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

Optimal. Leaf size=124 \[ \frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (\frac {\sqrt {a-b x^2}}{\sqrt {a}}+1\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}} \]

[Out]

1/2*arctan(a^(3/4)*(1-(-b*x^2+a)^(1/2)/a^(1/2))/x/(-b*x^2+a)^(1/4)/b^(1/2))/a^(3/4)/b^(1/2)+1/2*arctanh(a^(3/4
)*(1+(-b*x^2+a)^(1/2)/a^(1/2))/x/(-b*x^2+a)^(1/4)/b^(1/2))/a^(3/4)/b^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {397} \[ \frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (\frac {\sqrt {a-b x^2}}{\sqrt {a}}+1\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 397

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b^2/a, 4]}, -Simp[(b*ArcT
an[(b + q^2*Sqrt[a + b*x^2])/(q^3*x*(a + b*x^2)^(1/4))])/(2*a*d*q), x] - Simp[(b*ArcTanh[(b - q^2*Sqrt[a + b*x
^2])/(q^3*x*(a + b*x^2)^(1/4))])/(2*a*d*q), x]] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[b^2/a
]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right )} \, dx &=\frac {\tan ^{-1}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {a^{3/4} \left (1+\frac {\sqrt {a-b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt {b}}\\ \end {align*}

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Mathematica [C]  time = 0.16, size = 162, normalized size = 1.31 \[ \frac {6 a x F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};\frac {b x^2}{a},\frac {b x^2}{2 a}\right )}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right ) \left (b x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};\frac {b x^2}{a},\frac {b x^2}{2 a}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};\frac {b x^2}{a},\frac {b x^2}{2 a}\right )\right )+6 a F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};\frac {b x^2}{a},\frac {b x^2}{2 a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(6*a*x*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, (b*x^2)/(2*a)])/((a - b*x^2)^(1/4)*(2*a - b*x^2)*(6*a*AppellF1[1/
2, 1/4, 1, 3/2, (b*x^2)/a, (b*x^2)/(2*a)] + b*x^2*(2*AppellF1[3/2, 1/4, 2, 5/2, (b*x^2)/a, (b*x^2)/(2*a)] + Ap
pellF1[3/2, 5/4, 1, 5/2, (b*x^2)/a, (b*x^2)/(2*a)])))

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fricas [B]  time = 60.66, size = 343, normalized size = 2.77 \[ \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (\sqrt {\frac {1}{2}} {\left (2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {-b x^{2} + a} a \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}}\right )} \sqrt {a b \sqrt {-\frac {1}{a^{3} b^{2}}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}}\right )}}{x}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} - {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {-b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - {\left (-b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (-b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} - 2 \, a}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/4)^(1/4)*(-1/(a^3*b^2))^(1/4)*arctan(2*(sqrt(1/2)*(2*(1/4)^(3/4)*a^3*b*(-1/(a^3*b^2))^(3/4) - (1/4)^(1/4)*s
qrt(-b*x^2 + a)*a*(-1/(a^3*b^2))^(1/4))*sqrt(a*b*sqrt(-1/(a^3*b^2))) - (1/4)^(1/4)*(-b*x^2 + a)^(1/4)*a*(-1/(a
^3*b^2))^(1/4))/x) + 1/4*(1/4)^(1/4)*(-1/(a^3*b^2))^(1/4)*log(-(2*(1/4)^(3/4)*sqrt(-b*x^2 + a)*a^2*b^2*x*(-1/(
a^3*b^2))^(3/4) + (-b*x^2 + a)^(1/4)*a^2*b*sqrt(-1/(a^3*b^2)) + (1/4)^(1/4)*a*b*x*(-1/(a^3*b^2))^(1/4) - (-b*x
^2 + a)^(3/4))/(b*x^2 - 2*a)) - 1/4*(1/4)^(1/4)*(-1/(a^3*b^2))^(1/4)*log((2*(1/4)^(3/4)*sqrt(-b*x^2 + a)*a^2*b
^2*x*(-1/(a^3*b^2))^(3/4) - (-b*x^2 + a)^(1/4)*a^2*b*sqrt(-1/(a^3*b^2)) + (1/4)^(1/4)*a*b*x*(-1/(a^3*b^2))^(1/
4) + (-b*x^2 + a)^(3/4))/(b*x^2 - 2*a))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (b x^{2} - 2 \, a\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{4}} \left (-b \,x^{2}+2 a \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a-b\,x^2\right )}^{1/4}\,\left (2\,a-b\,x^2\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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