b x-\frac{a^6}{x}+\frac{2}{3} a b^5 x^9+\frac{b^6 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^6/x) + 6*a^5*b*x + 5*a^4*b^2*x^3 + 4*a^3*b^3*x^5 + (15*a^2*b^4*x^7)/7 + (2*a*b^5*x^9)/3 + (b^6*x^11)/11

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^2} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^2} \, dx}{b^6}\\ &=\frac{\int \left (6 a^5 b^7+\frac{a^6 b^6}{x^2}+15 a^4 b^8 x^2+20 a^3 b^9 x^4+15 a^2 b^{10} x^6+6 a b^{11} x^8+b^{12} x^{10}\right ) \, dx}{b^6}\\ &=-\frac{a^6}{x}+6 a^5 b x+5 a^4 b^2 x^3+4 a^3 b^3 x^5+\frac{15}{7} a^2 b^4 x^7+\frac{2}{3} a b^5 x^9+\frac{b^6 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0084472, size = 72, normalized size = 1. \[ \frac{15}{7} a^2 b^4 x^7+4 a^3 b^3 x^5+5 a^4 b^2 x^3+6 a^5 b x-\frac{a^6}{x}+\frac{2}{3} a b^5 x^9+\frac{b^6 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^6/x) + 6*a^5*b*x + 5*a^4*b^2*x^3 + 4*a^3*b^3*x^5 + (15*a^2*b^4*x^7)/7 + (2*a*b^5*x^9)/3 + (b^6*x^11)/11

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Maple [A]  time = 0.045, size = 67, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{x}}+6\,{a}^{5}bx+5\,{a}^{4}{b}^{2}{x}^{3}+4\,{a}^{3}{b}^{3}{x}^{5}+{\frac{15\,{a}^{2}{b}^{4}{x}^{7}}{7}}+{\frac{2\,a{b}^{5}{x}^{9}}{3}}+{\frac{{b}^{6}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^6/x+6*a^5*b*x+5*a^4*b^2*x^3+4*a^3*b^3*x^5+15/7*a^2*b^4*x^7+2/3*a*b^5*x^9+1/11*b^6*x^11

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Maxima [A]  time = 0.991793, size = 89, normalized size = 1.24 \begin{align*} \frac{1}{11} \, b^{6} x^{11} + \frac{2}{3} \, a b^{5} x^{9} + \frac{15}{7} \, a^{2} b^{4} x^{7} + 4 \, a^{3} b^{3} x^{5} + 5 \, a^{4} b^{2} x^{3} + 6 \, a^{5} b x - \frac{a^{6}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^6*x^11 + 2/3*a*b^5*x^9 + 15/7*a^2*b^4*x^7 + 4*a^3*b^3*x^5 + 5*a^4*b^2*x^3 + 6*a^5*b*x - a^6/x

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Fricas [A]  time = 1.71723, size = 165, normalized size = 2.29 \begin{align*} \frac{21 \, b^{6} x^{12} + 154 \, a b^{5} x^{10} + 495 \, a^{2} b^{4} x^{8} + 924 \, a^{3} b^{3} x^{6} + 1155 \, a^{4} b^{2} x^{4} + 1386 \, a^{5} b x^{2} - 231 \, a^{6}}{231 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/231*(21*b^6*x^12 + 154*a*b^5*x^10 + 495*a^2*b^4*x^8 + 924*a^3*b^3*x^6 + 1155*a^4*b^2*x^4 + 1386*a^5*b*x^2 -
231*a^6)/x

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Sympy [A]  time = 0.304902, size = 70, normalized size = 0.97 \begin{align*} - \frac{a^{6}}{x} + 6 a^{5} b x + 5 a^{4} b^{2} x^{3} + 4 a^{3} b^{3} x^{5} + \frac{15 a^{2} b^{4} x^{7}}{7} + \frac{2 a b^{5} x^{9}}{3} + \frac{b^{6} x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**6/x + 6*a**5*b*x + 5*a**4*b**2*x**3 + 4*a**3*b**3*x**5 + 15*a**2*b**4*x**7/7 + 2*a*b**5*x**9/3 + b**6*x**1
1/11

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Giac [A]  time = 1.14083, size = 89, normalized size = 1.24 \begin{align*} \frac{1}{11} \, b^{6} x^{11} + \frac{2}{3} \, a b^{5} x^{9} + \frac{15}{7} \, a^{2} b^{4} x^{7} + 4 \, a^{3} b^{3} x^{5} + 5 \, a^{4} b^{2} x^{3} + 6 \, a^{5} b x - \frac{a^{6}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*b^6*x^11 + 2/3*a*b^5*x^9 + 15/7*a^2*b^4*x^7 + 4*a^3*b^3*x^5 + 5*a^4*b^2*x^3 + 6*a^5*b*x - a^6/x