3.457 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^5} \, dx\)

Optimal. Leaf size=72 \[ \frac{15}{4} a^2 b^4 x^4+10 a^3 b^3 x^2+15 a^4 b^2 \log (x)-\frac{3 a^5 b}{x^2}-\frac{a^6}{4 x^4}+a b^5 x^6+\frac{b^6 x^8}{8} \]

[Out]

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

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Rubi [A]  time = 0.052438, antiderivative size = 72, 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} \[ \frac{15}{4} a^2 b^4 x^4+10 a^3 b^3 x^2+15 a^4 b^2 \log (x)-\frac{3 a^5 b}{x^2}-\frac{a^6}{4 x^4}+a b^5 x^6+\frac{b^6 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^5,x]

[Out]

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

Mathematica [A]  time = 0.0048738, size = 72, normalized size = 1. \[ \frac{15}{4} a^2 b^4 x^4+10 a^3 b^3 x^2+15 a^4 b^2 \log (x)-\frac{3 a^5 b}{x^2}-\frac{a^6}{4 x^4}+a b^5 x^6+\frac{b^6 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^5,x]

[Out]

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

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Maple [A]  time = 0.049, size = 67, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{4\,{x}^{4}}}-3\,{\frac{{a}^{5}b}{{x}^{2}}}+10\,{x}^{2}{a}^{3}{b}^{3}+{\frac{15\,{a}^{2}{b}^{4}{x}^{4}}{4}}+a{b}^{5}{x}^{6}+{\frac{{b}^{6}{x}^{8}}{8}}+15\,{a}^{4}{b}^{2}\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^5,x)

[Out]

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

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Maxima [A]  time = 1.01768, size = 93, normalized size = 1.29 \begin{align*} \frac{1}{8} \, b^{6} x^{8} + a b^{5} x^{6} + \frac{15}{4} \, a^{2} b^{4} x^{4} + 10 \, a^{3} b^{3} x^{2} + \frac{15}{2} \, a^{4} b^{2} \log \left (x^{2}\right ) - \frac{12 \, a^{5} b x^{2} + a^{6}}{4 \, x^{4}} \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^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.71567, size = 158, normalized size = 2.19 \begin{align*} \frac{b^{6} x^{12} + 8 \, a b^{5} x^{10} + 30 \, a^{2} b^{4} x^{8} + 80 \, a^{3} b^{3} x^{6} + 120 \, a^{4} b^{2} x^{4} \log \left (x\right ) - 24 \, a^{5} b x^{2} - 2 \, a^{6}}{8 \, x^{4}} \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^5,x, algorithm="fricas")

[Out]

1/8*(b^6*x^12 + 8*a*b^5*x^10 + 30*a^2*b^4*x^8 + 80*a^3*b^3*x^6 + 120*a^4*b^2*x^4*log(x) - 24*a^5*b*x^2 - 2*a^6
)/x^4

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Sympy [A]  time = 0.395663, size = 71, normalized size = 0.99 \begin{align*} 15 a^{4} b^{2} \log{\left (x \right )} + 10 a^{3} b^{3} x^{2} + \frac{15 a^{2} b^{4} x^{4}}{4} + a b^{5} x^{6} + \frac{b^{6} x^{8}}{8} - \frac{a^{6} + 12 a^{5} b x^{2}}{4 x^{4}} \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**5,x)

[Out]

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

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Giac [A]  time = 1.14632, size = 108, normalized size = 1.5 \begin{align*} \frac{1}{8} \, b^{6} x^{8} + a b^{5} x^{6} + \frac{15}{4} \, a^{2} b^{4} x^{4} + 10 \, a^{3} b^{3} x^{2} + \frac{15}{2} \, a^{4} b^{2} \log \left (x^{2}\right ) - \frac{45 \, a^{4} b^{2} x^{4} + 12 \, a^{5} b x^{2} + a^{6}}{4 \, x^{4}} \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^5,x, algorithm="giac")

[Out]

1/8*b^6*x^8 + a*b^5*x^6 + 15/4*a^2*b^4*x^4 + 10*a^3*b^3*x^2 + 15/2*a^4*b^2*log(x^2) - 1/4*(45*a^4*b^2*x^4 + 12
*a^5*b*x^2 + a^6)/x^4