3.64 \(\int \frac{(a+b x^2)^5}{x^{11}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{5 a^3 b^2}{3 x^6}-\frac{5 a^2 b^3}{2 x^4}-\frac{5 a^4 b}{8 x^8}-\frac{a^5}{10 x^{10}}-\frac{5 a b^4}{2 x^2}+b^5 \log (x) \]

[Out]

-a^5/(10*x^10) - (5*a^4*b)/(8*x^8) - (5*a^3*b^2)/(3*x^6) - (5*a^2*b^3)/(2*x^4) - (5*a*b^4)/(2*x^2) + b^5*Log[x
]

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Rubi [A]  time = 0.0310673, antiderivative size = 65, 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} \[ -\frac{5 a^3 b^2}{3 x^6}-\frac{5 a^2 b^3}{2 x^4}-\frac{5 a^4 b}{8 x^8}-\frac{a^5}{10 x^{10}}-\frac{5 a b^4}{2 x^2}+b^5 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^11,x]

[Out]

-a^5/(10*x^10) - (5*a^4*b)/(8*x^8) - (5*a^3*b^2)/(3*x^6) - (5*a^2*b^3)/(2*x^4) - (5*a*b^4)/(2*x^2) + b^5*Log[x
]

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^{11}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^6} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^5}{x^6}+\frac{5 a^4 b}{x^5}+\frac{10 a^3 b^2}{x^4}+\frac{10 a^2 b^3}{x^3}+\frac{5 a b^4}{x^2}+\frac{b^5}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5}{10 x^{10}}-\frac{5 a^4 b}{8 x^8}-\frac{5 a^3 b^2}{3 x^6}-\frac{5 a^2 b^3}{2 x^4}-\frac{5 a b^4}{2 x^2}+b^5 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0043998, size = 65, normalized size = 1. \[ -\frac{5 a^3 b^2}{3 x^6}-\frac{5 a^2 b^3}{2 x^4}-\frac{5 a^4 b}{8 x^8}-\frac{a^5}{10 x^{10}}-\frac{5 a b^4}{2 x^2}+b^5 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^11,x]

[Out]

-a^5/(10*x^10) - (5*a^4*b)/(8*x^8) - (5*a^3*b^2)/(3*x^6) - (5*a^2*b^3)/(2*x^4) - (5*a*b^4)/(2*x^2) + b^5*Log[x
]

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Maple [A]  time = 0.006, size = 56, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{10\,{x}^{10}}}-{\frac{5\,{a}^{4}b}{8\,{x}^{8}}}-{\frac{5\,{a}^{3}{b}^{2}}{3\,{x}^{6}}}-{\frac{5\,{a}^{2}{b}^{3}}{2\,{x}^{4}}}-{\frac{5\,a{b}^{4}}{2\,{x}^{2}}}+{b}^{5}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^11,x)

[Out]

-1/10*a^5/x^10-5/8*a^4*b/x^8-5/3*a^3*b^2/x^6-5/2*a^2*b^3/x^4-5/2*a*b^4/x^2+b^5*ln(x)

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Maxima [A]  time = 1.9581, size = 82, normalized size = 1.26 \begin{align*} \frac{1}{2} \, b^{5} \log \left (x^{2}\right ) - \frac{300 \, a b^{4} x^{8} + 300 \, a^{2} b^{3} x^{6} + 200 \, a^{3} b^{2} x^{4} + 75 \, a^{4} b x^{2} + 12 \, a^{5}}{120 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^11,x, algorithm="maxima")

[Out]

1/2*b^5*log(x^2) - 1/120*(300*a*b^4*x^8 + 300*a^2*b^3*x^6 + 200*a^3*b^2*x^4 + 75*a^4*b*x^2 + 12*a^5)/x^10

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Fricas [A]  time = 1.22703, size = 149, normalized size = 2.29 \begin{align*} \frac{120 \, b^{5} x^{10} \log \left (x\right ) - 300 \, a b^{4} x^{8} - 300 \, a^{2} b^{3} x^{6} - 200 \, a^{3} b^{2} x^{4} - 75 \, a^{4} b x^{2} - 12 \, a^{5}}{120 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^11,x, algorithm="fricas")

[Out]

1/120*(120*b^5*x^10*log(x) - 300*a*b^4*x^8 - 300*a^2*b^3*x^6 - 200*a^3*b^2*x^4 - 75*a^4*b*x^2 - 12*a^5)/x^10

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Sympy [A]  time = 0.561579, size = 60, normalized size = 0.92 \begin{align*} b^{5} \log{\left (x \right )} - \frac{12 a^{5} + 75 a^{4} b x^{2} + 200 a^{3} b^{2} x^{4} + 300 a^{2} b^{3} x^{6} + 300 a b^{4} x^{8}}{120 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**11,x)

[Out]

b**5*log(x) - (12*a**5 + 75*a**4*b*x**2 + 200*a**3*b**2*x**4 + 300*a**2*b**3*x**6 + 300*a*b**4*x**8)/(120*x**1
0)

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Giac [A]  time = 2.24977, size = 93, normalized size = 1.43 \begin{align*} \frac{1}{2} \, b^{5} \log \left (x^{2}\right ) - \frac{137 \, b^{5} x^{10} + 300 \, a b^{4} x^{8} + 300 \, a^{2} b^{3} x^{6} + 200 \, a^{3} b^{2} x^{4} + 75 \, a^{4} b x^{2} + 12 \, a^{5}}{120 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^11,x, algorithm="giac")

[Out]

1/2*b^5*log(x^2) - 1/120*(137*b^5*x^10 + 300*a*b^4*x^8 + 300*a^2*b^3*x^6 + 200*a^3*b^2*x^4 + 75*a^4*b*x^2 + 12
*a^5)/x^10