3.30.80 \(\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\)

Optimal. Leaf size=383 \[ -\frac {7 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [6]{c}}\right )}{6 \sqrt {3} a c^{13/6}}+\frac {7 \tan ^{-1}\left (\frac {2 \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [6]{c}}+\frac {1}{\sqrt {3}}\right )}{6 \sqrt {3} a c^{13/6}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [6]{c}}\right )}{9 a c^{13/6}}-\frac {7 \tanh ^{-1}\left (\frac {\frac {\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [6]{c}}+\sqrt [6]{c}}{\sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}\right )}{18 a c^{13/6}}+\frac {7 \sqrt [4]{\sqrt {a^2 x^2-b}+a x} \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{5/6}-6 c \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{5/6}}{3 a c^2 \sqrt {\sqrt {a^2 x^2-b}+a x}} \]

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Rubi [F]  time = 1.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(Sqrt[-b + a^2*x^2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)),x]

[Out]

Defer[Int][1/(Sqrt[-b + a^2*x^2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)),
x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}

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Mathematica [C]  time = 0.36, size = 76, normalized size = 0.20 \begin {gather*} -\frac {24 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{5/6} \, _2F_1\left (\frac {5}{6},3;\frac {11}{6};\frac {c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{c}\right )}{5 a c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-b + a^2*x^2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)),x
]

[Out]

(-24*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(5/6)*Hypergeometric2F1[5/6, 3, 11/6, (c + (a*x + Sqrt[-b + a^2*x^
2])^(1/4))/c])/(5*a*c^3)

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IntegrateAlgebraic [A]  time = 22.21, size = 355, normalized size = 0.93 \begin {gather*} \frac {\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{5/6} \left (-6 c+7 \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )}{3 a c^2 \sqrt {a x+\sqrt {-b+a^2 x^2}}}-\frac {7 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{6 \sqrt {3} a c^{13/6}}+\frac {7 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [6]{c}}\right )}{6 \sqrt {3} a c^{13/6}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}\right )}{9 a c^{13/6}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt [6]{c} \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}\right )}{18 a c^{13/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*Sqrt[a*x + Sqrt[-b + a^2*x^2]]*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))
^(1/6)),x]

[Out]

((c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(5/6)*(-6*c + 7*(a*x + Sqrt[-b + a^2*x^2])^(1/4)))/(3*a*c^2*Sqrt[a*x +
Sqrt[-b + a^2*x^2]]) - (7*ArcTan[1/Sqrt[3] - (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*c^(1/6
))])/(6*Sqrt[3]*a*c^(13/6)) + (7*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*
c^(1/6))])/(6*Sqrt[3]*a*c^(13/6)) - (7*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)/c^(1/6)])/(9*a*c^(
13/6)) - (7*ArcTanh[(c^(1/6)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/(c^(1/3) + (c + (a*x + Sqrt[-b + a^
2*x^2])^(1/4))^(1/3))])/(18*a*c^(13/6))

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fricas [B]  time = 0.75, size = 761, normalized size = 1.99

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6),x, algorit
hm="fricas")

[Out]

-1/36*(28*sqrt(3)*a*b*c^2*(1/(a^6*c^13))^(1/6)*arctan(2/3*sqrt(3)*sqrt(a^5*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4
))^(1/6)*c^11*(1/(a^6*c^13))^(5/6) + a^4*c^9*(1/(a^6*c^13))^(2/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3
))*a*c^2*(1/(a^6*c^13))^(1/6) - 2/3*sqrt(3)*a*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*c^2*(1/(a^6*c^13))^(
1/6) - 1/3*sqrt(3)) + 28*sqrt(3)*a*b*c^2*(1/(a^6*c^13))^(1/6)*arctan(2/3*sqrt(3)*sqrt(-a^5*(c + (a*x + sqrt(a^
2*x^2 - b))^(1/4))^(1/6)*c^11*(1/(a^6*c^13))^(5/6) + a^4*c^9*(1/(a^6*c^13))^(2/3) + (c + (a*x + sqrt(a^2*x^2 -
b))^(1/4))^(1/3))*a*c^2*(1/(a^6*c^13))^(1/6) - 2/3*sqrt(3)*a*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*c^2*
(1/(a^6*c^13))^(1/6) + 1/3*sqrt(3)) + 7*a*b*c^2*(1/(a^6*c^13))^(1/6)*log(a^5*(c + (a*x + sqrt(a^2*x^2 - b))^(1
/4))^(1/6)*c^11*(1/(a^6*c^13))^(5/6) + a^4*c^9*(1/(a^6*c^13))^(2/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1
/3)) - 7*a*b*c^2*(1/(a^6*c^13))^(1/6)*log(-a^5*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*c^11*(1/(a^6*c^13))
^(5/6) + a^4*c^9*(1/(a^6*c^13))^(2/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)) + 14*a*b*c^2*(1/(a^6*c^13
))^(1/6)*log(a^5*c^11*(1/(a^6*c^13))^(5/6) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)) - 14*a*b*c^2*(1/(a^6
*c^13))^(1/6)*log(-a^5*c^11*(1/(a^6*c^13))^(5/6) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)) - 12*(7*(a*x +
sqrt(a^2*x^2 - b))^(3/4)*(a*x - sqrt(a^2*x^2 - b)) - 6*(a*c*x - sqrt(a^2*x^2 - b)*c)*sqrt(a*x + sqrt(a^2*x^2
- b)))*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(5/6))/(a*b*c^2)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6),x, algorit
hm="giac")

[Out]

Timed out

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a^{2} x^{2}-b}\, \sqrt {a x +\sqrt {a^{2} x^{2}-b}}\, \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{6}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6),x)

[Out]

int(1/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a^{2} x^{2} - b} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/6),x, algorit
hm="maxima")

[Out]

integrate(1/(sqrt(a^2*x^2 - b)*sqrt(a*x + sqrt(a^2*x^2 - b))*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\sqrt {a\,x+\sqrt {a^2\,x^2-b}}\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/6}\,\sqrt {a^2\,x^2-b}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x + (a^2*x^2 - b)^(1/2))^(1/2)*(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(1/6)*(a^2*x^2 - b)^(1/2)),x)

[Out]

int(1/((a*x + (a^2*x^2 - b)^(1/2))^(1/2)*(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(1/6)*(a^2*x^2 - b)^(1/2)), x
)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [6]{c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {a^{2} x^{2} - b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a**2*x**2-b)**(1/2)/(a*x+(a**2*x**2-b)**(1/2))**(1/2)/(c+(a*x+(a**2*x**2-b)**(1/2))**(1/4))**(1/6
),x)

[Out]

Integral(1/((c + (a*x + sqrt(a**2*x**2 - b))**(1/4))**(1/6)*sqrt(a*x + sqrt(a**2*x**2 - b))*sqrt(a**2*x**2 - b
)), x)

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