3.28.55 \(\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\)

Optimal. Leaf size=257 \[ -\frac {4 \log \left (\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}-\sqrt [3]{c}\right )}{3 a c^{4/3}}+\frac {2 \log \left (\sqrt [3]{c} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+c^{2/3}\right )}{3 a c^{4/3}}-\frac {4 \tan ^{-1}\left (\frac {2 \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} a c^{4/3}}-\frac {4 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}}{a c \sqrt [4]{\sqrt {a^2 x^2-b}+a x}} \]

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Rubi [F]  time = 1.24, 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 [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{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]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)),x]

[Out]

Defer[Int][1/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)
), x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {1}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}

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Mathematica [C]  time = 0.30, size = 74, normalized size = 0.29 \begin {gather*} \frac {6 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3} \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {c+\sqrt [4]{a x+\sqrt {a^2 x^2-b}}}{c}\right )}{a c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))
,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.55, size = 257, normalized size = 1.00 \begin {gather*} -\frac {4 \left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}}{a c \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}-\frac {4 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} a c^{4/3}}-\frac {4 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{3 a c^{4/3}}+\frac {2 \log \left (c^{2/3}+\sqrt [3]{c} \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}+\left (c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}\right )^{2/3}\right )}{3 a c^{4/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4
))^(1/3)),x]

[Out]

(-4*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3))/(a*c*(a*x + Sqrt[-b + a^2*x^2])^(1/4)) - (4*ArcTan[1/Sqrt[3]
+ (2*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3))/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*a*c^(4/3)) - (4*Log[-c^(1/3)
+ (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)])/(3*a*c^(4/3)) + (2*Log[c^(2/3) + c^(1/3)*(c + (a*x + Sqrt[-b
+ a^2*x^2])^(1/4))^(1/3) + (c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(2/3)])/(3*a*c^(4/3))

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fricas [A]  time = 0.85, size = 758, normalized size = 2.95 \begin {gather*} \left [\frac {2 \, {\left (3 \, \sqrt {\frac {1}{3}} b c \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \log \left (-6 \, \sqrt {\frac {1}{3}} {\left (a \left (-c\right )^{\frac {2}{3}} x - \sqrt {a^{2} x^{2} - b} \left (-c\right )^{\frac {2}{3}}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} - 3 \, {\left (a \left (-c\right )^{\frac {2}{3}} x - \sqrt {\frac {1}{3}} {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} - \sqrt {a^{2} x^{2} - b} \left (-c\right )^{\frac {2}{3}}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} + 3 \, {\left (a c x - \sqrt {\frac {1}{3}} {\left (a \left (-c\right )^{\frac {1}{3}} c x - \sqrt {a^{2} x^{2} - b} \left (-c\right )^{\frac {1}{3}} c\right )} \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} - \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} + 2 \, b\right ) + b \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {2}{3}} - \left (-c\right )^{\frac {1}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right ) - 2 \, b \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {1}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) - 6 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}}{3 \, a b c^{2}}, -\frac {2 \, {\left (6 \, \sqrt {\frac {1}{3}} b c \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \arctan \left (-\sqrt {\frac {1}{3}} \left (-c\right )^{\frac {1}{3}} \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}} + 2 \, \sqrt {\frac {1}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}}\right ) - b \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {2}{3}} - \left (-c\right )^{\frac {1}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right ) + 2 \, b \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {1}{3}} + {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) + 6 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}}\right )}}{3 \, a b c^{2}}\right ] \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))^(1/4)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3),x, algorit
hm="fricas")

[Out]

[2/3*(3*sqrt(1/3)*b*c*sqrt((-c)^(1/3)/c)*log(-6*sqrt(1/3)*(a*(-c)^(2/3)*x - sqrt(a^2*x^2 - b)*(-c)^(2/3))*(a*x
+ sqrt(a^2*x^2 - b))^(3/4)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3)*sqrt((-c)^(1/3)/c) - 3*(a*(-c)^(2/3)*x
- sqrt(1/3)*(a*c*x - sqrt(a^2*x^2 - b)*c)*sqrt((-c)^(1/3)/c) - sqrt(a^2*x^2 - b)*(-c)^(2/3))*(a*x + sqrt(a^2*
x^2 - b))^(3/4)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) + 3*(a*c*x - sqrt(1/3)*(a*(-c)^(1/3)*c*x - sqrt(a^
2*x^2 - b)*(-c)^(1/3)*c)*sqrt((-c)^(1/3)/c) - sqrt(a^2*x^2 - b)*c)*(a*x + sqrt(a^2*x^2 - b))^(3/4) + 2*b) + b*
(-c)^(2/3)*log((-c)^(2/3) - (-c)^(1/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) + (c + (a*x + sqrt(a^2*x^2
- b))^(1/4))^(2/3)) - 2*b*(-c)^(2/3)*log((-c)^(1/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)) - 6*(a*c*x
- sqrt(a^2*x^2 - b)*c)*(a*x + sqrt(a^2*x^2 - b))^(3/4)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3))/(a*b*c^2),
-2/3*(6*sqrt(1/3)*b*c*sqrt(-(-c)^(1/3)/c)*arctan(-sqrt(1/3)*(-c)^(1/3)*sqrt(-(-c)^(1/3)/c) + 2*sqrt(1/3)*(c +
(a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)*sqrt(-(-c)^(1/3)/c)) - b*(-c)^(2/3)*log((-c)^(2/3) - (-c)^(1/3)*(c + (
a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3)) + 2*b*(-c)^(2/3)*log((-c)
^(1/3) + (c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)) + 6*(a*c*x - sqrt(a^2*x^2 - b)*c)*(a*x + sqrt(a^2*x^2 -
b))^(3/4)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3))/(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))^(1/4)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3),x, algorit
hm="giac")

[Out]

Timed out

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}} \left (c +\left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}\right )^{\frac {1}{3}}}\, dx\]

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))^(1/4)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3),x)

[Out]

int(1/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3),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} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}\,{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))^(1/4)/(c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3),x, algorit
hm="maxima")

[Out]

integrate(1/(sqrt(a^2*x^2 - b)*(a*x + sqrt(a^2*x^2 - b))^(1/4)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)), x
)

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mupad [B]  time = 2.60, size = 99, normalized size = 0.39 \begin {gather*} -\frac {3\,{\left (\frac {c}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}}+1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {4}{3};\ \frac {7}{3};\ -\frac {c}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}}\right )}{a\,{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x + (a^2*x^2 - b)^(1/2))^(1/4)*(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(1/3)*(a^2*x^2 - b)^(1/2)),x)

[Out]

-(3*(c/(a*x + (a^2*x^2 - b)^(1/2))^(1/4) + 1)^(1/3)*hypergeom([1/3, 4/3], 7/3, -c/(a*x + (a^2*x^2 - b)^(1/2))^
(1/4)))/(a*(a*x + (a^2*x^2 - b)^(1/2))^(1/4)*(c + (a*x + (a^2*x^2 - b)^(1/2))^(1/4))^(1/3))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{c + \sqrt [4]{a x + \sqrt {a^{2} x^{2} - b}}} \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(1/(a**2*x**2-b)**(1/2)/(a*x+(a**2*x**2-b)**(1/2))**(1/4)/(c+(a*x+(a**2*x**2-b)**(1/2))**(1/4))**(1/3
),x)

[Out]

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

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