Optimal. Leaf size=524 \[ \frac {\left (\sqrt {2} b \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \left (-\sqrt {p}\right ) \sqrt {q}+b}-i \sqrt {2} \sqrt {b} \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \left (-\sqrt {p}\right ) \sqrt {q}+b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}\right ) \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \left (-\sqrt {p}\right ) \sqrt {q}+b}}{\sqrt {a} \left (\sqrt {p x^4+q}+\sqrt {p} x^2+\sqrt {q}\right )}\right )}{2 a^{5/2} \sqrt {p} \sqrt {q}}-\frac {i \left (\sqrt {2} \sqrt {b} \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \sqrt {p} \sqrt {q}-b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}-i \sqrt {2} b \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \sqrt {p} \sqrt {q}-b}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {i \sqrt {b} \sqrt {2 a \sqrt {p} \sqrt {q}-b}+a \sqrt {p} \sqrt {q}-b}}{\sqrt {a} \left (\sqrt {p x^4+q}+\sqrt {p} x^2+\sqrt {q}\right )}\right )}{2 a^{5/2} \sqrt {p} \sqrt {q}}+\frac {\sqrt {p x^4+q}}{a x} \]
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Rubi [C] time = 3.47, antiderivative size = 1029, normalized size of antiderivative = 1.96, number of steps used = 22, number of rules used = 9, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {6728, 277, 305, 220, 1196, 1209, 1198, 1217, 1707} \begin {gather*} \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a} \sqrt {p x^4+q}}\right )}{a^{3/2}}-\frac {\left (-2 \sqrt {p} \sqrt {q} a+b-\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \sqrt {p x^4+q}}-\frac {\left (-2 \sqrt {p} \sqrt {q} a+b+\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \sqrt {p x^4+q}}-\frac {\sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{a \sqrt {p x^4+q}}+\frac {b \left (b-\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{2 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (-2 \sqrt {p} \sqrt {q} a+b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p x^4+q}}+\frac {b \left (b+\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{2 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (-2 \sqrt {p} \sqrt {q} a+b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p x^4+q}}-\frac {b \left (2 \sqrt {p} \sqrt {q} a+b-\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} \Pi \left (\frac {1}{4} \left (2-\frac {b}{a \sqrt {p} \sqrt {q}}\right );2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (-2 \sqrt {p} \sqrt {q} a+b-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p x^4+q}}-\frac {b \left (2 \sqrt {p} \sqrt {q} a+b+\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {p} x^2+\sqrt {q}\right ) \sqrt {\frac {p x^4+q}{\left (\sqrt {p} x^2+\sqrt {q}\right )^2}} \Pi \left (\frac {1}{4} \left (2-\frac {b}{a \sqrt {p} \sqrt {q}}\right );2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (-2 \sqrt {p} \sqrt {q} a+b+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {p x^4+q}}+\frac {\sqrt {p x^4+q}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 220
Rule 277
Rule 305
Rule 1196
Rule 1198
Rule 1209
Rule 1217
Rule 1707
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-q+p x^4\right ) \sqrt {q+p x^4}}{x^2 \left (a q+b x^2+a p x^4\right )} \, dx &=\int \left (-\frac {\sqrt {q+p x^4}}{a x^2}+\frac {\left (b+2 a p x^2\right ) \sqrt {q+p x^4}}{a \left (a q+b x^2+a p x^4\right )}\right ) \, dx\\ &=-\frac {\int \frac {\sqrt {q+p x^4}}{x^2} \, dx}{a}+\frac {\int \frac {\left (b+2 a p x^2\right ) \sqrt {q+p x^4}}{a q+b x^2+a p x^4} \, dx}{a}\\ &=\frac {\sqrt {q+p x^4}}{a x}+\frac {\int \left (\frac {2 a p \sqrt {q+p x^4}}{b-\sqrt {b^2-4 a^2 p q}+2 a p x^2}+\frac {2 a p \sqrt {q+p x^4}}{b+\sqrt {b^2-4 a^2 p q}+2 a p x^2}\right ) \, dx}{a}-\frac {(2 p) \int \frac {x^2}{\sqrt {q+p x^4}} \, dx}{a}\\ &=\frac {\sqrt {q+p x^4}}{a x}+(2 p) \int \frac {\sqrt {q+p x^4}}{b-\sqrt {b^2-4 a^2 p q}+2 a p x^2} \, dx+(2 p) \int \frac {\sqrt {q+p x^4}}{b+\sqrt {b^2-4 a^2 p q}+2 a p x^2} \, dx-\frac {\left (2 \sqrt {p} \sqrt {q}\right ) \int \frac {1}{\sqrt {q+p x^4}} \, dx}{a}+\frac {\left (2 \sqrt {p} \sqrt {q}\right ) \int \frac {1-\frac {\sqrt {p} x^2}{\sqrt {q}}}{\sqrt {q+p x^4}} \, dx}{a}\\ &=\frac {\sqrt {q+p x^4}}{a x}-\frac {2 \sqrt {p} x \sqrt {q+p x^4}}{a \left (\sqrt {q}+\sqrt {p} x^2\right )}+\frac {2 \sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{a \sqrt {q+p x^4}}-\frac {\sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{a \sqrt {q+p x^4}}-\frac {\int \frac {p \left (b-\sqrt {b^2-4 a^2 p q}\right )-2 a p^2 x^2}{\sqrt {q+p x^4}} \, dx}{2 a^2 p}-\frac {\int \frac {p \left (b+\sqrt {b^2-4 a^2 p q}\right )-2 a p^2 x^2}{\sqrt {q+p x^4}} \, dx}{2 a^2 p}+\frac {\left (b \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {1}{\left (b-\sqrt {b^2-4 a^2 p q}+2 a p x^2\right ) \sqrt {q+p x^4}} \, dx}{a^2}+\frac {\left (b \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {1}{\left (b+\sqrt {b^2-4 a^2 p q}+2 a p x^2\right ) \sqrt {q+p x^4}} \, dx}{a^2}\\ &=\frac {\sqrt {q+p x^4}}{a x}-\frac {2 \sqrt {p} x \sqrt {q+p x^4}}{a \left (\sqrt {q}+\sqrt {p} x^2\right )}+\frac {2 \sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{a \sqrt {q+p x^4}}-\frac {\sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{a \sqrt {q+p x^4}}-2 \frac {\left (\sqrt {p} \sqrt {q}\right ) \int \frac {1-\frac {\sqrt {p} x^2}{\sqrt {q}}}{\sqrt {q+p x^4}} \, dx}{a}+\frac {\left (b \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {1}{\sqrt {q+p x^4}} \, dx}{a^2 \left (b-2 a \sqrt {p} \sqrt {q}-\sqrt {b^2-4 a^2 p q}\right )}-\frac {\left (2 b \sqrt {p} \sqrt {q} \left (b-\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {1+\frac {\sqrt {p} x^2}{\sqrt {q}}}{\left (b-\sqrt {b^2-4 a^2 p q}+2 a p x^2\right ) \sqrt {q+p x^4}} \, dx}{a \left (b-2 a \sqrt {p} \sqrt {q}-\sqrt {b^2-4 a^2 p q}\right )}-\frac {\left (b-2 a \sqrt {p} \sqrt {q}-\sqrt {b^2-4 a^2 p q}\right ) \int \frac {1}{\sqrt {q+p x^4}} \, dx}{2 a^2}+\frac {\left (b \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {1}{\sqrt {q+p x^4}} \, dx}{a^2 \left (b-2 a \sqrt {p} \sqrt {q}+\sqrt {b^2-4 a^2 p q}\right )}-\frac {\left (2 b \sqrt {p} \sqrt {q} \left (b+\sqrt {b^2-4 a^2 p q}\right )\right ) \int \frac {1+\frac {\sqrt {p} x^2}{\sqrt {q}}}{\left (b+\sqrt {b^2-4 a^2 p q}+2 a p x^2\right ) \sqrt {q+p x^4}} \, dx}{a \left (b-2 a \sqrt {p} \sqrt {q}+\sqrt {b^2-4 a^2 p q}\right )}-\frac {\left (b-2 a \sqrt {p} \sqrt {q}+\sqrt {b^2-4 a^2 p q}\right ) \int \frac {1}{\sqrt {q+p x^4}} \, dx}{2 a^2}\\ &=\frac {\sqrt {q+p x^4}}{a x}-\frac {2 \sqrt {p} x \sqrt {q+p x^4}}{a \left (\sqrt {q}+\sqrt {p} x^2\right )}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a} \sqrt {q+p x^4}}\right )}{a^{3/2}}+\frac {2 \sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{a \sqrt {q+p x^4}}-2 \left (-\frac {\sqrt {p} x \sqrt {q+p x^4}}{a \left (\sqrt {q}+\sqrt {p} x^2\right )}+\frac {\sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{a \sqrt {q+p x^4}}\right )-\frac {\sqrt [4]{p} \sqrt [4]{q} \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{a \sqrt {q+p x^4}}+\frac {b \left (b-\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{2 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (b-2 a \sqrt {p} \sqrt {q}-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q+p x^4}}-\frac {\left (b-2 a \sqrt {p} \sqrt {q}-\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \sqrt {q+p x^4}}+\frac {b \left (b+\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{2 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (b-2 a \sqrt {p} \sqrt {q}+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q+p x^4}}-\frac {\left (b-2 a \sqrt {p} \sqrt {q}+\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \sqrt {q+p x^4}}-\frac {b \left (b+2 a \sqrt {p} \sqrt {q}-\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2-\frac {b}{a \sqrt {p} \sqrt {q}}\right );2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (b-2 a \sqrt {p} \sqrt {q}-\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q+p x^4}}-\frac {b \left (b+2 a \sqrt {p} \sqrt {q}+\sqrt {b^2-4 a^2 p q}\right ) \left (\sqrt {q}+\sqrt {p} x^2\right ) \sqrt {\frac {q+p x^4}{\left (\sqrt {q}+\sqrt {p} x^2\right )^2}} \Pi \left (\frac {1}{4} \left (2-\frac {b}{a \sqrt {p} \sqrt {q}}\right );2 \tan ^{-1}\left (\frac {\sqrt [4]{p} x}{\sqrt [4]{q}}\right )|\frac {1}{2}\right )}{4 a^2 \sqrt [4]{p} \sqrt [4]{q} \left (b-2 a \sqrt {p} \sqrt {q}+\sqrt {b^2-4 a^2 p q}\right ) \sqrt {q+p x^4}}\\ \end {align*}
