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