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

Optimal. Leaf size=71 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^8-b+c x^4}}\right )}{2 \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a x^8-b+c x^4}}\right )}{2 \sqrt [4]{c}} \]

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Rubi [F]  time = 0.95, 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/((S
qrt[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\\ &=(2 b) \int \frac {1}{\left (-b+a x^8\right ) \sqrt [4]{-b+c x^4+a x^8}} \, dx+\int \frac {1}{\sqrt [4]{-b+c x^4+a x^8}} \, dx\\ &=(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+\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 = 0.62, 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 = 2.86, size = 71, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^8}}\right )}{2 \sqrt [4]{c}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{-b+c x^4+a x^8}}\right )}{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[(c^(1/4)*x)/(-b + c*x^4 + a*x^8)^(1/4)]/c^(1/4) - ArcTanh[(c^(1/4)*x)/(-b + c*x^4 + a*x^8)^(1/4)]/
(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 = 0.02, 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 {a\,x^8+b}{\left (b-a\,x^8\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)*(a*x^8 - b + c*x^4)^(1/4)),x)

[Out]

int(-(b + a*x^8)/((b - a*x^8)*(a*x^8 - b + 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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