3.4.70 \(\int \frac {\sqrt {\frac {a}{x^{17}}}}{\sqrt {1+x^5}} \, dx\) [370]

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

[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)

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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, 277, 270} \begin {gather*} \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}}} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.83, size = 30, normalized size = 0.61 \begin {gather*} \frac {2}{15} \sqrt {\frac {a}{x^{17}}} x \sqrt {1+x^5} \left (-1+2 x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.20, size = 25, normalized size = 0.51

method	result	size
meijerg	\(-\frac {2 \sqrt {\frac {a}{x^{17}}}\, x \left (-2 x^{5}+1\right ) \sqrt {x^{5}+1}}{15}\)	\(25\)
risch	\(\frac {2 \sqrt {\frac {a}{x^{17}}}\, x \left (2 x^{10}+x^{5}-1\right )}{15 \sqrt {x^{5}+1}}\)	\(28\)
gosper	\(\frac {2 x \left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (2 x^{5}-1\right ) \sqrt {\frac {a}{x^{17}}}}{15 \sqrt {x^{5}+1}}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.67, size = 50, normalized size = 1.02 \begin {gather*} \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}}} \end {gather*}

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))

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Fricas [A]
time = 0.33, size = 25, normalized size = 0.51 \begin {gather*} \frac {2}{15} \, {\left (2 \, x^{6} - x\right )} \sqrt {x^{5} + 1} \sqrt {\frac {a}{x^{17}}} \end {gather*}

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)

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

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)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [B]
time = 2.67, size = 29, normalized size = 0.59 \begin {gather*} \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}} \end {gather*}

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)

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