3.23.19 \(\int \frac {x^4}{(b+a x^4)^2 \sqrt [4]{b x^2+a x^4}} \, dx\) [2219]

Optimal. Leaf size=165 \[ \frac {\left (-b+a x^2\right ) \left (b x^2+a x^4\right )^{3/4}}{4 a b (a+b) x \left (b+a x^4\right )}+\frac {\text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\& ,\frac {2 a \log (x)-2 a \log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-\text {$\#$1}^5}\& \right ]}{32 a (a+b)} \]

[Out]

Unintegrable

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Rubi [F]
time = 0.15, 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 {x^4}{\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[x^4/((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][x^8/((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 {x^4}{\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 {x^{7/2}}{\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 {x^8}{\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 = 0.00, size = 184, normalized size = 1.12 \begin {gather*} \frac {-8 b^2 x+8 a^2 x^5+\frac {1}{2} b \sqrt {x} \sqrt [4]{b+a x^2} \left (b+a x^4\right ) \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 ]}{32 a b (a+b) \sqrt [4]{x^2 \left (b+a x^2\right )} \left (b+a x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*b^2*x + 8*a^2*x^5 + (b*Sqrt[x]*(b + a*x^2)^(1/4)*(b + a*x^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)/(32*a*b*(a + b)*(x^2*(b + a*x^2))^(1/4)*(b + a*x^4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

Timed out

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

[Out]

integrate(x^4/((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 = -0.01 \begin {gather*} \int \frac {x^4}{{\left (a\,x^4+b\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(x^4/((b + a*x^4)^2*(a*x^4 + b*x^2)^(1/4)),x)

[Out]

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

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