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

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

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Rubi [F]  time = 0.25, 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 = 0.91, size = 486, normalized size = 2.95 \begin {gather*} \frac {x \left (\frac {4 \left (a^2 x^4-b^2\right )}{a x^4+b}-\frac {\sqrt {b} \sqrt [4]{a+\frac {b}{x^2}} \left (\sqrt {b} \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )-\sqrt {-a} \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )+\sqrt {b} \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )+\sqrt {-a} \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \tan ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )+\left (\sqrt {-a}-\sqrt {b}\right ) \sqrt [4]{\sqrt {-a} \sqrt {b}+a} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )-\left (\sqrt {-a}+\sqrt {b}\right ) \sqrt [4]{a-\sqrt {-a} \sqrt {b}} \tanh ^{-1}\left (\frac {\sqrt [4]{a+\frac {b}{x^2}}}{\sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )\right )}{\sqrt [4]{a-\sqrt {-a} \sqrt {b}} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )}{16 a b (a+b) \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*(-b^2 + a^2*x^4))/(b + a*x^4) - (Sqrt[b]*(a + b/x^2)^(1/4)*(-(Sqrt[-a]*(a + Sqrt[-a]*Sqrt[b])^(1/4)*Arc
Tan[(a + b/x^2)^(1/4)/(a - Sqrt[-a]*Sqrt[b])^(1/4)]) + (a + Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[b]*ArcTan[(a + b/x^2)
^(1/4)/(a - Sqrt[-a]*Sqrt[b])^(1/4)] + Sqrt[-a]*(a - Sqrt[-a]*Sqrt[b])^(1/4)*ArcTan[(a + b/x^2)^(1/4)/(a + Sqr
t[-a]*Sqrt[b])^(1/4)] + (a - Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[b]*ArcTan[(a + b/x^2)^(1/4)/(a + Sqrt[-a]*Sqrt[b])^(
1/4)] + (Sqrt[-a] - Sqrt[b])*(a + Sqrt[-a]*Sqrt[b])^(1/4)*ArcTanh[(a + b/x^2)^(1/4)/(a - Sqrt[-a]*Sqrt[b])^(1/
4)] - (Sqrt[-a] + Sqrt[b])*(a - Sqrt[-a]*Sqrt[b])^(1/4)*ArcTanh[(a + b/x^2)^(1/4)/(a + Sqrt[-a]*Sqrt[b])^(1/4)
]))/((a - Sqrt[-a]*Sqrt[b])^(1/4)*(a + Sqrt[-a]*Sqrt[b])^(1/4))))/(16*a*b*(a + b)*(x^2*(b + a*x^2))^(1/4))

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IntegrateAlgebraic [A]  time = 0.00, size = 164, 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 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)} \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*b*(a + 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 (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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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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