3.14.16 \(\int \frac {1}{x^{7/2} \sqrt {1+x^5}} \, dx\) [1316]

Optimal. Leaf size=18 \[ -\frac {2 \sqrt {1+x^5}}{5 x^{5/2}} \]

[Out]

-2/5*(x^5+1)^(1/2)/x^(5/2)

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Rubi [A]
time = 0.00, antiderivative size = 18, 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 {2 \sqrt {x^5+1}}{5 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*Sqrt[1 + x^5]),x]

[Out]

(-2*Sqrt[1 + x^5])/(5*x^(5/2))

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 {1}{x^{7/2} \sqrt {1+x^5}} \, dx &=-\frac {2 \sqrt {1+x^5}}{5 x^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {1+x^5}}{5 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(7/2)*Sqrt[1 + x^5]),x]

[Out]

(-2*Sqrt[1 + x^5])/(5*x^(5/2))

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Maple [A]
time = 0.17, size = 13, normalized size = 0.72

method	result	size
meijerg	\(-\frac {2 \sqrt {x^{5}+1}}{5 x^{\frac {5}{2}}}\)	\(13\)
risch	\(-\frac {2 \sqrt {x^{5}+1}}{5 x^{\frac {5}{2}}}\)	\(13\)
gosper	\(-\frac {2 \left (x +1\right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}{5 x^{\frac {5}{2}} \sqrt {x^{5}+1}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(x^5+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/5*(x^5+1)^(1/2)/x^(5/2)

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Maxima [A]
time = 0.29, size = 12, normalized size = 0.67 \begin {gather*} -\frac {2 \, \sqrt {x^{5} + 1}}{5 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(x^5+1)^(1/2),x, algorithm="maxima")

[Out]

-2/5*sqrt(x^5 + 1)/x^(5/2)

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Fricas [A]
time = 0.37, size = 12, normalized size = 0.67 \begin {gather*} -\frac {2 \, \sqrt {x^{5} + 1}}{5 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(x^5+1)^(1/2),x, algorithm="fricas")

[Out]

-2/5*sqrt(x^5 + 1)/x^(5/2)

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Sympy [A]
time = 3.45, size = 14, normalized size = 0.78 \begin {gather*} - \frac {2 \sqrt {1 + \frac {1}{x^{5}}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(7/2)/(x**5+1)**(1/2),x)

[Out]

-2*sqrt(1 + x**(-5))/5

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Giac [A]
time = 1.44, size = 11, normalized size = 0.61 \begin {gather*} -\frac {2}{5} \, \sqrt {\frac {1}{x^{5}} + 1} + \frac {2}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(x^5+1)^(1/2),x, algorithm="giac")

[Out]

-2/5*sqrt(1/x^5 + 1) + 2/5

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {1}{x^{7/2}\,\sqrt {x^5+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(7/2)*(x^5 + 1)^(1/2)),x)

[Out]

int(1/(x^(7/2)*(x^5 + 1)^(1/2)), x)

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