Optimal. Leaf size=457 \[ -\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {x \left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{a} \sqrt [8]{b}-\frac {2 \sqrt [8]{a} \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt {p x^4+q}}{-\sqrt [4]{a} p x^4-\sqrt [4]{a} q+\sqrt [4]{b} x^2}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {p x^4+q}}{\sqrt [4]{a} p x^4+\sqrt [4]{a} q-\sqrt [4]{b} x^2}\right )}{8 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt [8]{a} p x^4}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{a} q}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b} x^2}{\sqrt {2-\sqrt {2}} \sqrt [8]{a}}}{x \sqrt {p x^4+q}}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt [8]{a} p x^4}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{a} q}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}+\frac {\sqrt [8]{b} x^2}{\sqrt {2+\sqrt {2}} \sqrt [8]{a}}}{x \sqrt {p x^4+q}}\right )}{8 a^{3/8} b^{5/8}} \]
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Rubi [F] time = 2.62, 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 (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x^4 \left (-q+p x^4\right ) \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx &=\int \left (\frac {q x^4 \sqrt {q+p x^4}}{-a q^4-4 a p q^3 x^4-b \left (1+\frac {6 a p^2 q^2}{b}\right ) x^8-4 a p^3 q x^{12}-a p^4 x^{16}}+\frac {p x^8 \sqrt {q+p x^4}}{a q^4+4 a p q^3 x^4+b \left (1+\frac {6 a p^2 q^2}{b}\right ) x^8+4 a p^3 q x^{12}+a p^4 x^{16}}\right ) \, dx\\ &=p \int \frac {x^8 \sqrt {q+p x^4}}{a q^4+4 a p q^3 x^4+b \left (1+\frac {6 a p^2 q^2}{b}\right ) x^8+4 a p^3 q x^{12}+a p^4 x^{16}} \, dx+q \int \frac {x^4 \sqrt {q+p x^4}}{-a q^4-4 a p q^3 x^4-b \left (1+\frac {6 a p^2 q^2}{b}\right ) x^8-4 a p^3 q x^{12}-a p^4 x^{16}} \, dx\\ &=p \int \frac {x^8 \sqrt {q+p x^4}}{b x^8+a \left (q+p x^4\right )^4} \, dx+q \int \frac {x^4 \sqrt {q+p x^4}}{-b x^8-a \left (q+p x^4\right )^4} \, dx\\ \end {align*}
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Mathematica [C] time = 7.80, size = 15065, normalized size = 32.96 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 24.94, size = 425, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {2+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^4}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^4}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{b} x \sqrt {q+p x^4}}{\sqrt [4]{a} q-\sqrt [4]{b} x^2+\sqrt [4]{a} p x^4}\right )}{8 a^{3/8} b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt [8]{a} q}{\sqrt [8]{b}}+\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt [8]{b} x^2}{\sqrt [8]{a}}+\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \sqrt [8]{a} p x^4}{\sqrt [8]{b}}}{x \sqrt {q+p x^4}}\right )}{8 a^{3/8} b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} \tanh ^{-1}\left (\frac {\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt [8]{a} q}{\sqrt [8]{b}}+\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt [8]{b} x^2}{\sqrt [8]{a}}+\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \sqrt [8]{a} p x^4}{\sqrt [8]{b}}}{x \sqrt {q+p x^4}}\right )}{8 a^{3/8} b^{5/8}} \end {gather*}
Antiderivative was successfully verified.
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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.
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giac [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.
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maple [C] time = 0.21, size = 47, normalized size = 0.10
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (16 \textit {\_Z}^{8} a +b \right )}{\sum }\frac {\ln \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{2 x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right ) \sqrt {2}}{64 a}\) | \(47\) |
elliptic | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (16 \textit {\_Z}^{8} a +b \right )}{\sum }\frac {\ln \left (\frac {\sqrt {p \,x^{4}+q}\, \sqrt {2}}{2 x}-\textit {\_R} \right )}{\textit {\_R}^{5}}\right ) \sqrt {2}}{64 a}\) | \(47\) |
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*} \int \frac {\sqrt {p x^{4} + q} {\left (p x^{4} - q\right )} x^{4}}{b x^{8} + {\left (p x^{4} + q\right )}^{4} a}\,{d x} \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 = -0.00 \begin {gather*} -\int \frac {x^4\,\sqrt {p\,x^4+q}\,\left (q-p\,x^4\right )}{a\,{\left (p\,x^4+q\right )}^4+b\,x^8} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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