3.80 \(\int \frac{(a+b x^2)^5}{x^{12}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{10 a^3 b^2}{7 x^7}-\frac{2 a^2 b^3}{x^5}-\frac{5 a^4 b}{9 x^9}-\frac{a^5}{11 x^{11}}-\frac{5 a b^4}{3 x^3}-\frac{b^5}{x} \]

[Out]

-a^5/(11*x^11) - (5*a^4*b)/(9*x^9) - (10*a^3*b^2)/(7*x^7) - (2*a^2*b^3)/x^5 - (5*a*b^4)/(3*x^3) - b^5/x

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Rubi [A]  time = 0.0223313, antiderivative size = 65, 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} \[ -\frac{10 a^3 b^2}{7 x^7}-\frac{2 a^2 b^3}{x^5}-\frac{5 a^4 b}{9 x^9}-\frac{a^5}{11 x^{11}}-\frac{5 a b^4}{3 x^3}-\frac{b^5}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^12,x]

[Out]

-a^5/(11*x^11) - (5*a^4*b)/(9*x^9) - (10*a^3*b^2)/(7*x^7) - (2*a^2*b^3)/x^5 - (5*a*b^4)/(3*x^3) - 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^{12}} \, dx &=\int \left (\frac{a^5}{x^{12}}+\frac{5 a^4 b}{x^{10}}+\frac{10 a^3 b^2}{x^8}+\frac{10 a^2 b^3}{x^6}+\frac{5 a b^4}{x^4}+\frac{b^5}{x^2}\right ) \, dx\\ &=-\frac{a^5}{11 x^{11}}-\frac{5 a^4 b}{9 x^9}-\frac{10 a^3 b^2}{7 x^7}-\frac{2 a^2 b^3}{x^5}-\frac{5 a b^4}{3 x^3}-\frac{b^5}{x}\\ \end{align*}

Mathematica [A]  time = 0.003967, size = 65, normalized size = 1. \[ -\frac{10 a^3 b^2}{7 x^7}-\frac{2 a^2 b^3}{x^5}-\frac{5 a^4 b}{9 x^9}-\frac{a^5}{11 x^{11}}-\frac{5 a b^4}{3 x^3}-\frac{b^5}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^12,x]

[Out]

-a^5/(11*x^11) - (5*a^4*b)/(9*x^9) - (10*a^3*b^2)/(7*x^7) - (2*a^2*b^3)/x^5 - (5*a*b^4)/(3*x^3) - b^5/x

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Maple [A]  time = 0.006, size = 58, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{11\,{x}^{11}}}-{\frac{5\,{a}^{4}b}{9\,{x}^{9}}}-{\frac{10\,{a}^{3}{b}^{2}}{7\,{x}^{7}}}-2\,{\frac{{a}^{2}{b}^{3}}{{x}^{5}}}-{\frac{5\,a{b}^{4}}{3\,{x}^{3}}}-{\frac{{b}^{5}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^12,x)

[Out]

-1/11*a^5/x^11-5/9*a^4*b/x^9-10/7*a^3*b^2/x^7-2*a^2*b^3/x^5-5/3*a*b^4/x^3-b^5/x

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Maxima [A]  time = 2.13639, size = 80, normalized size = 1.23 \begin{align*} -\frac{693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^12,x, algorithm="maxima")

[Out]

-1/693*(693*b^5*x^10 + 1155*a*b^4*x^8 + 1386*a^2*b^3*x^6 + 990*a^3*b^2*x^4 + 385*a^4*b*x^2 + 63*a^5)/x^11

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Fricas [A]  time = 1.19324, size = 144, normalized size = 2.22 \begin{align*} -\frac{693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^12,x, algorithm="fricas")

[Out]

-1/693*(693*b^5*x^10 + 1155*a*b^4*x^8 + 1386*a^2*b^3*x^6 + 990*a^3*b^2*x^4 + 385*a^4*b*x^2 + 63*a^5)/x^11

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Sympy [A]  time = 0.583955, size = 63, normalized size = 0.97 \begin{align*} - \frac{63 a^{5} + 385 a^{4} b x^{2} + 990 a^{3} b^{2} x^{4} + 1386 a^{2} b^{3} x^{6} + 1155 a b^{4} x^{8} + 693 b^{5} x^{10}}{693 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**12,x)

[Out]

-(63*a**5 + 385*a**4*b*x**2 + 990*a**3*b**2*x**4 + 1386*a**2*b**3*x**6 + 1155*a*b**4*x**8 + 693*b**5*x**10)/(6
93*x**11)

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Giac [A]  time = 1.33366, size = 80, normalized size = 1.23 \begin{align*} -\frac{693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^12,x, algorithm="giac")

[Out]

-1/693*(693*b^5*x^10 + 1155*a*b^4*x^8 + 1386*a^2*b^3*x^6 + 990*a^3*b^2*x^4 + 385*a^4*b*x^2 + 63*a^5)/x^11