3.1317 \(\int \frac {1}{x^{17/2} \sqrt {1+x^5}} \, dx\)

Optimal. Leaf size=37 \[ \frac {4 \sqrt {x^5+1}}{15 x^{5/2}}-\frac {2 \sqrt {x^5+1}}{15 x^{15/2}} \]

[Out]

-2/15*(x^5+1)^(1/2)/x^(15/2)+4/15*(x^5+1)^(1/2)/x^(5/2)

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Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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} \[ \frac {4 \sqrt {x^5+1}}{15 x^{5/2}}-\frac {2 \sqrt {x^5+1}}{15 x^{15/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(17/2)*Sqrt[1 + x^5]),x]

[Out]

(-2*Sqrt[1 + x^5])/(15*x^(15/2)) + (4*Sqrt[1 + x^5])/(15*x^(5/2))

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]

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]

Rubi steps

\begin {align*} \int \frac {1}{x^{17/2} \sqrt {1+x^5}} \, dx &=-\frac {2 \sqrt {1+x^5}}{15 x^{15/2}}-\frac {2}{3} \int \frac {1}{x^{7/2} \sqrt {1+x^5}} \, dx\\ &=-\frac {2 \sqrt {1+x^5}}{15 x^{15/2}}+\frac {4 \sqrt {1+x^5}}{15 x^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.68 \[ -\frac {2 \left (1-2 x^5\right ) \sqrt {x^5+1}}{15 x^{15/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(17/2)*Sqrt[1 + x^5]),x]

[Out]

(-2*(1 - 2*x^5)*Sqrt[1 + x^5])/(15*x^(15/2))

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fricas [A]  time = 0.85, size = 19, normalized size = 0.51 \[ \frac {2 \, {\left (2 \, x^{5} - 1\right )} \sqrt {x^{5} + 1}}{15 \, x^{\frac {15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(17/2)/(x^5+1)^(1/2),x, algorithm="fricas")

[Out]

2/15*(2*x^5 - 1)*sqrt(x^5 + 1)/x^(15/2)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(17/2)/(x^5+1)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is
real):Check [abs(t_nostep)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argum
ent Value

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maple [A]  time = 0.00, size = 39, normalized size = 1.05 \[ \frac {2 \left (x +1\right ) \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (2 x^{5}-1\right )}{15 \sqrt {x^{5}+1}\, x^{\frac {15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(17/2)/(x^5+1)^(1/2),x)

[Out]

2/15*(x+1)*(x^4-x^3+x^2-x+1)*(2*x^5-1)/x^(15/2)/(x^5+1)^(1/2)

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maxima [A]  time = 0.95, size = 25, normalized size = 0.68 \[ \frac {2 \, \sqrt {x^{5} + 1}}{5 \, x^{\frac {5}{2}}} - \frac {2 \, {\left (x^{5} + 1\right )}^{\frac {3}{2}}}{15 \, x^{\frac {15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(17/2)/(x^5+1)^(1/2),x, algorithm="maxima")

[Out]

2/5*sqrt(x^5 + 1)/x^(5/2) - 2/15*(x^5 + 1)^(3/2)/x^(15/2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{x^{17/2}\,\sqrt {x^5+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(17/2)*(x^5 + 1)^(1/2)),x)

[Out]

int(1/(x^(17/2)*(x^5 + 1)^(1/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(17/2)/(x**5+1)**(1/2),x)

[Out]

Timed out

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