3.773 \(\int \frac{\sqrt{a+c x^4}}{x^7} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+c x^4\right )^{3/2}}{6 a x^6} \]

[Out]

-(a + c*x^4)^(3/2)/(6*a*x^6)

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Rubi [A]  time = 0.0042621, 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} \[ -\frac{\left (a+c x^4\right )^{3/2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^4]/x^7,x]

[Out]

-(a + c*x^4)^(3/2)/(6*a*x^6)

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^7} \, dx &=-\frac{\left (a+c x^4\right )^{3/2}}{6 a x^6}\\ \end{align*}

Mathematica [A]  time = 0.0045265, size = 21, normalized size = 1. \[ -\frac{\left (a+c x^4\right )^{3/2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^4]/x^7,x]

[Out]

-(a + c*x^4)^(3/2)/(6*a*x^6)

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Maple [A]  time = 0.003, size = 18, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,{x}^{6}a} \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^7,x)

[Out]

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

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Maxima [A]  time = 0.978639, size = 23, normalized size = 1.1 \begin{align*} -\frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{6 \, a x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^7,x, algorithm="maxima")

[Out]

-1/6*(c*x^4 + a)^(3/2)/(a*x^6)

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Fricas [A]  time = 1.74164, size = 43, normalized size = 2.05 \begin{align*} -\frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{6 \, a x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^7,x, algorithm="fricas")

[Out]

-1/6*(c*x^4 + a)^(3/2)/(a*x^6)

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Sympy [B]  time = 0.86008, size = 42, normalized size = 2. \begin{align*} - \frac{\sqrt{c} \sqrt{\frac{a}{c x^{4}} + 1}}{6 x^{4}} - \frac{c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{4}} + 1}}{6 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(1/2)/x**7,x)

[Out]

-sqrt(c)*sqrt(a/(c*x**4) + 1)/(6*x**4) - c**(3/2)*sqrt(a/(c*x**4) + 1)/(6*a)

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Giac [A]  time = 1.27869, size = 19, normalized size = 0.9 \begin{align*} -\frac{{\left (c + \frac{a}{x^{4}}\right )}^{\frac{3}{2}}}{6 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^7,x, algorithm="giac")

[Out]

-1/6*(c + a/x^4)^(3/2)/a