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

Optimal. Leaf size=76 \[ \frac{15}{8} a^2 b^4 x^8+\frac{10}{3} a^3 b^3 x^6+\frac{15}{4} a^4 b^2 x^4+3 a^5 b x^2+a^6 \log (x)+\frac{3}{5} a b^5 x^{10}+\frac{b^6 x^{12}}{12} \]

[Out]

3*a^5*b*x^2 + (15*a^4*b^2*x^4)/4 + (10*a^3*b^3*x^6)/3 + (15*a^2*b^4*x^8)/8 + (3*a*b^5*x^10)/5 + (b^6*x^12)/12
+ a^6*Log[x]

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Rubi [A]  time = 0.0545137, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ \frac{15}{8} a^2 b^4 x^8+\frac{10}{3} a^3 b^3 x^6+\frac{15}{4} a^4 b^2 x^4+3 a^5 b x^2+a^6 \log (x)+\frac{3}{5} a b^5 x^{10}+\frac{b^6 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

3*a^5*b*x^2 + (15*a^4*b^2*x^4)/4 + (10*a^3*b^3*x^6)/3 + (15*a^2*b^4*x^8)/8 + (3*a*b^5*x^10)/5 + (b^6*x^12)/12
+ a^6*Log[x]

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0044944, size = 76, normalized size = 1. \[ \frac{15}{8} a^2 b^4 x^8+\frac{10}{3} a^3 b^3 x^6+\frac{15}{4} a^4 b^2 x^4+3 a^5 b x^2+a^6 \log (x)+\frac{3}{5} a b^5 x^{10}+\frac{b^6 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

3*a^5*b*x^2 + (15*a^4*b^2*x^4)/4 + (10*a^3*b^3*x^6)/3 + (15*a^2*b^4*x^8)/8 + (3*a*b^5*x^10)/5 + (b^6*x^12)/12
+ a^6*Log[x]

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Maple [A]  time = 0.043, size = 67, normalized size = 0.9 \begin{align*} 3\,{a}^{5}b{x}^{2}+{\frac{15\,{a}^{4}{b}^{2}{x}^{4}}{4}}+{\frac{10\,{a}^{3}{b}^{3}{x}^{6}}{3}}+{\frac{15\,{a}^{2}{b}^{4}{x}^{8}}{8}}+{\frac{3\,a{b}^{5}{x}^{10}}{5}}+{\frac{{b}^{6}{x}^{12}}{12}}+{a}^{6}\ln \left ( x \right ) \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,x)

[Out]

3*a^5*b*x^2+15/4*a^4*b^2*x^4+10/3*a^3*b^3*x^6+15/8*a^2*b^4*x^8+3/5*a*b^5*x^10+1/12*b^6*x^12+a^6*ln(x)

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Maxima [A]  time = 1.01517, size = 93, normalized size = 1.22 \begin{align*} \frac{1}{12} \, b^{6} x^{12} + \frac{3}{5} \, a b^{5} x^{10} + \frac{15}{8} \, a^{2} b^{4} x^{8} + \frac{10}{3} \, a^{3} b^{3} x^{6} + \frac{15}{4} \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + \frac{1}{2} \, a^{6} \log \left (x^{2}\right ) \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,x, algorithm="maxima")

[Out]

1/12*b^6*x^12 + 3/5*a*b^5*x^10 + 15/8*a^2*b^4*x^8 + 10/3*a^3*b^3*x^6 + 15/4*a^4*b^2*x^4 + 3*a^5*b*x^2 + 1/2*a^
6*log(x^2)

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Fricas [A]  time = 1.75273, size = 157, normalized size = 2.07 \begin{align*} \frac{1}{12} \, b^{6} x^{12} + \frac{3}{5} \, a b^{5} x^{10} + \frac{15}{8} \, a^{2} b^{4} x^{8} + \frac{10}{3} \, a^{3} b^{3} x^{6} + \frac{15}{4} \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6} \log \left (x\right ) \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,x, algorithm="fricas")

[Out]

1/12*b^6*x^12 + 3/5*a*b^5*x^10 + 15/8*a^2*b^4*x^8 + 10/3*a^3*b^3*x^6 + 15/4*a^4*b^2*x^4 + 3*a^5*b*x^2 + a^6*lo
g(x)

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Sympy [A]  time = 0.30263, size = 76, normalized size = 1. \begin{align*} a^{6} \log{\left (x \right )} + 3 a^{5} b x^{2} + \frac{15 a^{4} b^{2} x^{4}}{4} + \frac{10 a^{3} b^{3} x^{6}}{3} + \frac{15 a^{2} b^{4} x^{8}}{8} + \frac{3 a b^{5} x^{10}}{5} + \frac{b^{6} x^{12}}{12} \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,x)

[Out]

a**6*log(x) + 3*a**5*b*x**2 + 15*a**4*b**2*x**4/4 + 10*a**3*b**3*x**6/3 + 15*a**2*b**4*x**8/8 + 3*a*b**5*x**10
/5 + b**6*x**12/12

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Giac [A]  time = 1.12874, size = 93, normalized size = 1.22 \begin{align*} \frac{1}{12} \, b^{6} x^{12} + \frac{3}{5} \, a b^{5} x^{10} + \frac{15}{8} \, a^{2} b^{4} x^{8} + \frac{10}{3} \, a^{3} b^{3} x^{6} + \frac{15}{4} \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + \frac{1}{2} \, a^{6} \log \left (x^{2}\right ) \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,x, algorithm="giac")

[Out]

1/12*b^6*x^12 + 3/5*a*b^5*x^10 + 15/8*a^2*b^4*x^8 + 10/3*a^3*b^3*x^6 + 15/4*a^4*b^2*x^4 + 3*a^5*b*x^2 + 1/2*a^
6*log(x^2)