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Mathematica [C] time = 1.51, size = 293, normalized size = 0.56 \begin {gather*} \frac {-i b x \sqrt {\frac {p x^4}{q}+1} \Pi \left (\frac {2 i a \sqrt {p} \sqrt {q}}{\sqrt {b^2-4 a^2 p q}-b};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} x\right )\right |-1\right )-i b x \sqrt {\frac {p x^4}{q}+1} \Pi \left (-\frac {2 i a \sqrt {p} \sqrt {q}}{b+\sqrt {b^2-4 a^2 p q}};\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} x\right )\right |-1\right )+a p x^4 \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}+a q \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}+i b x \sqrt {\frac {p x^4}{q}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} x\right )\right |-1\right )}{a^2 x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}} \sqrt {p x^4+q}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.92, size = 54, normalized size = 0.10 \begin {gather*} \frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a} \sqrt {p x^4+q}}\right )}{a^{3/2}}+\frac {\sqrt {p x^4+q}}{a x} \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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {p x^{4} + q} {\left (p x^{4} - q\right )}}{{\left (a p x^{4} + b x^{2} + a q\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 528, normalized size = 1.01 \begin {gather*} -\frac {-\frac {\sqrt {p \,x^{4}+q}}{x}+\frac {2 i \sqrt {p}\, \sqrt {q}\, \sqrt {1-\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \sqrt {1+\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}\, \sqrt {p \,x^{4}+q}}}{a}+\frac {-\frac {b \sqrt {1-\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \sqrt {1+\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, i\right )}{a \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}\, \sqrt {p \,x^{4}+q}}+\frac {2 i \sqrt {p}\, \sqrt {q}\, \sqrt {1-\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \sqrt {1+\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \left (\EllipticF \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, i\right )-\EllipticE \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}\, \sqrt {p \,x^{4}+q}}+\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a p \,\textit {\_Z}^{4}+b \,\textit {\_Z}^{2}+a q \right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} b +2 a q \right ) \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (-\underline {\hspace {1.25 ex}}\alpha ^{2} a p +a p \,x^{2}-b \right )}{a \sqrt {-\frac {b \,\underline {\hspace {1.25 ex}}\alpha ^{2}}{a}}\, \sqrt {p \,x^{4}+q}}\right )}{\sqrt {-\frac {b \,\underline {\hspace {1.25 ex}}\alpha ^{2}}{a}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a p +b \right ) \sqrt {1-\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \sqrt {1+\frac {i \sqrt {p}\, x^{2}}{\sqrt {q}}}\, \EllipticPi \left (x \sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}, \frac {i \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a p +b \right )}{\sqrt {q}\, \sqrt {p}\, a}, \frac {\sqrt {-\frac {i \sqrt {p}}{\sqrt {q}}}}{\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}}\right )}{\sqrt {\frac {i \sqrt {p}}{\sqrt {q}}}\, a q \sqrt {p \,x^{4}+q}}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (2 \underline {\hspace {1.25 ex}}\alpha ^{2} a p +b \right )}\right )}{4 a}}{a} \end {gather*}
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 )}}{{\left (a p x^{4} + b x^{2} + a q\right )} x^{2}}\,{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 {\sqrt {p\,x^4+q}\,\left (q-p\,x^4\right )}{x^2\,\left (a\,p\,x^4+b\,x^2+a\,q\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (p x^{4} - q\right ) \sqrt {p x^{4} + q}}{x^{2} \left (a p x^{4} + a q + b x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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