3.3.93 \(\int x^9 \sqrt {\frac {5-7 x^5}{7+5 x^5}} \, dx\) [293]

Optimal. Leaf size=106 \[ -\frac {27}{350} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )+\frac {1}{250} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )^2+\frac {2257 \tan ^{-1}\left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^5}{7+5 x^5}}\right )}{875 \sqrt {35}} \]

[Out]

2257/30625*arctan(1/7*35^(1/2)*((-7*x^5+5)/(5*x^5+7))^(1/2))*35^(1/2)-27/350*(5*x^5+7)*((-7*x^5+5)/(5*x^5+7))^
(1/2)+1/250*(5*x^5+7)^2*((-7*x^5+5)/(5*x^5+7))^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1981, 1980, 466, 393, 210} \begin {gather*} \frac {2257 \text {ArcTan}\left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^5}{5 x^5+7}}\right )}{875 \sqrt {35}}+\frac {1}{250} \sqrt {\frac {5-7 x^5}{5 x^5+7}} \left (5 x^5+7\right )^2-\frac {27}{350} \sqrt {\frac {5-7 x^5}{5 x^5+7}} \left (5 x^5+7\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^9*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)],x]

[Out]

(-27*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]*(7 + 5*x^5))/350 + (Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]*(7 + 5*x^5)^2)/250 + (225
7*ArcTan[Sqrt[5/7]*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]])/(875*Sqrt[35])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 466

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1980

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_Symbol] :> With[{q = Denominator[p]}
, Dist[q*e*(b*c - a*d), Subst[Int[x^(q*(p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a
+ b*x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]

Rule 1981

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Dist[1/n, Sub
st[Int[x^(Simplify[(m + 1)/n] - 1)*(e*((a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n,
p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int x^9 \sqrt {\frac {5-7 x^5}{7+5 x^5}} \, dx &=-\left (\frac {148}{5} \text {Subst}\left (\int \frac {x^2 \left (-5+7 x^2\right )}{\left (-7-5 x^2\right )^3} \, dx,x,\sqrt {\frac {5-7 x^5}{7+5 x^5}}\right )\right )\\ &=\frac {1}{250} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )^2+\frac {37}{125} \text {Subst}\left (\int \frac {-74+140 x^2}{\left (-7-5 x^2\right )^2} \, dx,x,\sqrt {\frac {5-7 x^5}{7+5 x^5}}\right )\\ &=-\frac {27}{350} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )+\frac {1}{250} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )^2-\frac {2257}{875} \text {Subst}\left (\int \frac {1}{-7-5 x^2} \, dx,x,\sqrt {\frac {5-7 x^5}{7+5 x^5}}\right )\\ &=-\frac {27}{350} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )+\frac {1}{250} \sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (7+5 x^5\right )^2+\frac {2257 \tan ^{-1}\left (\sqrt {\frac {5}{7}} \sqrt {\frac {5-7 x^5}{7+5 x^5}}\right )}{875 \sqrt {35}}\\ \end {align*}

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Mathematica [A]
time = 0.63, size = 106, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\frac {5-7 x^5}{7+5 x^5}} \left (35 \sqrt {5-7 x^5} \left (-602-185 x^5+175 x^{10}\right )+4514 \sqrt {35} \sqrt {7+5 x^5} \tan ^{-1}\left (\frac {\sqrt {\frac {25}{7}-5 x^5}}{\sqrt {7+5 x^5}}\right )\right )}{61250 \sqrt {5-7 x^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^9*Sqrt[(5 - 7*x^5)/(7 + 5*x^5)],x]

[Out]

(Sqrt[(5 - 7*x^5)/(7 + 5*x^5)]*(35*Sqrt[5 - 7*x^5]*(-602 - 185*x^5 + 175*x^10) + 4514*Sqrt[35]*Sqrt[7 + 5*x^5]
*ArcTan[Sqrt[25/7 - 5*x^5]/Sqrt[7 + 5*x^5]]))/(61250*Sqrt[5 - 7*x^5])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.42, size = 114, normalized size = 1.08

method	result	size
trager	\(\frac {\left (5 x^{5}+7\right ) \left (35 x^{5}-86\right ) \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}}{1750}+\frac {2257 \RootOf \left (\textit {\_Z}^{2}+35\right ) \ln \left (35 \RootOf \left (\textit {\_Z}^{2}+35\right ) x^{5}+175 \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}\, x^{5}+12 \RootOf \left (\textit {\_Z}^{2}+35\right )+245 \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}\right )}{61250}\)	\(114\)
risch	\(\frac {\left (5 x^{5}+7\right ) \left (35 x^{5}-86\right ) \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}}{1750}+\frac {2257 \RootOf \left (\textit {\_Z}^{2}+35\right ) \ln \left (-35 \RootOf \left (\textit {\_Z}^{2}+35\right ) x^{5}-12 \RootOf \left (\textit {\_Z}^{2}+35\right )+35 \sqrt {-35 x^{10}-24 x^{5}+35}\right ) \sqrt {-\frac {7 x^{5}-5}{5 x^{5}+7}}\, \sqrt {-\left (5 x^{5}+7\right ) \left (7 x^{5}-5\right )}}{61250 \left (7 x^{5}-5\right )}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*((-7*x^5+5)/(5*x^5+7))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/1750*(5*x^5+7)*(35*x^5-86)*(-(7*x^5-5)/(5*x^5+7))^(1/2)+2257/61250*RootOf(_Z^2+35)*ln(35*RootOf(_Z^2+35)*x^5
+175*(-(7*x^5-5)/(5*x^5+7))^(1/2)*x^5+12*RootOf(_Z^2+35)+245*(-(7*x^5-5)/(5*x^5+7))^(1/2))

