3.28.29 \(\int \frac {1}{(-b+a x^4)^2 \sqrt [4]{-b x^2+a x^4}} \, dx\) [2729]

Optimal. Leaf size=251 \[ \frac {\left (-b-a x^2\right ) \left (-b x^2+a x^4\right )^{3/4}}{4 b^2 (-a+b) x \left (b-a x^4\right )}-\frac {\text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {-8 a^2 \log (x)+6 a b \log (x)+8 a^2 \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )-6 a b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right )+8 a \log (x) \text {$\#$1}^4-7 b \log (x) \text {$\#$1}^4-8 a \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+7 b \log \left (\sqrt [4]{-b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-\text {$\#$1}^5}\& \right ]}{32 (a-b) b^2} \]

[Out]

Unintegrable

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Rubi [F]
time = 0.08, 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 {1}{\left (-b+a x^4\right )^2 \sqrt [4]{-b x^2+a x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.74, size = 268, normalized size = 1.07 \begin {gather*} \frac {\sqrt {x} \left (-8 \sqrt [4]{-b+a x^2} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]+\frac {\frac {16 \sqrt {x} \left (b^2-a^2 x^4\right )}{-b+a x^4}+b \sqrt [4]{-b+a x^2} \text {RootSum}\left [a^2-a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a \log (x)-4 a \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+2 \log \left (\sqrt [4]{-b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-\text {$\#$1}^5}\&\right ]}{2 (a-b)}\right )}{32 b^2 \sqrt [4]{-b x^2+a x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(-8*(-b + a*x^2)^(1/4)*RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (-Log[Sqrt[x]] + Log[(-b + a*x^2)^(1/4
) - Sqrt[x]*#1])/#1 & ] + ((16*Sqrt[x]*(b^2 - a^2*x^4))/(-b + a*x^4) + b*(-b + a*x^2)^(1/4)*RootSum[a^2 - a*b
- 2*a*#1^4 + #1^8 & , (2*a*Log[x] - 4*a*Log[(-b + a*x^2)^(1/4) - Sqrt[x]*#1] - Log[x]*#1^4 + 2*Log[(-b + a*x^2
)^(1/4) - Sqrt[x]*#1]*#1^4)/(a*#1 - #1^5) & ])/(2*(a - b))))/(32*b^2*(-(b*x^2) + a*x^4)^(1/4))

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a \,x^{4}-b \right )^{2} \left (a \,x^{4}-b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
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(1/(a*x^4-b)^2/(a*x^4-b*x^2)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{2} \left (a x^{2} - b\right )} \left (a x^{4} - b\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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