3.14.11 \(\int \frac {1}{x^{7/2} \sqrt {a+b x^5}} \, dx\) [1311]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {270} \begin {gather*} -\frac {2 \sqrt {a+b x^5}}{5 a x^{5/2}} \end {gather*}

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

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Mathematica [A]
time = 0.24, size = 23, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {a+b x^5}}{5 a x^{5/2}} \end {gather*}

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 = 0.16, size = 18, normalized size = 0.78

method	result	size
gosper	\(-\frac {2 \sqrt {b \,x^{5}+a}}{5 a \,x^{\frac {5}{2}}}\)	\(18\)
risch	\(-\frac {2 \sqrt {b \,x^{5}+a}}{5 a \,x^{\frac {5}{2}}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 17, normalized size = 0.74 \begin {gather*} -\frac {2 \, \sqrt {b x^{5} + a}}{5 \, a x^{\frac {5}{2}}} \end {gather*}

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 = 0.37, size = 17, normalized size = 0.74 \begin {gather*} -\frac {2 \, \sqrt {b x^{5} + a}}{5 \, a x^{\frac {5}{2}}} \end {gather*}

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 = 3.66, size = 22, normalized size = 0.96 \begin {gather*} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x^{5}} + 1}}{5 a} \end {gather*}

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 = 1.10, size = 23, normalized size = 1.00 \begin {gather*} -\frac {2 \, \sqrt {b + \frac {a}{x^{5}}}}{5 \, a} + \frac {2 \, \sqrt {b}}{5 \, a} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{x^{7/2}\,\sqrt {b\,x^5+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(7/2)*(a + b*x^5)^(1/2)),x)

[Out]

int(1/(x^(7/2)*(a + b*x^5)^(1/2)), x)

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