3.69 \(\int \frac{(a+b x^2)^5}{x^{21}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{5 a^3 b^2}{8 x^{16}}-\frac{5 a^2 b^3}{7 x^{14}}-\frac{5 a^4 b}{18 x^{18}}-\frac{a^5}{20 x^{20}}-\frac{5 a b^4}{12 x^{12}}-\frac{b^5}{10 x^{10}} \]

[Out]

-a^5/(20*x^20) - (5*a^4*b)/(18*x^18) - (5*a^3*b^2)/(8*x^16) - (5*a^2*b^3)/(7*x^14) - (5*a*b^4)/(12*x^12) - b^5
/(10*x^10)

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Rubi [A]  time = 0.0305198, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{5 a^3 b^2}{8 x^{16}}-\frac{5 a^2 b^3}{7 x^{14}}-\frac{5 a^4 b}{18 x^{18}}-\frac{a^5}{20 x^{20}}-\frac{5 a b^4}{12 x^{12}}-\frac{b^5}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^21,x]

[Out]

-a^5/(20*x^20) - (5*a^4*b)/(18*x^18) - (5*a^3*b^2)/(8*x^16) - (5*a^2*b^3)/(7*x^14) - (5*a*b^4)/(12*x^12) - b^5
/(10*x^10)

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+b x^2\right )^5}{x^{21}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{11}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^5}{x^{11}}+\frac{5 a^4 b}{x^{10}}+\frac{10 a^3 b^2}{x^9}+\frac{10 a^2 b^3}{x^8}+\frac{5 a b^4}{x^7}+\frac{b^5}{x^6}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5}{20 x^{20}}-\frac{5 a^4 b}{18 x^{18}}-\frac{5 a^3 b^2}{8 x^{16}}-\frac{5 a^2 b^3}{7 x^{14}}-\frac{5 a b^4}{12 x^{12}}-\frac{b^5}{10 x^{10}}\\ \end{align*}

Mathematica [A]  time = 0.0042634, size = 69, normalized size = 1. \[ -\frac{5 a^3 b^2}{8 x^{16}}-\frac{5 a^2 b^3}{7 x^{14}}-\frac{5 a^4 b}{18 x^{18}}-\frac{a^5}{20 x^{20}}-\frac{5 a b^4}{12 x^{12}}-\frac{b^5}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^21,x]

[Out]

-a^5/(20*x^20) - (5*a^4*b)/(18*x^18) - (5*a^3*b^2)/(8*x^16) - (5*a^2*b^3)/(7*x^14) - (5*a*b^4)/(12*x^12) - b^5
/(10*x^10)

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Maple [A]  time = 0.006, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{20\,{x}^{20}}}-{\frac{5\,{a}^{4}b}{18\,{x}^{18}}}-{\frac{5\,{a}^{3}{b}^{2}}{8\,{x}^{16}}}-{\frac{5\,{a}^{2}{b}^{3}}{7\,{x}^{14}}}-{\frac{5\,a{b}^{4}}{12\,{x}^{12}}}-{\frac{{b}^{5}}{10\,{x}^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^21,x)

[Out]

-1/20*a^5/x^20-5/18*a^4*b/x^18-5/8*a^3*b^2/x^16-5/7*a^2*b^3/x^14-5/12*a*b^4/x^12-1/10*b^5/x^10

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Maxima [A]  time = 2.77342, size = 80, normalized size = 1.16 \begin{align*} -\frac{252 \, b^{5} x^{10} + 1050 \, a b^{4} x^{8} + 1800 \, a^{2} b^{3} x^{6} + 1575 \, a^{3} b^{2} x^{4} + 700 \, a^{4} b x^{2} + 126 \, a^{5}}{2520 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^21,x, algorithm="maxima")

[Out]

-1/2520*(252*b^5*x^10 + 1050*a*b^4*x^8 + 1800*a^2*b^3*x^6 + 1575*a^3*b^2*x^4 + 700*a^4*b*x^2 + 126*a^5)/x^20

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Fricas [A]  time = 1.26214, size = 149, normalized size = 2.16 \begin{align*} -\frac{252 \, b^{5} x^{10} + 1050 \, a b^{4} x^{8} + 1800 \, a^{2} b^{3} x^{6} + 1575 \, a^{3} b^{2} x^{4} + 700 \, a^{4} b x^{2} + 126 \, a^{5}}{2520 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^21,x, algorithm="fricas")

[Out]

-1/2520*(252*b^5*x^10 + 1050*a*b^4*x^8 + 1800*a^2*b^3*x^6 + 1575*a^3*b^2*x^4 + 700*a^4*b*x^2 + 126*a^5)/x^20

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Sympy [A]  time = 0.874056, size = 63, normalized size = 0.91 \begin{align*} - \frac{126 a^{5} + 700 a^{4} b x^{2} + 1575 a^{3} b^{2} x^{4} + 1800 a^{2} b^{3} x^{6} + 1050 a b^{4} x^{8} + 252 b^{5} x^{10}}{2520 x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**21,x)

[Out]

-(126*a**5 + 700*a**4*b*x**2 + 1575*a**3*b**2*x**4 + 1800*a**2*b**3*x**6 + 1050*a*b**4*x**8 + 252*b**5*x**10)/
(2520*x**20)

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Giac [A]  time = 2.51817, size = 80, normalized size = 1.16 \begin{align*} -\frac{252 \, b^{5} x^{10} + 1050 \, a b^{4} x^{8} + 1800 \, a^{2} b^{3} x^{6} + 1575 \, a^{3} b^{2} x^{4} + 700 \, a^{4} b x^{2} + 126 \, a^{5}}{2520 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^21,x, algorithm="giac")

[Out]

-1/2520*(252*b^5*x^10 + 1050*a*b^4*x^8 + 1800*a^2*b^3*x^6 + 1575*a^3*b^2*x^4 + 700*a^4*b*x^2 + 126*a^5)/x^20