\(\int \frac {1}{x^{17/2} \sqrt {a+b x^5}} \, dx\) [1312]

Optimal result

Integrand size = 17, antiderivative size = 48 \[ \int \frac {1}{x^{17/2} \sqrt {a+b x^5}} \, dx=-\frac {2 \sqrt {a+b x^5}}{15 a x^{15/2}}+\frac {4 b \sqrt {a+b x^5}}{15 a^2 x^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {277, 270} \[ \int \frac {1}{x^{17/2} \sqrt {a+b x^5}} \, dx=\frac {4 b \sqrt {a+b x^5}}{15 a^2 x^{5/2}}-\frac {2 \sqrt {a+b x^5}}{15 a x^{15/2}} \]

[In]

Int[1/(x^(17/2)*Sqrt[a + b*x^5]),x]

[Out]

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

Rule 277

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*} \text {integral}& = -\frac {2 \sqrt {a+b x^5}}{15 a x^{15/2}}-\frac {(2 b) \int \frac {1}{x^{7/2} \sqrt {a+b x^5}} \, dx}{3 a} \\ & = -\frac {2 \sqrt {a+b x^5}}{15 a x^{15/2}}+\frac {4 b \sqrt {a+b x^5}}{15 a^2 x^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.80 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.65 \[ \int \frac {1}{x^{17/2} \sqrt {a+b x^5}} \, dx=-\frac {2 \left (a-2 b x^5\right ) \sqrt {a+b x^5}}{15 a^2 x^{15/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 4.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.54

method	result	size
gosper	\(-\frac {2 \sqrt {b \,x^{5}+a}\, \left (-2 b \,x^{5}+a \right )}{15 x^{\frac {15}{2}} a^{2}}\)	\(26\)
risch	\(-\frac {2 \sqrt {b \,x^{5}+a}\, \left (-2 b \,x^{5}+a \right )}{15 x^{\frac {15}{2}} a^{2}}\)	\(26\)

[In]

int(1/x^(17/2)/(b*x^5+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.56 \[ \int \frac {1}{x^{17/2} \sqrt {a+b x^5}} \, dx=\frac {2 \, {\left (2 \, b x^{5} - a\right )} \sqrt {b x^{5} + a}}{15 \, a^{2} x^{\frac {15}{2}}} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^{17/2} \sqrt {a+b x^5}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.73 \[ \int \frac {1}{x^{17/2} \sqrt {a+b x^5}} \, dx=\frac {2 \, {\left (\frac {3 \, \sqrt {b x^{5} + a} b}{x^{\frac {5}{2}}} - \frac {{\left (b x^{5} + a\right )}^{\frac {3}{2}}}{x^{\frac {15}{2}}}\right )}}{15 \, a^{2}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^{17/2} \sqrt {a+b x^5}} \, dx=-\frac {2 \, {\left (b + \frac {a}{x^{5}}\right )}^{\frac {3}{2}}}{15 \, a^{2}} + \frac {2 \, \sqrt {b + \frac {a}{x^{5}}} b}{5 \, a^{2}} - \frac {4 \, b^{\frac {3}{2}}}{15 \, a^{2}} \]

[In]

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

[Out]

-2/15*(b + a/x^5)^(3/2)/a^2 + 2/5*sqrt(b + a/x^5)*b/a^2 - 4/15*b^(3/2)/a^2

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^{17/2} \sqrt {a+b x^5}} \, dx=\int \frac {1}{x^{17/2}\,\sqrt {b\,x^5+a}} \,d x \]

[In]

int(1/(x^(17/2)*(a + b*x^5)^(1/2)),x)

[Out]

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