3.471 \(\int \frac{(a^2+2 a b x^2+b^2 x^4)^3}{x^{19}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{b^2 \left (a+b x^2\right )^7}{504 a^3 x^{14}}+\frac{b \left (a+b x^2\right )^7}{72 a^2 x^{16}}-\frac{\left (a+b x^2\right )^7}{18 a x^{18}} \]

[Out]

-(a + b*x^2)^7/(18*a*x^18) + (b*(a + b*x^2)^7)/(72*a^2*x^16) - (b^2*(a + b*x^2)^7)/(504*a^3*x^14)

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Rubi [A]  time = 0.0386362, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {28, 266, 45, 37} \[ -\frac{b^2 \left (a+b x^2\right )^7}{504 a^3 x^{14}}+\frac{b \left (a+b x^2\right )^7}{72 a^2 x^{16}}-\frac{\left (a+b x^2\right )^7}{18 a x^{18}} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^19,x]

[Out]

-(a + b*x^2)^7/(18*a*x^18) + (b*(a + b*x^2)^7)/(72*a^2*x^16) - (b^2*(a + b*x^2)^7)/(504*a^3*x^14)

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{19}} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^{19}} \, dx}{b^6}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^{10}} \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac{\left (a+b x^2\right )^7}{18 a x^{18}}-\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^9} \, dx,x,x^2\right )}{9 a b^5}\\ &=-\frac{\left (a+b x^2\right )^7}{18 a x^{18}}+\frac{b \left (a+b x^2\right )^7}{72 a^2 x^{16}}+\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^8} \, dx,x,x^2\right )}{72 a^2 b^4}\\ &=-\frac{\left (a+b x^2\right )^7}{18 a x^{18}}+\frac{b \left (a+b x^2\right )^7}{72 a^2 x^{16}}-\frac{b^2 \left (a+b x^2\right )^7}{504 a^3 x^{14}}\\ \end{align*}

Mathematica [A]  time = 0.0043377, size = 82, normalized size = 1.32 \[ -\frac{15 a^4 b^2}{14 x^{14}}-\frac{5 a^3 b^3}{3 x^{12}}-\frac{3 a^2 b^4}{2 x^{10}}-\frac{3 a^5 b}{8 x^{16}}-\frac{a^6}{18 x^{18}}-\frac{3 a b^5}{4 x^8}-\frac{b^6}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^19,x]

[Out]

-a^6/(18*x^18) - (3*a^5*b)/(8*x^16) - (15*a^4*b^2)/(14*x^14) - (5*a^3*b^3)/(3*x^12) - (3*a^2*b^4)/(2*x^10) - (
3*a*b^5)/(4*x^8) - b^6/(6*x^6)

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Maple [A]  time = 0.049, size = 69, normalized size = 1.1 \begin{align*} -{\frac{{a}^{6}}{18\,{x}^{18}}}-{\frac{3\,{a}^{5}b}{8\,{x}^{16}}}-{\frac{15\,{a}^{4}{b}^{2}}{14\,{x}^{14}}}-{\frac{3\,{a}^{2}{b}^{4}}{2\,{x}^{10}}}-{\frac{3\,a{b}^{5}}{4\,{x}^{8}}}-{\frac{{b}^{6}}{6\,{x}^{6}}}-{\frac{5\,{a}^{3}{b}^{3}}{3\,{x}^{12}}} \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^19,x)

[Out]

-1/18*a^6/x^18-3/8*a^5*b/x^16-15/14*a^4*b^2/x^14-3/2*a^2*b^4/x^10-3/4*a*b^5/x^8-1/6*b^6/x^6-5/3*a^3*b^3/x^12

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Maxima [A]  time = 0.977026, size = 95, normalized size = 1.53 \begin{align*} -\frac{84 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} + 756 \, a^{2} b^{4} x^{8} + 840 \, a^{3} b^{3} x^{6} + 540 \, a^{4} b^{2} x^{4} + 189 \, a^{5} b x^{2} + 28 \, a^{6}}{504 \, x^{18}} \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^19,x, algorithm="maxima")

[Out]

-1/504*(84*b^6*x^12 + 378*a*b^5*x^10 + 756*a^2*b^4*x^8 + 840*a^3*b^3*x^6 + 540*a^4*b^2*x^4 + 189*a^5*b*x^2 + 2
8*a^6)/x^18

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Fricas [A]  time = 1.65314, size = 166, normalized size = 2.68 \begin{align*} -\frac{84 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} + 756 \, a^{2} b^{4} x^{8} + 840 \, a^{3} b^{3} x^{6} + 540 \, a^{4} b^{2} x^{4} + 189 \, a^{5} b x^{2} + 28 \, a^{6}}{504 \, x^{18}} \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^19,x, algorithm="fricas")

[Out]

-1/504*(84*b^6*x^12 + 378*a*b^5*x^10 + 756*a^2*b^4*x^8 + 840*a^3*b^3*x^6 + 540*a^4*b^2*x^4 + 189*a^5*b*x^2 + 2
8*a^6)/x^18

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Sympy [A]  time = 0.861803, size = 75, normalized size = 1.21 \begin{align*} - \frac{28 a^{6} + 189 a^{5} b x^{2} + 540 a^{4} b^{2} x^{4} + 840 a^{3} b^{3} x^{6} + 756 a^{2} b^{4} x^{8} + 378 a b^{5} x^{10} + 84 b^{6} x^{12}}{504 x^{18}} \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**19,x)

[Out]

-(28*a**6 + 189*a**5*b*x**2 + 540*a**4*b**2*x**4 + 840*a**3*b**3*x**6 + 756*a**2*b**4*x**8 + 378*a*b**5*x**10
+ 84*b**6*x**12)/(504*x**18)

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Giac [A]  time = 1.12034, size = 95, normalized size = 1.53 \begin{align*} -\frac{84 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} + 756 \, a^{2} b^{4} x^{8} + 840 \, a^{3} b^{3} x^{6} + 540 \, a^{4} b^{2} x^{4} + 189 \, a^{5} b x^{2} + 28 \, a^{6}}{504 \, x^{18}} \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^19,x, algorithm="giac")

[Out]

-1/504*(84*b^6*x^12 + 378*a*b^5*x^10 + 756*a^2*b^4*x^8 + 840*a^3*b^3*x^6 + 540*a^4*b^2*x^4 + 189*a^5*b*x^2 + 2
8*a^6)/x^18