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

Optimal. Leaf size=171 \[ \frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+a^2-a b\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))-2 a \log \left (\sqrt [4]{a x^4+b x^2}-\text {$\#$1} x\right )+2 a \log (x)}{\text {$\#$1} a-\text {$\#$1}^5}\& \right ]}{32 a (a-b)}+\frac {\left (a x^4+b x^2\right )^{3/4} \left (b-a x^2\right )}{4 a b x (a-b) \left (a x^4-b\right )} \]

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Rubi [F]  time = 0.26, 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 ) \operatorname {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 [B]  time = 1.15, size = 445, normalized size = 2.60 \begin {gather*} \frac {x \left (\frac {4 \sqrt [8]{a} \left (a^2 x^4-b^2\right )}{a x^4-b}+\frac {\sqrt {b} \sqrt [4]{a+\frac {b}{x^2}} \left (-\sqrt {a} \sqrt [4]{\sqrt {a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}\right )+\sqrt {b} \sqrt [4]{\sqrt {a}-\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}\right )+\sqrt {b} \sqrt [4]{\sqrt {a}+\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}\right )+\sqrt {a} \sqrt [4]{\sqrt {a}+\sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}\right )+\left (\sqrt {a}-\sqrt {b}\right )^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}+\sqrt {b}}}\right )-\left (\sqrt {a}+\sqrt {b}\right )^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [8]{a} \sqrt [4]{\sqrt {a}-\sqrt {b}}}\right )\right )}{\sqrt [4]{\sqrt {a}-\sqrt {b}} \sqrt [4]{\sqrt {a}+\sqrt {b}}}\right )}{16 a^{9/8} b (b-a) \sqrt [4]{x^2 \left (a x^2+b\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]

(x*((4*a^(1/8)*(-b^2 + a^2*x^4))/(-b + a*x^4) + (Sqrt[b]*(a + b/x^2)^(1/4)*(Sqrt[a]*(Sqrt[a] + Sqrt[b])^(1/4)*
ArcTan[(a + b/x^2)^(1/4)/(a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4))] + (Sqrt[a] + Sqrt[b])^(1/4)*Sqrt[b]*ArcTan[(a +
b/x^2)^(1/4)/(a^(1/8)*(Sqrt[a] - Sqrt[b])^(1/4))] - Sqrt[a]*(Sqrt[a] - Sqrt[b])^(1/4)*ArcTan[(a + b/x^2)^(1/4)
/(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4))] + (Sqrt[a] - Sqrt[b])^(1/4)*Sqrt[b]*ArcTan[(a + b/x^2)^(1/4)/(a^(1/8)*(S
qrt[a] + Sqrt[b])^(1/4))] - (Sqrt[a] + Sqrt[b])^(5/4)*ArcTanh[(a + b/x^2)^(1/4)/(a^(1/8)*(Sqrt[a] - Sqrt[b])^(
1/4))] + (Sqrt[a] - Sqrt[b])^(5/4)*ArcTanh[(a + b/x^2)^(1/4)/(a^(1/8)*(Sqrt[a] + Sqrt[b])^(1/4))]))/((Sqrt[a]
- Sqrt[b])^(1/4)*(Sqrt[a] + Sqrt[b])^(1/4))))/(16*a^(9/8)*b*(-a + b)*(x^2*(b + a*x^2))^(1/4))

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IntegrateAlgebraic [A]  time = 0.00, size = 170, normalized size = 0.99 \begin {gather*} \frac {\left (b-a x^2\right ) \left (b x^2+a x^4\right )^{3/4}}{4 a (a-b) 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)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b - a*x^2)*(b*x^2 + a*x^4)^(3/4))/(4*a*(a - b)*b*x*(-b + a*x^4)) + RootSum[a^2 - a*b - 2*a*#1^4 + #1^8 & , (
-2*a*Log[x] + 2*a*Log[(b*x^2 + a*x^4)^(1/4) - x*#1] + Log[x]*#1^4 - Log[(b*x^2 + a*x^4)^(1/4) - x*#1]*#1^4)/(-
(a*#1) + #1^5) & ]/(32*a*(a - b))

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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(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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}^{2}}\,{d x} \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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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*} \int \frac {x^{4}}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} - b\right )}^{2}}\,{d x} \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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{{\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(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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sympy [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(x**4/(a*x**4-b)**2/(a*x**4+b*x**2)**(1/4),x)

[Out]

Timed out

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