3.369 \(\int \frac {\sqrt {\frac {a}{x^7}}}{\sqrt {1+x^5}} \, dx\)

Optimal. Leaf size=23 \[ -\frac {2}{5} x \sqrt {x^5+1} \sqrt {\frac {a}{x^7}} \]

[Out]

-2/5*x*(a/x^7)^(1/2)*(x^5+1)^(1/2)

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Rubi [A]  time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {15, 264} \[ -\frac {2}{5} x \sqrt {x^5+1} \sqrt {\frac {a}{x^7}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a/x^7]/Sqrt[1 + x^5],x]

[Out]

(-2*Sqrt[a/x^7]*x*Sqrt[1 + x^5])/5

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

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 {\sqrt {\frac {a}{x^7}}}{\sqrt {1+x^5}} \, dx &=\left (\sqrt {\frac {a}{x^7}} x^{7/2}\right ) \int \frac {1}{x^{7/2} \sqrt {1+x^5}} \, dx\\ &=-\frac {2}{5} \sqrt {\frac {a}{x^7}} x \sqrt {1+x^5}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \[ -\frac {2}{5} x \sqrt {x^5+1} \sqrt {\frac {a}{x^7}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a/x^7]/Sqrt[1 + x^5],x]

[Out]

(-2*Sqrt[a/x^7]*x*Sqrt[1 + x^5])/5

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fricas [A]  time = 0.45, size = 17, normalized size = 0.74 \[ -\frac {2}{5} \, \sqrt {x^{5} + 1} x \sqrt {\frac {a}{x^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*sqrt(x^5 + 1)*x*sqrt(a/x^7)

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giac [A]  time = 0.24, size = 28, normalized size = 1.22 \[ -\frac {2 \, a^{4} {\left (\frac {\sqrt {a + \frac {a}{x^{5}}}}{a^{3}} - \frac {1}{a^{\frac {5}{2}}}\right )}}{5 \, {\left | a \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*a^4*(sqrt(a + a/x^5)/a^3 - 1/a^(5/2))/abs(a)

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maple [B]  time = 0.00, size = 37, normalized size = 1.61 \[ -\frac {2 \left (x +1\right ) \left (x^{4}-x^{3}+x^{2}-x +1\right ) \sqrt {\frac {a}{x^{7}}}\, x}{5 \sqrt {x^{5}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*x*(x+1)*(x^4-x^3+x^2-x+1)*(a/x^7)^(1/2)/(x^5+1)^(1/2)

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maxima [B]  time = 2.22, size = 41, normalized size = 1.78 \[ -\frac {2 \, {\left (\sqrt {a} x^{6} + \sqrt {a} x\right )}}{5 \, \sqrt {x^{4} - x^{3} + x^{2} - x + 1} \sqrt {x + 1} x^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(sqrt(a)*x^6 + sqrt(a)*x)/(sqrt(x^4 - x^3 + x^2 - x + 1)*sqrt(x + 1)*x^(7/2))

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mupad [B]  time = 2.69, size = 17, normalized size = 0.74 \[ -\frac {2\,x\,\sqrt {x^5+1}\,\sqrt {\frac {a}{x^7}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/x^7)^(1/2)/(x^5 + 1)^(1/2),x)

[Out]

-(2*x*(x^5 + 1)^(1/2)*(a/x^7)^(1/2))/5

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a}{x^{7}}}}{\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a/x**7)/sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)

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