3.79 \(\int \frac{(a+b x^2)^5}{x^{10}} \, dx\)

Optimal. Leaf size=60 \[ -\frac{2 a^3 b^2}{x^5}-\frac{10 a^2 b^3}{3 x^3}-\frac{5 a^4 b}{7 x^7}-\frac{a^5}{9 x^9}-\frac{5 a b^4}{x}+b^5 x \]

[Out]

-a^5/(9*x^9) - (5*a^4*b)/(7*x^7) - (2*a^3*b^2)/x^5 - (10*a^2*b^3)/(3*x^3) - (5*a*b^4)/x + b^5*x

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Rubi [A]  time = 0.0219145, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ -\frac{2 a^3 b^2}{x^5}-\frac{10 a^2 b^3}{3 x^3}-\frac{5 a^4 b}{7 x^7}-\frac{a^5}{9 x^9}-\frac{5 a b^4}{x}+b^5 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^10,x]

[Out]

-a^5/(9*x^9) - (5*a^4*b)/(7*x^7) - (2*a^3*b^2)/x^5 - (10*a^2*b^3)/(3*x^3) - (5*a*b^4)/x + b^5*x

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0052029, size = 60, normalized size = 1. \[ -\frac{2 a^3 b^2}{x^5}-\frac{10 a^2 b^3}{3 x^3}-\frac{5 a^4 b}{7 x^7}-\frac{a^5}{9 x^9}-\frac{5 a b^4}{x}+b^5 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^10,x]

[Out]

-a^5/(9*x^9) - (5*a^4*b)/(7*x^7) - (2*a^3*b^2)/x^5 - (10*a^2*b^3)/(3*x^3) - (5*a*b^4)/x + b^5*x

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Maple [A]  time = 0.006, size = 55, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{9\,{x}^{9}}}-{\frac{5\,{a}^{4}b}{7\,{x}^{7}}}-2\,{\frac{{a}^{3}{b}^{2}}{{x}^{5}}}-{\frac{10\,{a}^{2}{b}^{3}}{3\,{x}^{3}}}-5\,{\frac{a{b}^{4}}{x}}+{b}^{5}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^10,x)

[Out]

-1/9*a^5/x^9-5/7*a^4*b/x^7-2*a^3*b^2/x^5-10/3*a^2*b^3/x^3-5*a*b^4/x+b^5*x

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Maxima [A]  time = 1.97457, size = 77, normalized size = 1.28 \begin{align*} b^{5} x - \frac{315 \, a b^{4} x^{8} + 210 \, a^{2} b^{3} x^{6} + 126 \, a^{3} b^{2} x^{4} + 45 \, a^{4} b x^{2} + 7 \, a^{5}}{63 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^10,x, algorithm="maxima")

[Out]

b^5*x - 1/63*(315*a*b^4*x^8 + 210*a^2*b^3*x^6 + 126*a^3*b^2*x^4 + 45*a^4*b*x^2 + 7*a^5)/x^9

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Fricas [A]  time = 1.29122, size = 134, normalized size = 2.23 \begin{align*} \frac{63 \, b^{5} x^{10} - 315 \, a b^{4} x^{8} - 210 \, a^{2} b^{3} x^{6} - 126 \, a^{3} b^{2} x^{4} - 45 \, a^{4} b x^{2} - 7 \, a^{5}}{63 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^10,x, algorithm="fricas")

[Out]

1/63*(63*b^5*x^10 - 315*a*b^4*x^8 - 210*a^2*b^3*x^6 - 126*a^3*b^2*x^4 - 45*a^4*b*x^2 - 7*a^5)/x^9

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Sympy [A]  time = 0.513978, size = 58, normalized size = 0.97 \begin{align*} b^{5} x - \frac{7 a^{5} + 45 a^{4} b x^{2} + 126 a^{3} b^{2} x^{4} + 210 a^{2} b^{3} x^{6} + 315 a b^{4} x^{8}}{63 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**10,x)

[Out]

b**5*x - (7*a**5 + 45*a**4*b*x**2 + 126*a**3*b**2*x**4 + 210*a**2*b**3*x**6 + 315*a*b**4*x**8)/(63*x**9)

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Giac [A]  time = 2.80845, size = 77, normalized size = 1.28 \begin{align*} b^{5} x - \frac{315 \, a b^{4} x^{8} + 210 \, a^{2} b^{3} x^{6} + 126 \, a^{3} b^{2} x^{4} + 45 \, a^{4} b x^{2} + 7 \, a^{5}}{63 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^10,x, algorithm="giac")

[Out]

b^5*x - 1/63*(315*a*b^4*x^8 + 210*a^2*b^3*x^6 + 126*a^3*b^2*x^4 + 45*a^4*b*x^2 + 7*a^5)/x^9