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Maxima [A]
time = 0.48, size = 121, normalized size = 1.14 \begin {gather*} \frac {2257}{30625} \, \sqrt {35} \arctan \left (\frac {1}{7} \, \sqrt {35} \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}}\right ) - \frac {37 \, {\left (675 \, \left (-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}\right )^{\frac {3}{2}} + 427 \, \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}}\right )}}{875 \, {\left (\frac {25 \, {\left (7 \, x^{5} - 5\right )}^{2}}{{\left (5 \, x^{5} + 7\right )}^{2}} - \frac {70 \, {\left (7 \, x^{5} - 5\right )}}{5 \, x^{5} + 7} + 49\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*((-7*x^5+5)/(5*x^5+7))^(1/2),x, algorithm="maxima")

[Out]

2257/30625*sqrt(35)*arctan(1/7*sqrt(35)*sqrt(-(7*x^5 - 5)/(5*x^5 + 7))) - 37/875*(675*(-(7*x^5 - 5)/(5*x^5 + 7
))^(3/2) + 427*sqrt(-(7*x^5 - 5)/(5*x^5 + 7)))/(25*(7*x^5 - 5)^2/(5*x^5 + 7)^2 - 70*(7*x^5 - 5)/(5*x^5 + 7) +
49)

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Fricas [A]
time = 0.38, size = 82, normalized size = 0.77 \begin {gather*} \frac {1}{1750} \, {\left (175 \, x^{10} - 185 \, x^{5} - 602\right )} \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}} + \frac {2257}{61250} \, \sqrt {35} \arctan \left (\frac {\sqrt {35} {\left (35 \, x^{5} + 12\right )} \sqrt {-\frac {7 \, x^{5} - 5}{5 \, x^{5} + 7}}}{35 \, {\left (7 \, x^{5} - 5\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*((-7*x^5+5)/(5*x^5+7))^(1/2),x, algorithm="fricas")

[Out]

1/1750*(175*x^10 - 185*x^5 - 602)*sqrt(-(7*x^5 - 5)/(5*x^5 + 7)) + 2257/61250*sqrt(35)*arctan(1/35*sqrt(35)*(3
5*x^5 + 12)*sqrt(-(7*x^5 - 5)/(5*x^5 + 7))/(7*x^5 - 5))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*((-7*x**5+5)/(5*x**5+7))**(1/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3656 deep

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Giac [A]
time = 4.36, size = 47, normalized size = 0.44 \begin {gather*} \frac {1}{61250} \, {\left (35 \, \sqrt {-35 \, x^{10} - 24 \, x^{5} + 35} {\left (35 \, x^{5} - 86\right )} - 2257 \, \sqrt {35} \arcsin \left (\frac {35}{37} \, x^{5} + \frac {12}{37}\right )\right )} \mathrm {sgn}\left (5 \, x^{5} + 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*((-7*x^5+5)/(5*x^5+7))^(1/2),x, algorithm="giac")

[Out]

1/61250*(35*sqrt(-35*x^10 - 24*x^5 + 35)*(35*x^5 - 86) - 2257*sqrt(35)*arcsin(35/37*x^5 + 12/37))*sgn(5*x^5 +
7)

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Mupad [B]
time = 2.99, size = 134, normalized size = 1.26 \begin {gather*} \frac {2257\,\sqrt {35}\,\mathrm {atan}\left (\frac {\sqrt {5}\,\sqrt {7}\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{7}\right )}{30625}-\frac {43\,\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{4375}-\frac {37\,\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,x^5\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{12250}+\frac {\sqrt {5}\,\sqrt {7}\,\sqrt {35}\,x^{10}\,\sqrt {-\frac {7\,x^5-5}{5\,x^5+7}}}{350} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(-(7*x^5 - 5)/(5*x^5 + 7))^(1/2),x)

[Out]

(2257*35^(1/2)*atan((5^(1/2)*7^(1/2)*(-(7*x^5 - 5)/(5*x^5 + 7))^(1/2))/7))/30625 - (43*5^(1/2)*7^(1/2)*35^(1/2
)*(-(7*x^5 - 5)/(5*x^5 + 7))^(1/2))/4375 - (37*5^(1/2)*7^(1/2)*35^(1/2)*x^5*(-(7*x^5 - 5)/(5*x^5 + 7))^(1/2))/
12250 + (5^(1/2)*7^(1/2)*35^(1/2)*x^10*(-(7*x^5 - 5)/(5*x^5 + 7))^(1/2))/350

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