3.8.73 \(\int \frac {\sqrt {a+c x^4}}{x^7} \, dx\) [773]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 = {270} \begin {gather*} -\frac {\left (a+c x^4\right )^{3/2}}{6 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^4]/x^7,x]

[Out]

-1/6*(a + c*x^4)^(3/2)/(a*x^6)

Rule 270

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*}

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Mathematica [A]
time = 0.11, size = 21, normalized size = 1.00 \begin {gather*} -\frac {\left (a+c x^4\right )^{3/2}}{6 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^4]/x^7,x]

[Out]

-1/6*(a + c*x^4)^(3/2)/(a*x^6)

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Maple [A]
time = 0.15, size = 18, normalized size = 0.86

method	result	size
gosper	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{6 x^{6} a}\)	\(18\)
default	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{6 x^{6} a}\)	\(18\)
trager	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{6 x^{6} a}\)	\(18\)
risch	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{6 x^{6} a}\)	\(18\)
elliptic	\(-\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{6 x^{6} a}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(1/2)/x^7,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, a x^{6}} \end {gather*}

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 = 0.36, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, a x^{6}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (17) = 34\).
time = 0.41, size = 42, normalized size = 2.00 \begin {gather*} - \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (17) = 34\).
time = 0.52, size = 63, normalized size = 3.00 \begin {gather*} \frac {3 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{4} c^{\frac {3}{2}} + a^{2} c^{\frac {3}{2}}}{3 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + a}\right )}^{2} - a\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.22, size = 17, normalized size = 0.81 \begin {gather*} -\frac {{\left (c\,x^4+a\right )}^{3/2}}{6\,a\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^4)^(1/2)/x^7,x)

[Out]

-(a + c*x^4)^(3/2)/(6*a*x^6)

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