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

Optimal. Leaf size=289 \[ \frac {24 \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{a}+\frac {4 \sqrt {3} \sqrt [6]{c} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [6]{c}}\right )}{a}-\frac {4 \sqrt {3} \sqrt [6]{c} \tan ^{-1}\left (\frac {2 \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [6]{c}}+\frac {1}{\sqrt {3}}\right )}{a}-\frac {8 \sqrt [6]{c} \tanh ^{-1}\left (\frac {\sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [6]{c}}\right )}{a}-\frac {4 \sqrt [6]{c} \tanh ^{-1}\left (\frac {\frac {\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt [6]{c}}+\sqrt [6]{c}}{\sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}\right )}{a} \]

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Rubi [F]  time = 0.31, 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 {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.18, size = 73, normalized size = 0.25 \begin {gather*} -\frac {24 \sqrt [6]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c} \left (\, _2F_1\left (\frac {1}{6},1;\frac {7}{6};\frac {c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{c}\right )-1\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-24*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)*(-1 + Hypergeometric2F1[1/6, 1, 7/6, (c + (a*x + Sqrt[-b + a
^2*x^2])^(1/4))/c]))/a

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IntegrateAlgebraic [A]  time = 1.22, size = 289, normalized size = 1.00 \begin {gather*} \frac {24 \sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{a}+\frac {4 \sqrt {3} \sqrt [6]{c} \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 )}{a}-\frac {4 \sqrt {3} \sqrt [6]{c} \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 )}{a}-\frac {8 \sqrt [6]{c} \tanh ^{-1}\left (\frac {\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}\right )}{a}-\frac {4 \sqrt [6]{c} \tanh ^{-1}\left (\frac {\sqrt [6]{c}+\frac {\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt [6]{c}}}{\sqrt [6]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(24*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6))/a + (4*Sqrt[3]*c^(1/6)*ArcTan[1/Sqrt[3] - (2*(c + (a*x + Sqr
t[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*c^(1/6))])/a - (4*Sqrt[3]*c^(1/6)*ArcTan[1/Sqrt[3] + (2*(c + (a*x + Sq
rt[-b + a^2*x^2])^(1/4))^(1/6))/(Sqrt[3]*c^(1/6))])/a - (8*c^(1/6)*ArcTanh[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/
4))^(1/6)/c^(1/6)])/a - (4*c^(1/6)*ArcTanh[(c^(1/6) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)/c^(1/6))/(c
+ (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/6)])/a

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fricas [B]  time = 0.65, size = 584, normalized size = 2.02 \begin {gather*} \frac {2 \, {\left (4 \, \sqrt {3} a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {a^{2} \left (\frac {c}{a^{6}}\right )^{\frac {1}{3}} + a {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}} a^{5} \left (\frac {c}{a^{6}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} a^{5} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} \left (\frac {c}{a^{6}}\right )^{\frac {5}{6}} - \sqrt {3} c}{3 \, c}\right ) + 4 \, \sqrt {3} a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {a^{2} \left (\frac {c}{a^{6}}\right )^{\frac {1}{3}} - a {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}} a^{5} \left (\frac {c}{a^{6}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} a^{5} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} \left (\frac {c}{a^{6}}\right )^{\frac {5}{6}} + \sqrt {3} c}{3 \, c}\right ) - a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \log \left (64 \, a^{2} \left (\frac {c}{a^{6}}\right )^{\frac {1}{3}} + 64 \, a {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + 64 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) + a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \log \left (64 \, a^{2} \left (\frac {c}{a^{6}}\right )^{\frac {1}{3}} - 64 \, a {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}} \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + 64 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) - 2 \, a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \log \left (4 \, a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + 4 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) + 2 \, a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} \log \left (-4 \, a \left (\frac {c}{a^{6}}\right )^{\frac {1}{6}} + 4 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right ) + 12 \, {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{6}}\right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(4*sqrt(3)*a*(c/a^6)^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt(a^2*(c/a^6)^(1/3) + a*(c + (a*x + sqrt(a^2*x^2 - b))^(
1/4))^(1/6)*(c/a^6)^(1/6) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3))*a^5*(c/a^6)^(5/6) - 2*sqrt(3)*a^5*(c
+ (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*(c/a^6)^(5/6) - sqrt(3)*c)/c) + 4*sqrt(3)*a*(c/a^6)^(1/6)*arctan(1/3*
(2*sqrt(3)*sqrt(a^2*(c/a^6)^(1/3) - a*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*(c/a^6)^(1/6) + (c + (a*x +
sqrt(a^2*x^2 - b))^(1/4))^(1/3))*a^5*(c/a^6)^(5/6) - 2*sqrt(3)*a^5*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)
*(c/a^6)^(5/6) + sqrt(3)*c)/c) - a*(c/a^6)^(1/6)*log(64*a^2*(c/a^6)^(1/3) + 64*a*(c + (a*x + sqrt(a^2*x^2 - b)
)^(1/4))^(1/6)*(c/a^6)^(1/6) + 64*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)) + a*(c/a^6)^(1/6)*log(64*a^2*(c
/a^6)^(1/3) - 64*a*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)*(c/a^6)^(1/6) + 64*(c + (a*x + sqrt(a^2*x^2 - b
))^(1/4))^(1/3)) - 2*a*(c/a^6)^(1/6)*log(4*a*(c/a^6)^(1/6) + 4*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)) +
2*a*(c/a^6)^(1/6)*log(-4*a*(c/a^6)^(1/6) + 4*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/6)) + 12*(c + (a*x + sqr
t(a^2*x^2 - b))^(1/4))^(1/6))/a

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

[Out]

Timed out

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{6}}}{\sqrt {a^{2} x^{2}-b}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

int((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 {\sqrt [6]{c + \sqrt [4]{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((c+(a*x+(a**2*x**2-b)**(1/2))**(1/4))**(1/6)/(a**2*x**2-b)**(1/2),x)

[Out]

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

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