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

Optimal. Leaf size=254 \[ \frac {4 \log \left (\sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}-\sqrt [3]{c}\right )}{3 a c^{2/3}}-\frac {2 \log \left (\sqrt [3]{c} \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}+\left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{2/3}+c^{2/3}\right )}{3 a c^{2/3}}-\frac {4 \tan ^{-1}\left (\frac {2 \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{\sqrt {3} \sqrt [3]{c}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} a c^{2/3}}-\frac {4 \sqrt [3]{\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c}}{a \sqrt [4]{\sqrt {a^2 x^2-b}+a x}} \]

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Rubi [F]  time = 1.09, 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 [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {-b+a^2 x^2} \sqrt [4]{a x+\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))^(1/3)/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)),x]

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.34, size = 74, normalized size = 0.29 \begin {gather*} \frac {3 \left (\sqrt [4]{\sqrt {a^2 x^2-b}+a x}+c\right )^{4/3} \, _2F_1\left (\frac {4}{3},2;\frac {7}{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[(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(1/3)/(Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)),x
]

[Out]

(3*(c + (a*x + Sqrt[-b + a^2*x^2])^(1/4))^(4/3)*Hypergeometric2F1[4/3, 2, 7/3, (c + (a*x + Sqrt[-b + a^2*x^2])
^(1/4))/c])/(a*c^2)

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IntegrateAlgebraic [A]  time = 0.90, size = 254, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}}{a \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^{2/3}}+\frac {4 \log \left (-\sqrt [3]{c}+\sqrt [3]{c+\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{3 a c^{2/3}}-\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^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 255, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {3} b {\left (c^{2}\right )}^{\frac {1}{6}} c \arctan \left (\frac {\sqrt {3} \sqrt {c^{2}} c + 2 \, \sqrt {3} {\left (c^{2}\right )}^{\frac {5}{6}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}}{3 \, c^{2}}\right ) + b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {2}{3}} c + {\left (c^{2}\right )}^{\frac {1}{3}} c + {\left (c^{2}\right )}^{\frac {2}{3}} {\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}}\right ) - 2 \, b {\left (c^{2}\right )}^{\frac {2}{3}} \log \left ({\left (c + {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}}\right )}^{\frac {1}{3}} c - {\left (c^{2}\right )}^{\frac {2}{3}}\right ) + 6 \, {\left (a c^{2} x - \sqrt {a^{2} x^{2} - b} c^{2}\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}}\right )}}{3 \, a b c^{2}} \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))^(1/3)/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x, algorithm
="fricas")

[Out]

-2/3*(2*sqrt(3)*b*(c^2)^(1/6)*c*arctan(1/3*(sqrt(3)*sqrt(c^2)*c + 2*sqrt(3)*(c^2)^(5/6)*(c + (a*x + sqrt(a^2*x
^2 - b))^(1/4))^(1/3))/c^2) + b*(c^2)^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(2/3)*c + (c^2)^(1/3)*c
+ (c^2)^(2/3)*(c + (a*x + sqrt(a^2*x^2 - b))^(1/4))^(1/3)) - 2*b*(c^2)^(2/3)*log((c + (a*x + sqrt(a^2*x^2 - b)
)^(1/4))^(1/3)*c - (c^2)^(2/3)) + 6*(a*c^2*x - sqrt(a^2*x^2 - b)*c^2)*(a*x + sqrt(a^2*x^2 - b))^(3/4)*(c + (a*
x + sqrt(a^2*x^2 - b))^(1/4))^(1/3))/(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((c+(a*x+(a^2*x^2-b)^(1/2))^(1/4))^(1/3)/(a^2*x^2-b)^(1/2)/(a*x+(a^2*x^2-b)^(1/2))^(1/4),x, algorithm
="giac")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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mupad [B]  time = 2.19, size = 99, normalized size = 0.39 \begin {gather*} -\frac {6\,{\left (c+{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},\frac {2}{3};\ \frac {5}{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 (\frac {c}{{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}}+1\right )}^{1/3}} \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))^(1/3)/((a*x + (a^2*x^2 - b)^(1/2))^(1/4)*(a^2*x^2 - b)^(1/2)),x)

[Out]

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

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

[Out]

Integral((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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