3.78 \(\int \frac{(a+b x^2)^5}{x^8} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0216362, 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} \[ -\frac{10 a^3 b^2}{3 x^3}-\frac{10 a^2 b^3}{x}-\frac{a^4 b}{x^5}-\frac{a^5}{7 x^7}+5 a b^4 x+\frac{b^5 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^8,x]

[Out]

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

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

Mathematica [A]  time = 0.0038928, size = 61, normalized size = 1. \[ -\frac{10 a^3 b^2}{3 x^3}-\frac{10 a^2 b^3}{x}-\frac{a^4 b}{x^5}-\frac{a^5}{7 x^7}+5 a b^4 x+\frac{b^5 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^8,x]

[Out]

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

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Maple [A]  time = 0.006, size = 56, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{7\,{x}^{7}}}-{\frac{{a}^{4}b}{{x}^{5}}}-{\frac{10\,{a}^{3}{b}^{2}}{3\,{x}^{3}}}-10\,{\frac{{a}^{2}{b}^{3}}{x}}+5\,a{b}^{4}x+{\frac{{b}^{5}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^8,x)

[Out]

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

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Maxima [A]  time = 1.87512, size = 78, normalized size = 1.28 \begin{align*} \frac{1}{3} \, b^{5} x^{3} + 5 \, a b^{4} x - \frac{210 \, a^{2} b^{3} x^{6} + 70 \, a^{3} b^{2} x^{4} + 21 \, a^{4} b x^{2} + 3 \, a^{5}}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^8,x, algorithm="maxima")

[Out]

1/3*b^5*x^3 + 5*a*b^4*x - 1/21*(210*a^2*b^3*x^6 + 70*a^3*b^2*x^4 + 21*a^4*b*x^2 + 3*a^5)/x^7

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Fricas [A]  time = 1.22507, size = 131, normalized size = 2.15 \begin{align*} \frac{7 \, b^{5} x^{10} + 105 \, a b^{4} x^{8} - 210 \, a^{2} b^{3} x^{6} - 70 \, a^{3} b^{2} x^{4} - 21 \, a^{4} b x^{2} - 3 \, a^{5}}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^8,x, algorithm="fricas")

[Out]

1/21*(7*b^5*x^10 + 105*a*b^4*x^8 - 210*a^2*b^3*x^6 - 70*a^3*b^2*x^4 - 21*a^4*b*x^2 - 3*a^5)/x^7

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Sympy [A]  time = 0.451072, size = 60, normalized size = 0.98 \begin{align*} 5 a b^{4} x + \frac{b^{5} x^{3}}{3} - \frac{3 a^{5} + 21 a^{4} b x^{2} + 70 a^{3} b^{2} x^{4} + 210 a^{2} b^{3} x^{6}}{21 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**8,x)

[Out]

5*a*b**4*x + b**5*x**3/3 - (3*a**5 + 21*a**4*b*x**2 + 70*a**3*b**2*x**4 + 210*a**2*b**3*x**6)/(21*x**7)

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Giac [A]  time = 2.19656, size = 78, normalized size = 1.28 \begin{align*} \frac{1}{3} \, b^{5} x^{3} + 5 \, a b^{4} x - \frac{210 \, a^{2} b^{3} x^{6} + 70 \, a^{3} b^{2} x^{4} + 21 \, a^{4} b x^{2} + 3 \, a^{5}}{21 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^8,x, algorithm="giac")

[Out]

1/3*b^5*x^3 + 5*a*b^4*x - 1/21*(210*a^2*b^3*x^6 + 70*a^3*b^2*x^4 + 21*a^4*b*x^2 + 3*a^5)/x^7