3.774 \(\int \frac{\sqrt{a+c x^4}}{x^{11}} \, dx\)

Optimal. Leaf size=44 \[ \frac{c \left (a+c x^4\right )^{3/2}}{15 a^2 x^6}-\frac{\left (a+c x^4\right )^{3/2}}{10 a x^{10}} \]

[Out]

-(a + c*x^4)^(3/2)/(10*a*x^10) + (c*(a + c*x^4)^(3/2))/(15*a^2*x^6)

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Rubi [A]  time = 0.0106655, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{c \left (a+c x^4\right )^{3/2}}{15 a^2 x^6}-\frac{\left (a+c x^4\right )^{3/2}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^4]/x^11,x]

[Out]

-(a + c*x^4)^(3/2)/(10*a*x^10) + (c*(a + c*x^4)^(3/2))/(15*a^2*x^6)

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+c x^4}}{x^{11}} \, dx &=-\frac{\left (a+c x^4\right )^{3/2}}{10 a x^{10}}-\frac{(2 c) \int \frac{\sqrt{a+c x^4}}{x^7} \, dx}{5 a}\\ &=-\frac{\left (a+c x^4\right )^{3/2}}{10 a x^{10}}+\frac{c \left (a+c x^4\right )^{3/2}}{15 a^2 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0098735, size = 31, normalized size = 0.7 \[ \frac{\left (a+c x^4\right )^{3/2} \left (2 c x^4-3 a\right )}{30 a^2 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^4]/x^11,x]

[Out]

((a + c*x^4)^(3/2)*(-3*a + 2*c*x^4))/(30*a^2*x^10)

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Maple [A]  time = 0.003, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-2\,c{x}^{4}+3\,a}{30\,{x}^{10}{a}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(1/2)/x^11,x)

[Out]

-1/30*(c*x^4+a)^(3/2)*(-2*c*x^4+3*a)/x^10/a^2

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Maxima [A]  time = 0.962047, size = 47, normalized size = 1.07 \begin{align*} \frac{\frac{5 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} c}{x^{6}} - \frac{3 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}}}{x^{10}}}{30 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^11,x, algorithm="maxima")

[Out]

1/30*(5*(c*x^4 + a)^(3/2)*c/x^6 - 3*(c*x^4 + a)^(5/2)/x^10)/a^2

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Fricas [A]  time = 1.64673, size = 85, normalized size = 1.93 \begin{align*} \frac{{\left (2 \, c^{2} x^{8} - a c x^{4} - 3 \, a^{2}\right )} \sqrt{c x^{4} + a}}{30 \, a^{2} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^11,x, algorithm="fricas")

[Out]

1/30*(2*c^2*x^8 - a*c*x^4 - 3*a^2)*sqrt(c*x^4 + a)/(a^2*x^10)

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Sympy [A]  time = 1.5732, size = 66, normalized size = 1.5 \begin{align*} - \frac{\sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{10 x^{8}} - \frac{c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{30 a x^{4}} + \frac{c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{15 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(1/2)/x**11,x)

[Out]

-sqrt(c)*sqrt(a/(c*x**4) + 1)/(10*x**8) - c**(3/2)*sqrt(a/(c*x**4) + 1)/(30*a*x**4) + c**(5/2)*sqrt(a/(c*x**4)
+ 1)/(15*a**2)

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Giac [A]  time = 1.11378, size = 39, normalized size = 0.89 \begin{align*} -\frac{3 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{5}{2}} - 5 \,{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} c}{30 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^11,x, algorithm="giac")

[Out]

-1/30*(3*(c + a/x^4)^(5/2) - 5*(c + a/x^4)^(3/2)*c)/a^2