3.793 \(\int \frac{(a+c x^4)^{3/2}}{x^{11}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+c x^4\right )^{5/2}}{10 a x^{10}} \]

[Out]

-(a + c*x^4)^(5/2)/(10*a*x^10)

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Rubi [A]  time = 0.0047694, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{\left (a+c x^4\right )^{5/2}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^(3/2)/x^11,x]

[Out]

-(a + c*x^4)^(5/2)/(10*a*x^10)

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

Mathematica [A]  time = 0.0053433, size = 21, normalized size = 1. \[ -\frac{\left (a+c x^4\right )^{5/2}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^(3/2)/x^11,x]

[Out]

-(a + c*x^4)^(5/2)/(10*a*x^10)

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Maple [A]  time = 0.002, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{10\,a{x}^{10}} \left ( c{x}^{4}+a \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(3/2)/x^11,x)

[Out]

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

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Maxima [A]  time = 0.959087, size = 23, normalized size = 1.1 \begin{align*} -\frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}}}{10 \, a x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^11,x, algorithm="maxima")

[Out]

-1/10*(c*x^4 + a)^(5/2)/(a*x^10)

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Fricas [B]  time = 1.48456, size = 81, normalized size = 3.86 \begin{align*} -\frac{{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \sqrt{c x^{4} + a}}{10 \, a x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^11,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 2.63643, size = 66, normalized size = 3.14 \begin{align*} - \frac{a \sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{10 x^{8}} - \frac{c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{5 x^{4}} - \frac{c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{10 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(3/2)/x**11,x)

[Out]

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

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Giac [A]  time = 1.1187, size = 19, normalized size = 0.9 \begin{align*} -\frac{{\left (c + \frac{a}{x^{4}}\right )}^{\frac{5}{2}}}{10 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(3/2)/x^11,x, algorithm="giac")

[Out]

-1/10*(c + a/x^4)^(5/2)/a