3.370 \(\int \frac {\sqrt {\frac {a}{x^{17}}}}{\sqrt {1+x^5}} \, dx\)

Optimal. Leaf size=49 \[ \frac {4}{15} x^6 \sqrt {x^5+1} \sqrt {\frac {a}{x^{17}}}-\frac {2}{15} x \sqrt {x^5+1} \sqrt {\frac {a}{x^{17}}} \]

[Out]

-2/15*x*(a/x^17)^(1/2)*(x^5+1)^(1/2)+4/15*x^6*(a/x^17)^(1/2)*(x^5+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {15, 271, 264} \[ \frac {4}{15} x^6 \sqrt {x^5+1} \sqrt {\frac {a}{x^{17}}}-\frac {2}{15} x \sqrt {x^5+1} \sqrt {\frac {a}{x^{17}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a/x^17]/Sqrt[1 + x^5],x]

[Out]

(-2*Sqrt[a/x^17]*x*Sqrt[1 + x^5])/15 + (4*Sqrt[a/x^17]*x^6*Sqrt[1 + x^5])/15

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]

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 {\sqrt {\frac {a}{x^{17}}}}{\sqrt {1+x^5}} \, dx &=\left (\sqrt {\frac {a}{x^{17}}} x^{17/2}\right ) \int \frac {1}{x^{17/2} \sqrt {1+x^5}} \, dx\\ &=-\frac {2}{15} \sqrt {\frac {a}{x^{17}}} x \sqrt {1+x^5}-\frac {1}{3} \left (2 \sqrt {\frac {a}{x^{17}}} x^{17/2}\right ) \int \frac {1}{x^{7/2} \sqrt {1+x^5}} \, dx\\ &=-\frac {2}{15} \sqrt {\frac {a}{x^{17}}} x \sqrt {1+x^5}+\frac {4}{15} \sqrt {\frac {a}{x^{17}}} x^6 \sqrt {1+x^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 30, normalized size = 0.61 \[ -\frac {2}{15} x \left (1-2 x^5\right ) \sqrt {x^5+1} \sqrt {\frac {a}{x^{17}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a/x^17]/Sqrt[1 + x^5],x]

[Out]

(-2*Sqrt[a/x^17]*x*(1 - 2*x^5)*Sqrt[1 + x^5])/15

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 25, normalized size = 0.51 \[ \frac {2}{15} \, {\left (2 \, x^{6} - x\right )} \sqrt {x^{5} + 1} \sqrt {\frac {a}{x^{17}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(2*x^6 - x)*sqrt(x^5 + 1)*sqrt(a/x^17)

________________________________________________________________________________________

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((a/x^17)^(1/2)/(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(x)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):
Check [abs(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the ar
gument is real):Check [abs(t_nostep)]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error:
Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.00, size = 44, normalized size = 0.90 \[ \frac {2 \left (x +1\right ) \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (2 x^{5}-1\right ) \sqrt {\frac {a}{x^{17}}}\, x}{15 \sqrt {x^{5}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*x*(x+1)*(x^4-x^3+x^2-x+1)*(2*x^5-1)*(a/x^17)^(1/2)/(x^5+1)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 2.87, size = 50, normalized size = 1.02 \[ \frac {2 \, {\left (2 \, \sqrt {a} x^{11} + \sqrt {a} x^{6} - \sqrt {a} x\right )}}{15 \, \sqrt {x^{4} - x^{3} + x^{2} - x + 1} \sqrt {x + 1} x^{\frac {17}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(2*sqrt(a)*x^11 + sqrt(a)*x^6 - sqrt(a)*x)/(sqrt(x^4 - x^3 + x^2 - x + 1)*sqrt(x + 1)*x^(17/2))

________________________________________________________________________________________

mupad [B]  time = 2.67, size = 29, normalized size = 0.59 \[ \frac {\sqrt {\frac {a}{x^{17}}}\,\left (\frac {4\,x^{11}}{15}+\frac {2\,x^6}{15}-\frac {2\,x}{15}\right )}{\sqrt {x^5+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/x^17)^(1/2)/(x^5 + 1)^(1/2),x)

[Out]

((a/x^17)^(1/2)*((2*x^6)/15 - (2*x)/15 + (4*x^11)/15))/(x^5 + 1)^(1/2)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a}{x^{17}}}}{\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a/x**17)/sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)

________________________________________________________________________________________