3.68 \(\int \frac{(a+b x^2)^5}{x^{19}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{5 a^3 b^2}{7 x^{14}}-\frac{5 a^2 b^3}{6 x^{12}}-\frac{5 a^4 b}{16 x^{16}}-\frac{a^5}{18 x^{18}}-\frac{a b^4}{2 x^{10}}-\frac{b^5}{8 x^8} \]

[Out]

-a^5/(18*x^18) - (5*a^4*b)/(16*x^16) - (5*a^3*b^2)/(7*x^14) - (5*a^2*b^3)/(6*x^12) - (a*b^4)/(2*x^10) - b^5/(8
*x^8)

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Rubi [A]  time = 0.0316203, 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} \[ -\frac{5 a^3 b^2}{7 x^{14}}-\frac{5 a^2 b^3}{6 x^{12}}-\frac{5 a^4 b}{16 x^{16}}-\frac{a^5}{18 x^{18}}-\frac{a b^4}{2 x^{10}}-\frac{b^5}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^19,x]

[Out]

-a^5/(18*x^18) - (5*a^4*b)/(16*x^16) - (5*a^3*b^2)/(7*x^14) - (5*a^2*b^3)/(6*x^12) - (a*b^4)/(2*x^10) - b^5/(8
*x^8)

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^{19}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^5}{x^{10}}+\frac{5 a^4 b}{x^9}+\frac{10 a^3 b^2}{x^8}+\frac{10 a^2 b^3}{x^7}+\frac{5 a b^4}{x^6}+\frac{b^5}{x^5}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5}{18 x^{18}}-\frac{5 a^4 b}{16 x^{16}}-\frac{5 a^3 b^2}{7 x^{14}}-\frac{5 a^2 b^3}{6 x^{12}}-\frac{a b^4}{2 x^{10}}-\frac{b^5}{8 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0040041, size = 69, normalized size = 1. \[ -\frac{5 a^3 b^2}{7 x^{14}}-\frac{5 a^2 b^3}{6 x^{12}}-\frac{5 a^4 b}{16 x^{16}}-\frac{a^5}{18 x^{18}}-\frac{a b^4}{2 x^{10}}-\frac{b^5}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^19,x]

[Out]

-a^5/(18*x^18) - (5*a^4*b)/(16*x^16) - (5*a^3*b^2)/(7*x^14) - (5*a^2*b^3)/(6*x^12) - (a*b^4)/(2*x^10) - b^5/(8
*x^8)

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Maple [A]  time = 0.006, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{18\,{x}^{18}}}-{\frac{5\,{a}^{4}b}{16\,{x}^{16}}}-{\frac{5\,{a}^{3}{b}^{2}}{7\,{x}^{14}}}-{\frac{5\,{a}^{2}{b}^{3}}{6\,{x}^{12}}}-{\frac{a{b}^{4}}{2\,{x}^{10}}}-{\frac{{b}^{5}}{8\,{x}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^19,x)

[Out]

-1/18*a^5/x^18-5/16*a^4*b/x^16-5/7*a^3*b^2/x^14-5/6*a^2*b^3/x^12-1/2*a*b^4/x^10-1/8*b^5/x^8

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Maxima [A]  time = 2.14467, size = 80, normalized size = 1.16 \begin{align*} -\frac{126 \, b^{5} x^{10} + 504 \, a b^{4} x^{8} + 840 \, a^{2} b^{3} x^{6} + 720 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} + 56 \, a^{5}}{1008 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^19,x, algorithm="maxima")

[Out]

-1/1008*(126*b^5*x^10 + 504*a*b^4*x^8 + 840*a^2*b^3*x^6 + 720*a^3*b^2*x^4 + 315*a^4*b*x^2 + 56*a^5)/x^18

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Fricas [A]  time = 1.22489, size = 143, normalized size = 2.07 \begin{align*} -\frac{126 \, b^{5} x^{10} + 504 \, a b^{4} x^{8} + 840 \, a^{2} b^{3} x^{6} + 720 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} + 56 \, a^{5}}{1008 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^19,x, algorithm="fricas")

[Out]

-1/1008*(126*b^5*x^10 + 504*a*b^4*x^8 + 840*a^2*b^3*x^6 + 720*a^3*b^2*x^4 + 315*a^4*b*x^2 + 56*a^5)/x^18

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Sympy [A]  time = 0.772806, size = 63, normalized size = 0.91 \begin{align*} - \frac{56 a^{5} + 315 a^{4} b x^{2} + 720 a^{3} b^{2} x^{4} + 840 a^{2} b^{3} x^{6} + 504 a b^{4} x^{8} + 126 b^{5} x^{10}}{1008 x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**19,x)

[Out]

-(56*a**5 + 315*a**4*b*x**2 + 720*a**3*b**2*x**4 + 840*a**2*b**3*x**6 + 504*a*b**4*x**8 + 126*b**5*x**10)/(100
8*x**18)

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Giac [A]  time = 1.60327, size = 80, normalized size = 1.16 \begin{align*} -\frac{126 \, b^{5} x^{10} + 504 \, a b^{4} x^{8} + 840 \, a^{2} b^{3} x^{6} + 720 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} + 56 \, a^{5}}{1008 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^19,x, algorithm="giac")

[Out]

-1/1008*(126*b^5*x^10 + 504*a*b^4*x^8 + 840*a^2*b^3*x^6 + 720*a^3*b^2*x^4 + 315*a^4*b*x^2 + 56*a^5)/x^18