3.1311 \(\int \frac{1}{x^{7/2} \sqrt{a+b x^5}} \, dx\)

Optimal. Leaf size=23 \[ -\frac{2 \sqrt{a+b x^5}}{5 a x^{5/2}} \]

[Out]

(-2*Sqrt[a + b*x^5])/(5*a*x^(5/2))

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Rubi [A]  time = 0.0047669, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {264} \[ -\frac{2 \sqrt{a+b x^5}}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*Sqrt[a + b*x^5]),x]

[Out]

(-2*Sqrt[a + b*x^5])/(5*a*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]

Rubi steps

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

Mathematica [A]  time = 0.0048548, size = 23, normalized size = 1. \[ -\frac{2 \sqrt{a+b x^5}}{5 a x^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[a + b*x^5])/(5*a*x^(5/2))

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Maple [A]  time = 0.003, size = 18, normalized size = 0.8 \begin{align*} -{\frac{2}{5\,a}\sqrt{b{x}^{5}+a}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(b*x^5+a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.96917, size = 23, normalized size = 1. \begin{align*} -\frac{2 \, \sqrt{b x^{5} + a}}{5 \, a x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*sqrt(b*x^5 + a)/(a*x^(5/2))

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Fricas [A]  time = 1.66232, size = 46, normalized size = 2. \begin{align*} -\frac{2 \, \sqrt{b x^{5} + a}}{5 \, a x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*sqrt(b*x^5 + a)/(a*x^(5/2))

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Sympy [A]  time = 27.6844, size = 22, normalized size = 0.96 \begin{align*} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x^{5}} + 1}}{5 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b)*sqrt(a/(b*x**5) + 1)/(5*a)

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Giac [A]  time = 1.16133, size = 31, normalized size = 1.35 \begin{align*} -\frac{2 \, \sqrt{b + \frac{a}{x^{5}}}}{5 \, a} + \frac{2 \, \sqrt{b}}{5 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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