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

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

[Out]

-1/4*a^6/x^4-3*a^5*b/x^2+10*a^3*b^3*x^2+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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Rubi [A]  time = 0.05, antiderivative size = 72, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.00, size = 72, normalized size = 1.00 \[ -\frac {a^6}{4 x^4}-\frac {3 a^5 b}{x^2}+15 a^4 b^2 \log (x)+10 a^3 b^3 x^2+\frac {15}{4} a^2 b^4 x^4+a b^5 x^6+\frac {b^6 x^8}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*a^6/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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fricas [A]  time = 0.95, size = 71, normalized size = 0.99 \[ \frac {b^{6} x^{12} + 8 \, a b^{5} x^{10} + 30 \, a^{2} b^{4} x^{8} + 80 \, a^{3} b^{3} x^{6} + 120 \, a^{4} b^{2} x^{4} \log \relax (x) - 24 \, a^{5} b x^{2} - 2 \, a^{6}}{8 \, x^{4}} \]

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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giac [A]  time = 0.16, size = 80, normalized size = 1.11 \[ \frac {1}{8} \, b^{6} x^{8} + a b^{5} x^{6} + \frac {15}{4} \, a^{2} b^{4} x^{4} + 10 \, a^{3} b^{3} x^{2} + \frac {15}{2} \, a^{4} b^{2} \log \left (x^{2}\right ) - \frac {45 \, a^{4} b^{2} x^{4} + 12 \, a^{5} b x^{2} + a^{6}}{4 \, x^{4}} \]

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

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maple [A]  time = 0.01, size = 67, normalized size = 0.93 \[ \frac {b^{6} x^{8}}{8}+a \,b^{5} x^{6}+\frac {15 a^{2} b^{4} x^{4}}{4}+10 a^{3} b^{3} x^{2}+15 a^{4} b^{2} \ln \relax (x )-\frac {3 a^{5} b}{x^{2}}-\frac {a^{6}}{4 x^{4}} \]

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*a^3*b^3*x^2+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 = 1.37, size = 69, normalized size = 0.96 \[ \frac {1}{8} \, b^{6} x^{8} + a b^{5} x^{6} + \frac {15}{4} \, a^{2} b^{4} x^{4} + 10 \, a^{3} b^{3} x^{2} + \frac {15}{2} \, a^{4} b^{2} \log \left (x^{2}\right ) - \frac {12 \, a^{5} b x^{2} + a^{6}}{4 \, x^{4}} \]

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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mupad [B]  time = 0.04, size = 69, normalized size = 0.96 \[ \frac {b^6\,x^8}{8}-\frac {\frac {a^6}{4}+3\,b\,a^5\,x^2}{x^4}+a\,b^5\,x^6+10\,a^3\,b^3\,x^2+\frac {15\,a^2\,b^4\,x^4}{4}+15\,a^4\,b^2\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^3/x^5,x)

[Out]

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

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sympy [A]  time = 0.25, size = 73, normalized size = 1.01 \[ 15 a^{4} b^{2} \log {\relax (x )} + 10 a^{3} b^{3} x^{2} + \frac {15 a^{2} b^{4} x^{4}}{4} + a b^{5} x^{6} + \frac {b^{6} x^{8}}{8} + \frac {- a^{6} - 12 a^{5} b x^{2}}{4 x^{4}} \]

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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