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

Optimal. Leaf size=61 \[ 2 a^2 b^3 x^5+\frac{10}{3} a^3 b^2 x^3+5 a^4 b x-\frac{a^5}{x}+\frac{5}{7} a b^4 x^7+\frac{b^5 x^9}{9} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0210491, antiderivative size = 61, 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} \[ 2 a^2 b^3 x^5+\frac{10}{3} a^3 b^2 x^3+5 a^4 b x-\frac{a^5}{x}+\frac{5}{7} a b^4 x^7+\frac{b^5 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0036962, size = 61, normalized size = 1. \[ 2 a^2 b^3 x^5+\frac{10}{3} a^3 b^2 x^3+5 a^4 b x-\frac{a^5}{x}+\frac{5}{7} a b^4 x^7+\frac{b^5 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 56, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{x}}+5\,{a}^{4}bx+{\frac{10\,{a}^{3}{b}^{2}{x}^{3}}{3}}+2\,{a}^{2}{b}^{3}{x}^{5}+{\frac{5\,a{b}^{4}{x}^{7}}{7}}+{\frac{{b}^{5}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.75965, size = 74, normalized size = 1.21 \begin{align*} \frac{1}{9} \, b^{5} x^{9} + \frac{5}{7} \, a b^{4} x^{7} + 2 \, a^{2} b^{3} x^{5} + \frac{10}{3} \, a^{3} b^{2} x^{3} + 5 \, a^{4} b x - \frac{a^{5}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.18136, size = 131, normalized size = 2.15 \begin{align*} \frac{7 \, b^{5} x^{10} + 45 \, a b^{4} x^{8} + 126 \, a^{2} b^{3} x^{6} + 210 \, a^{3} b^{2} x^{4} + 315 \, a^{4} b x^{2} - 63 \, a^{5}}{63 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.279188, size = 58, normalized size = 0.95 \begin{align*} - \frac{a^{5}}{x} + 5 a^{4} b x + \frac{10 a^{3} b^{2} x^{3}}{3} + 2 a^{2} b^{3} x^{5} + \frac{5 a b^{4} x^{7}}{7} + \frac{b^{5} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 2.72171, size = 74, normalized size = 1.21 \begin{align*} \frac{1}{9} \, b^{5} x^{9} + \frac{5}{7} \, a b^{4} x^{7} + 2 \, a^{2} b^{3} x^{5} + \frac{10}{3} \, a^{3} b^{2} x^{3} + 5 \, a^{4} b x - \frac{a^{5}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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