3.459 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^7} \, dx\)

Optimal. Leaf size=79 \[ -\frac{15 a^4 b^2}{2 x^2}+\frac{15}{2} a^2 b^4 x^2+20 a^3 b^3 \log (x)-\frac{3 a^5 b}{2 x^4}-\frac{a^6}{6 x^6}+\frac{3}{2} a b^5 x^4+\frac{b^6 x^6}{6} \]

[Out]

-a^6/(6*x^6) - (3*a^5*b)/(2*x^4) - (15*a^4*b^2)/(2*x^2) + (15*a^2*b^4*x^2)/2 + (3*a*b^5*x^4)/2 + (b^6*x^6)/6 +
20*a^3*b^3*Log[x]

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Rubi [A]  time = 0.0505286, antiderivative size = 79, 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} \[ -\frac{15 a^4 b^2}{2 x^2}+\frac{15}{2} a^2 b^4 x^2+20 a^3 b^3 \log (x)-\frac{3 a^5 b}{2 x^4}-\frac{a^6}{6 x^6}+\frac{3}{2} a b^5 x^4+\frac{b^6 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^7,x]

[Out]

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

Mathematica [A]  time = 0.0049735, size = 79, normalized size = 1. \[ -\frac{15 a^4 b^2}{2 x^2}+\frac{15}{2} a^2 b^4 x^2+20 a^3 b^3 \log (x)-\frac{3 a^5 b}{2 x^4}-\frac{a^6}{6 x^6}+\frac{3}{2} a b^5 x^4+\frac{b^6 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^7,x]

[Out]

-a^6/(6*x^6) - (3*a^5*b)/(2*x^4) - (15*a^4*b^2)/(2*x^2) + (15*a^2*b^4*x^2)/2 + (3*a*b^5*x^4)/2 + (b^6*x^6)/6 +
20*a^3*b^3*Log[x]

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Maple [A]  time = 0.05, size = 68, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{6\,{x}^{6}}}-{\frac{3\,{a}^{5}b}{2\,{x}^{4}}}-{\frac{15\,{a}^{4}{b}^{2}}{2\,{x}^{2}}}+{\frac{15\,{a}^{2}{b}^{4}{x}^{2}}{2}}+{\frac{3\,a{b}^{5}{x}^{4}}{2}}+{\frac{{b}^{6}{x}^{6}}{6}}+20\,{a}^{3}{b}^{3}\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^7,x)

[Out]

-1/6*a^6/x^6-3/2*a^5*b/x^4-15/2*a^4*b^2/x^2+15/2*a^2*b^4*x^2+3/2*a*b^5*x^4+1/6*b^6*x^6+20*a^3*b^3*ln(x)

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Maxima [A]  time = 0.989, size = 95, normalized size = 1.2 \begin{align*} \frac{1}{6} \, b^{6} x^{6} + \frac{3}{2} \, a b^{5} x^{4} + \frac{15}{2} \, a^{2} b^{4} x^{2} + 10 \, a^{3} b^{3} \log \left (x^{2}\right ) - \frac{45 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{6 \, x^{6}} \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^7,x, algorithm="maxima")

[Out]

1/6*b^6*x^6 + 3/2*a*b^5*x^4 + 15/2*a^2*b^4*x^2 + 10*a^3*b^3*log(x^2) - 1/6*(45*a^4*b^2*x^4 + 9*a^5*b*x^2 + a^6
)/x^6

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Fricas [A]  time = 1.79303, size = 154, normalized size = 1.95 \begin{align*} \frac{b^{6} x^{12} + 9 \, a b^{5} x^{10} + 45 \, a^{2} b^{4} x^{8} + 120 \, a^{3} b^{3} x^{6} \log \left (x\right ) - 45 \, a^{4} b^{2} x^{4} - 9 \, a^{5} b x^{2} - a^{6}}{6 \, x^{6}} \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^7,x, algorithm="fricas")

[Out]

1/6*(b^6*x^12 + 9*a*b^5*x^10 + 45*a^2*b^4*x^8 + 120*a^3*b^3*x^6*log(x) - 45*a^4*b^2*x^4 - 9*a^5*b*x^2 - a^6)/x
^6

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Sympy [A]  time = 0.478979, size = 75, normalized size = 0.95 \begin{align*} 20 a^{3} b^{3} \log{\left (x \right )} + \frac{15 a^{2} b^{4} x^{2}}{2} + \frac{3 a b^{5} x^{4}}{2} + \frac{b^{6} x^{6}}{6} - \frac{a^{6} + 9 a^{5} b x^{2} + 45 a^{4} b^{2} x^{4}}{6 x^{6}} \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**7,x)

[Out]

20*a**3*b**3*log(x) + 15*a**2*b**4*x**2/2 + 3*a*b**5*x**4/2 + b**6*x**6/6 - (a**6 + 9*a**5*b*x**2 + 45*a**4*b*
*2*x**4)/(6*x**6)

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Giac [A]  time = 1.14009, size = 109, normalized size = 1.38 \begin{align*} \frac{1}{6} \, b^{6} x^{6} + \frac{3}{2} \, a b^{5} x^{4} + \frac{15}{2} \, a^{2} b^{4} x^{2} + 10 \, a^{3} b^{3} \log \left (x^{2}\right ) - \frac{110 \, a^{3} b^{3} x^{6} + 45 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{6 \, x^{6}} \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^7,x, algorithm="giac")

[Out]

1/6*b^6*x^6 + 3/2*a*b^5*x^4 + 15/2*a^2*b^4*x^2 + 10*a^3*b^3*log(x^2) - 1/6*(110*a^3*b^3*x^6 + 45*a^4*b^2*x^4 +
9*a^5*b*x^2 + a^6)/x^6