3.22.30 \(\int \frac {-b+a x^8}{(b+a x^8) \sqrt [4]{b-c x^4+a x^8}} \, dx\)

Optimal. Leaf size=154 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a x^8+b-c x^4}}{\sqrt {a x^8+b-c x^4}-\sqrt {c} x^2}\right )}{2 \sqrt {2} \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt {a x^8+b-c x^4}}{\sqrt {2} \sqrt [4]{c}}+\frac {\sqrt [4]{c} x^2}{\sqrt {2}}}{x \sqrt [4]{a x^8+b-c x^4}}\right )}{2 \sqrt {2} \sqrt [4]{c}} \]

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Rubi [F]  time = 0.73, 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 {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-b + a*x^8)/((b + a*x^8)*(b - c*x^4 + a*x^8)^(1/4)),x]

[Out]

(x*(1 - (2*a*x^4)/(c - Sqrt[-4*a*b + c^2]))^(1/4)*(1 - (2*a*x^4)/(c + Sqrt[-4*a*b + c^2]))^(1/4)*AppellF1[1/4,
1/4, 1/4, 5/4, (2*a*x^4)/(c + Sqrt[-4*a*b + c^2]), (2*a*x^4)/(c - Sqrt[-4*a*b + c^2])])/(b - c*x^4 + a*x^8)^(
1/4) - Sqrt[b]*Defer[Int][1/((Sqrt[b] - Sqrt[-a]*x^4)*(b - c*x^4 + a*x^8)^(1/4)), x] - Sqrt[b]*Defer[Int][1/((
Sqrt[b] + Sqrt[-a]*x^4)*(b - c*x^4 + a*x^8)^(1/4)), x]

Rubi steps

\begin {align*} \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx &=\int \left (\frac {1}{\sqrt [4]{b-c x^4+a x^8}}-\frac {2 b}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}}\right ) \, dx\\ &=-\left ((2 b) \int \frac {1}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx\right )+\int \frac {1}{\sqrt [4]{b-c x^4+a x^8}} \, dx\\ &=-\left ((2 b) \int \left (\frac {1}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^4\right ) \sqrt [4]{b-c x^4+a x^8}}+\frac {1}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^4\right ) \sqrt [4]{b-c x^4+a x^8}}\right ) \, dx\right )+\frac {\left (\sqrt [4]{1+\frac {2 a x^4}{-c-\sqrt {-4 a b+c^2}}} \sqrt [4]{1+\frac {2 a x^4}{-c+\sqrt {-4 a b+c^2}}}\right ) \int \frac {1}{\sqrt [4]{1+\frac {2 a x^4}{-c-\sqrt {-4 a b+c^2}}} \sqrt [4]{1+\frac {2 a x^4}{-c+\sqrt {-4 a b+c^2}}}} \, dx}{\sqrt [4]{b-c x^4+a x^8}}\\ &=\frac {x \sqrt [4]{1-\frac {2 a x^4}{c-\sqrt {-4 a b+c^2}}} \sqrt [4]{1-\frac {2 a x^4}{c+\sqrt {-4 a b+c^2}}} F_1\left (\frac {1}{4};\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {2 a x^4}{c+\sqrt {-4 a b+c^2}},\frac {2 a x^4}{c-\sqrt {-4 a b+c^2}}\right )}{\sqrt [4]{b-c x^4+a x^8}}-\sqrt {b} \int \frac {1}{\left (\sqrt {b}-\sqrt {-a} x^4\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx-\sqrt {b} \int \frac {1}{\left (\sqrt {b}+\sqrt {-a} x^4\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.66, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b+a x^8}{\left (b+a x^8\right ) \sqrt [4]{b-c x^4+a x^8}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-b + a*x^8)/((b + a*x^8)*(b - c*x^4 + a*x^8)^(1/4)),x]

[Out]

Integrate[(-b + a*x^8)/((b + a*x^8)*(b - c*x^4 + a*x^8)^(1/4)), x]

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IntegrateAlgebraic [A]  time = 3.09, size = 154, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{b-c x^4+a x^8}}{-\sqrt {c} x^2+\sqrt {b-c x^4+a x^8}}\right )}{2 \sqrt {2} \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\frac {\sqrt [4]{c} x^2}{\sqrt {2}}+\frac {\sqrt {b-c x^4+a x^8}}{\sqrt {2} \sqrt [4]{c}}}{x \sqrt [4]{b-c x^4+a x^8}}\right )}{2 \sqrt {2} \sqrt [4]{c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-b + a*x^8)/((b + a*x^8)*(b - c*x^4 + a*x^8)^(1/4)),x]

[Out]

-1/2*ArcTan[(Sqrt[2]*c^(1/4)*x*(b - c*x^4 + a*x^8)^(1/4))/(-(Sqrt[c]*x^2) + Sqrt[b - c*x^4 + a*x^8])]/(Sqrt[2]
*c^(1/4)) - ArcTanh[((c^(1/4)*x^2)/Sqrt[2] + Sqrt[b - c*x^4 + a*x^8]/(Sqrt[2]*c^(1/4)))/(x*(b - c*x^4 + a*x^8)
^(1/4))]/(2*Sqrt[2]*c^(1/4))

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fricas [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((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{8} - b}{{\left (a x^{8} - c x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{8} + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x, algorithm="giac")

[Out]

integrate((a*x^8 - b)/((a*x^8 - c*x^4 + b)^(1/4)*(a*x^8 + b)), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{8}-b}{\left (a \,x^{8}+b \right ) \left (a \,x^{8}-c \,x^{4}+b \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x)

[Out]

int((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{8} - b}{{\left (a x^{8} - c x^{4} + b\right )}^{\frac {1}{4}} {\left (a x^{8} + b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^8-b)/(a*x^8+b)/(a*x^8-c*x^4+b)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^8 - b)/((a*x^8 - c*x^4 + b)^(1/4)*(a*x^8 + b)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {b-a\,x^8}{\left (a\,x^8+b\right )\,{\left (a\,x^8-c\,x^4+b\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b - a*x^8)/((b + a*x^8)*(b + a*x^8 - c*x^4)^(1/4)),x)

[Out]

int(-(b - a*x^8)/((b + a*x^8)*(b + a*x^8 - c*x^4)^(1/4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{8} - b}{\left (a x^{8} + b\right ) \sqrt [4]{a x^{8} + b - c x^{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**8-b)/(a*x**8+b)/(a*x**8-c*x**4+b)**(1/4),x)

[Out]

Integral((a*x**8 - b)/((a*x**8 + b)*(a*x**8 + b - c*x**4)**(1/4)), x)

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