3.81 \(\int \frac{(a+b x^2)^5}{x^{14}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{10 a^3 b^2}{9 x^9}-\frac{10 a^2 b^3}{7 x^7}-\frac{5 a^4 b}{11 x^{11}}-\frac{a^5}{13 x^{13}}-\frac{a b^4}{x^5}-\frac{b^5}{3 x^3} \]

[Out]

-a^5/(13*x^13) - (5*a^4*b)/(11*x^11) - (10*a^3*b^2)/(9*x^9) - (10*a^2*b^3)/(7*x^7) - (a*b^4)/x^5 - b^5/(3*x^3)

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Rubi [A]  time = 0.0230642, antiderivative size = 67, 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} \[ -\frac{10 a^3 b^2}{9 x^9}-\frac{10 a^2 b^3}{7 x^7}-\frac{5 a^4 b}{11 x^{11}}-\frac{a^5}{13 x^{13}}-\frac{a b^4}{x^5}-\frac{b^5}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^14,x]

[Out]

-a^5/(13*x^13) - (5*a^4*b)/(11*x^11) - (10*a^3*b^2)/(9*x^9) - (10*a^2*b^3)/(7*x^7) - (a*b^4)/x^5 - b^5/(3*x^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^{14}} \, dx &=\int \left (\frac{a^5}{x^{14}}+\frac{5 a^4 b}{x^{12}}+\frac{10 a^3 b^2}{x^{10}}+\frac{10 a^2 b^3}{x^8}+\frac{5 a b^4}{x^6}+\frac{b^5}{x^4}\right ) \, dx\\ &=-\frac{a^5}{13 x^{13}}-\frac{5 a^4 b}{11 x^{11}}-\frac{10 a^3 b^2}{9 x^9}-\frac{10 a^2 b^3}{7 x^7}-\frac{a b^4}{x^5}-\frac{b^5}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0039706, size = 67, normalized size = 1. \[ -\frac{10 a^3 b^2}{9 x^9}-\frac{10 a^2 b^3}{7 x^7}-\frac{5 a^4 b}{11 x^{11}}-\frac{a^5}{13 x^{13}}-\frac{a b^4}{x^5}-\frac{b^5}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^14,x]

[Out]

-a^5/(13*x^13) - (5*a^4*b)/(11*x^11) - (10*a^3*b^2)/(9*x^9) - (10*a^2*b^3)/(7*x^7) - (a*b^4)/x^5 - b^5/(3*x^3)

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Maple [A]  time = 0.005, size = 58, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{13\,{x}^{13}}}-{\frac{5\,{a}^{4}b}{11\,{x}^{11}}}-{\frac{10\,{a}^{3}{b}^{2}}{9\,{x}^{9}}}-{\frac{10\,{a}^{2}{b}^{3}}{7\,{x}^{7}}}-{\frac{a{b}^{4}}{{x}^{5}}}-{\frac{{b}^{5}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^14,x)

[Out]

-1/13*a^5/x^13-5/11*a^4*b/x^11-10/9*a^3*b^2/x^9-10/7*a^2*b^3/x^7-a*b^4/x^5-1/3*b^5/x^3

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Maxima [A]  time = 2.49029, size = 80, normalized size = 1.19 \begin{align*} -\frac{3003 \, b^{5} x^{10} + 9009 \, a b^{4} x^{8} + 12870 \, a^{2} b^{3} x^{6} + 10010 \, a^{3} b^{2} x^{4} + 4095 \, a^{4} b x^{2} + 693 \, a^{5}}{9009 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^14,x, algorithm="maxima")

[Out]

-1/9009*(3003*b^5*x^10 + 9009*a*b^4*x^8 + 12870*a^2*b^3*x^6 + 10010*a^3*b^2*x^4 + 4095*a^4*b*x^2 + 693*a^5)/x^
13

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Fricas [A]  time = 1.32342, size = 154, normalized size = 2.3 \begin{align*} -\frac{3003 \, b^{5} x^{10} + 9009 \, a b^{4} x^{8} + 12870 \, a^{2} b^{3} x^{6} + 10010 \, a^{3} b^{2} x^{4} + 4095 \, a^{4} b x^{2} + 693 \, a^{5}}{9009 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^14,x, algorithm="fricas")

[Out]

-1/9009*(3003*b^5*x^10 + 9009*a*b^4*x^8 + 12870*a^2*b^3*x^6 + 10010*a^3*b^2*x^4 + 4095*a^4*b*x^2 + 693*a^5)/x^
13

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Sympy [A]  time = 0.645281, size = 63, normalized size = 0.94 \begin{align*} - \frac{693 a^{5} + 4095 a^{4} b x^{2} + 10010 a^{3} b^{2} x^{4} + 12870 a^{2} b^{3} x^{6} + 9009 a b^{4} x^{8} + 3003 b^{5} x^{10}}{9009 x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**14,x)

[Out]

-(693*a**5 + 4095*a**4*b*x**2 + 10010*a**3*b**2*x**4 + 12870*a**2*b**3*x**6 + 9009*a*b**4*x**8 + 3003*b**5*x**
10)/(9009*x**13)

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Giac [A]  time = 2.81248, size = 80, normalized size = 1.19 \begin{align*} -\frac{3003 \, b^{5} x^{10} + 9009 \, a b^{4} x^{8} + 12870 \, a^{2} b^{3} x^{6} + 10010 \, a^{3} b^{2} x^{4} + 4095 \, a^{4} b x^{2} + 693 \, a^{5}}{9009 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^14,x, algorithm="giac")

[Out]

-1/9009*(3003*b^5*x^10 + 9009*a*b^4*x^8 + 12870*a^2*b^3*x^6 + 10010*a^3*b^2*x^4 + 4095*a^4*b*x^2 + 693*a^5)/x^
13