3.77 \(\int \frac{(a+b x^2)^5}{x^6} \, dx\)

Optimal. Leaf size=63 \[ -\frac{10 a^3 b^2}{x}+10 a^2 b^3 x-\frac{5 a^4 b}{3 x^3}-\frac{a^5}{5 x^5}+\frac{5}{3} a b^4 x^3+\frac{b^5 x^5}{5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0224724, antiderivative size = 63, 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} \[ -\frac{10 a^3 b^2}{x}+10 a^2 b^3 x-\frac{5 a^4 b}{3 x^3}-\frac{a^5}{5 x^5}+\frac{5}{3} a b^4 x^3+\frac{b^5 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^6,x]

[Out]

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

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

Mathematica [A]  time = 0.0039618, size = 63, normalized size = 1. \[ -\frac{10 a^3 b^2}{x}+10 a^2 b^3 x-\frac{5 a^4 b}{3 x^3}-\frac{a^5}{5 x^5}+\frac{5}{3} a b^4 x^3+\frac{b^5 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^6,x]

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^6,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.80023, size = 78, normalized size = 1.24 \begin{align*} \frac{1}{5} \, b^{5} x^{5} + \frac{5}{3} \, a b^{4} x^{3} + 10 \, a^{2} b^{3} x - \frac{150 \, a^{3} b^{2} x^{4} + 25 \, a^{4} b x^{2} + 3 \, a^{5}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^6,x, algorithm="maxima")

[Out]

1/5*b^5*x^5 + 5/3*a*b^4*x^3 + 10*a^2*b^3*x - 1/15*(150*a^3*b^2*x^4 + 25*a^4*b*x^2 + 3*a^5)/x^5

________________________________________________________________________________________

Fricas [A]  time = 1.21474, size = 131, normalized size = 2.08 \begin{align*} \frac{3 \, b^{5} x^{10} + 25 \, a b^{4} x^{8} + 150 \, a^{2} b^{3} x^{6} - 150 \, a^{3} b^{2} x^{4} - 25 \, a^{4} b x^{2} - 3 \, a^{5}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^6,x, algorithm="fricas")

[Out]

1/15*(3*b^5*x^10 + 25*a*b^4*x^8 + 150*a^2*b^3*x^6 - 150*a^3*b^2*x^4 - 25*a^4*b*x^2 - 3*a^5)/x^5

________________________________________________________________________________________

Sympy [A]  time = 0.373617, size = 61, normalized size = 0.97 \begin{align*} 10 a^{2} b^{3} x + \frac{5 a b^{4} x^{3}}{3} + \frac{b^{5} x^{5}}{5} - \frac{3 a^{5} + 25 a^{4} b x^{2} + 150 a^{3} b^{2} x^{4}}{15 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**6,x)

[Out]

10*a**2*b**3*x + 5*a*b**4*x**3/3 + b**5*x**5/5 - (3*a**5 + 25*a**4*b*x**2 + 150*a**3*b**2*x**4)/(15*x**5)

________________________________________________________________________________________

Giac [A]  time = 1.91505, size = 78, normalized size = 1.24 \begin{align*} \frac{1}{5} \, b^{5} x^{5} + \frac{5}{3} \, a b^{4} x^{3} + 10 \, a^{2} b^{3} x - \frac{150 \, a^{3} b^{2} x^{4} + 25 \, a^{4} b x^{2} + 3 \, a^{5}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^6,x, algorithm="giac")

[Out]

1/5*b^5*x^5 + 5/3*a*b^4*x^3 + 10*a^2*b^3*x - 1/15*(150*a^3*b^2*x^4 + 25*a^4*b*x^2 + 3*a^5)/x^5