3.25.55 \(\int \frac {b+a x^4}{(-b+a x^4) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx\)

Optimal. Leaf size=199 \[ \frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \tan ^{-1}\left (\frac {(1+i) x \sqrt [4]{2 a b+c} \sqrt [4]{a^2 x^8+b^2+c x^4}}{x^2 \sqrt {2 a b+c}-i \sqrt {a^2 x^8+b^2+c x^4}}\right )}{\sqrt [4]{2 a b+c}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \tanh ^{-1}\left (\frac {\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a^2 x^8+b^2+c x^4}}{\sqrt [4]{2 a b+c}}+\left (\frac {1}{2}-\frac {i}{2}\right ) x^2 \sqrt [4]{2 a b+c}}{x \sqrt [4]{a^2 x^8+b^2+c x^4}}\right )}{\sqrt [4]{2 a b+c}} \]

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Rubi [F]  time = 0.62, 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^4}{\left (-b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(x*(1 + (2*a^2*x^4)/(c - Sqrt[-4*a^2*b^2 + c^2]))^(1/4)*(1 + (2*a^2*x^4)/(c + Sqrt[-4*a^2*b^2 + c^2]))^(1/4)*A
ppellF1[1/4, 1/4, 1/4, 5/4, (-2*a^2*x^4)/(c - Sqrt[-4*a^2*b^2 + c^2]), (-2*a^2*x^4)/(c + Sqrt[-4*a^2*b^2 + c^2
])])/(b^2 + c*x^4 + a^2*x^8)^(1/4) + 2*b*Defer[Int][1/((-b + a*x^4)*(b^2 + c*x^4 + a^2*x^8)^(1/4)), x]

Rubi steps

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

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Mathematica [F]  time = 0.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+a x^4}{\left (-b+a x^4\right ) \sqrt [4]{b^2+c x^4+a^2 x^8}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(b + a*x^4)/((-b + a*x^4)*(b^2 + c*x^4 + a^2*x^8)^(1/4)),x]

[Out]

Integrate[(b + a*x^4)/((-b + a*x^4)*(b^2 + c*x^4 + a^2*x^8)^(1/4)), x]

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IntegrateAlgebraic [A]  time = 3.14, size = 207, normalized size = 1.04 \begin {gather*} -\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \tan ^{-1}\left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{2 a b+c} x^2-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b^2+c x^4+a^2 x^8}}{\sqrt [4]{2 a b+c}}}{x \sqrt [4]{b^2+c x^4+a^2 x^8}}\right )}{\sqrt [4]{2 a b+c}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \tanh ^{-1}\left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{2 a b+c} x^2+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b^2+c x^4+a^2 x^8}}{\sqrt [4]{2 a b+c}}}{x \sqrt [4]{b^2+c x^4+a^2 x^8}}\right )}{\sqrt [4]{2 a b+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b + a*x^4)/((-b + a*x^4)*(b^2 + c*x^4 + a^2*x^8)^(1/4)),x]

[Out]

((-1/4 - I/4)*ArcTan[((1/2 - I/2)*(2*a*b + c)^(1/4)*x^2 - ((1/2 + I/2)*Sqrt[b^2 + c*x^4 + a^2*x^8])/(2*a*b + c
)^(1/4))/(x*(b^2 + c*x^4 + a^2*x^8)^(1/4))])/(2*a*b + c)^(1/4) - ((1/4 + I/4)*ArcTanh[((1/2 - I/2)*(2*a*b + c)
^(1/4)*x^2 + ((1/2 + I/2)*Sqrt[b^2 + c*x^4 + a^2*x^8])/(2*a*b + c)^(1/4))/(x*(b^2 + c*x^4 + a^2*x^8)^(1/4))])/
(2*a*b + 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^4+b)/(a*x^4-b)/(a^2*x^8+c*x^4+b^2)^(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^{4} + b}{{\left (a^{2} x^{8} + c x^{4} + b^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4+b)/(a*x^4-b)/(a^2*x^8+c*x^4+b^2)^(1/4),x, algorithm="giac")

[Out]

integrate((a*x^4 + b)/((a^2*x^8 + c*x^4 + b^2)^(1/4)*(a*x^4 - b)), x)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}+b}{\left (a \,x^{4}-b \right ) \left (a^{2} x^{8}+c \,x^{4}+b^{2}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^4+b)/(a*x^4-b)/(a^2*x^8+c*x^4+b^2)^(1/4),x)

[Out]

int((a*x^4+b)/(a*x^4-b)/(a^2*x^8+c*x^4+b^2)^(1/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} + b}{{\left (a^{2} x^{8} + c x^{4} + b^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4+b)/(a*x^4-b)/(a^2*x^8+c*x^4+b^2)^(1/4),x, algorithm="maxima")

[Out]

integrate((a*x^4 + b)/((a^2*x^8 + c*x^4 + b^2)^(1/4)*(a*x^4 - b)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {a\,x^4+b}{\left (b-a\,x^4\right )\,{\left (a^2\,x^8+b^2+c\,x^4\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b + a*x^4)/((b - a*x^4)*(c*x^4 + b^2 + a^2*x^8)^(1/4)),x)

[Out]

int(-(b + a*x^4)/((b - a*x^4)*(c*x^4 + b^2 + a^2*x^8)^(1/4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} + b}{\left (a x^{4} - b\right ) \sqrt [4]{a^{2} x^{8} + b^{2} + c x^{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**4+b)/(a*x**4-b)/(a**2*x**8+c*x**4+b**2)**(1/4),x)

[Out]

Integral((a*x**4 + b)/((a*x**4 - b)*(a**2*x**8 + b**2 + c*x**4)**(1/4)), x)

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