Optimal. Leaf size=804 \[ \frac{\sqrt{d x^4+c} x}{b \sqrt{d} \left (\sqrt{d} x^2+\sqrt{c}\right )}-\frac{a \sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{-\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 b (b c-a d)}-\frac{a \sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} \tan ^{-1}\left (\frac{\sqrt{\frac{b c-a d}{\sqrt{-a} \sqrt{b}}} x}{\sqrt{d x^4+c}}\right )}{4 b (b c-a d)}-\frac{\sqrt [4]{c} \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{b d^{3/4} \sqrt{d x^4+c}}+\frac{\sqrt [4]{c} (b c+2 a d) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac{1}{2}\right )}{2 b d^{3/4} (b c+a d) \sqrt{d x^4+c}}+\frac{a \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \left (\sqrt{-a} \sqrt{b} \sqrt{c}-a \sqrt{d}\right ) \sqrt [4]{d} \sqrt{d x^4+c}}-\frac{a \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} \Pi \left (-\frac{\sqrt{c} \left (\sqrt{b}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{c}}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \left (\sqrt{d} a+\sqrt{-a} \sqrt{b} \sqrt{c}\right ) \sqrt [4]{d} \sqrt{d x^4+c}} \]
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Rubi [A] time = 0.962957, antiderivative size = 982, normalized size of antiderivative = 1.22, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {483, 305, 220, 1196, 490, 1217, 1707} \[ -\frac{\sqrt{-a} \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right ) \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{8 b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{d x^4+c}}+\frac{(-a)^{3/4} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^4+c}}\right )}{4 b^{5/4} \sqrt{b c-a d}}-\frac{(-a)^{3/4} \tan ^{-1}\left (\frac{\sqrt{a d-b c} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^4+c}}\right )}{4 b^{5/4} \sqrt{a d-b c}}-\frac{\sqrt [4]{c} \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{b d^{3/4} \sqrt{d x^4+c}}+\frac{\sqrt [4]{c} \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{2 b d^{3/4} \sqrt{d x^4+c}}+\frac{a \left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt [4]{d} \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 b \sqrt [4]{c} (b c+a d) \sqrt{d x^4+c}}+\frac{a \left (\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt [4]{d} \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 b \sqrt [4]{c} (b c+a d) \sqrt{d x^4+c}}+\frac{\sqrt{-a} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 \left (\sqrt{d} x^2+\sqrt{c}\right ) \sqrt{\frac{d x^4+c}{\left (\sqrt{d} x^2+\sqrt{c}\right )^2}} \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{d x^4+c}}+\frac{x \sqrt{d x^4+c}}{b \sqrt{d} \left (\sqrt{d} x^2+\sqrt{c}\right )} \]
Warning: Unable to verify antiderivative.
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Rule 483
Rule 305
Rule 220
Rule 1196
Rule 490
Rule 1217
Rule 1707
Rubi steps
\begin{align*} \int \frac{x^6}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx &=\frac{\int \frac{x^2}{\sqrt{c+d x^4}} \, dx}{b}-\frac{a \int \frac{x^2}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx}{b}\\ &=\frac{a \int \frac{1}{\left (\sqrt{-a}-\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx}{2 b^{3/2}}-\frac{a \int \frac{1}{\left (\sqrt{-a}+\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx}{2 b^{3/2}}+\frac{\sqrt{c} \int \frac{1}{\sqrt{c+d x^4}} \, dx}{b \sqrt{d}}-\frac{\sqrt{c} \int \frac{1-\frac{\sqrt{d} x^2}{\sqrt{c}}}{\sqrt{c+d x^4}} \, dx}{b \sqrt{d}}\\ &=\frac{x \sqrt{c+d x^4}}{b \sqrt{d} \left (\sqrt{c}+\sqrt{d} x^2\right )}-\frac{\sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{b d^{3/4} \sqrt{c+d x^4}}+\frac{\sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{2 b d^{3/4} \sqrt{c+d x^4}}+\frac{\left (a \sqrt{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )\right ) \int \frac{1+\frac{\sqrt{d} x^2}{\sqrt{c}}}{\left (\sqrt{-a}-\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx}{2 b (b c+a d)}-\frac{\left (a \sqrt{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )\right ) \int \frac{1+\frac{\sqrt{d} x^2}{\sqrt{c}}}{\left (\sqrt{-a}+\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx}{2 b (b c+a d)}+\frac{\left (a \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt{d}\right ) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{2 b^{3/2} (b c+a d)}+\frac{\left (a \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt{d}\right ) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{2 b^{3/2} (b c+a d)}\\ &=\frac{x \sqrt{c+d x^4}}{b \sqrt{d} \left (\sqrt{c}+\sqrt{d} x^2\right )}+\frac{(-a)^{3/4} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{4 b^{5/4} \sqrt{b c-a d}}-\frac{(-a)^{3/4} \tan ^{-1}\left (\frac{\sqrt{-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{4 b^{5/4} \sqrt{-b c+a d}}-\frac{\sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{b d^{3/4} \sqrt{c+d x^4}}+\frac{\sqrt [4]{c} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{2 b d^{3/4} \sqrt{c+d x^4}}+\frac{a \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 b^{3/2} \sqrt [4]{c} (b c+a d) \sqrt{c+d x^4}}+\frac{a \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 b^{3/2} \sqrt [4]{c} (b c+a d) \sqrt{c+d x^4}}+\frac{\sqrt{-a} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \Pi \left (-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{c+d x^4}}-\frac{\sqrt{-a} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 \left (\sqrt{c}+\sqrt{d} x^2\right ) \sqrt{\frac{c+d x^4}{\left (\sqrt{c}+\sqrt{d} x^2\right )^2}} \Pi \left (\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2}{4 \sqrt{-a} \sqrt{b} \sqrt{c} \sqrt{d}};2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{8 b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c+a d) \sqrt{c+d x^4}}\\ \end{align*}
Mathematica [C] time = 0.0348157, size = 65, normalized size = 0.08 \[ \frac{x^7 \sqrt{\frac{c+d x^4}{c}} F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )}{7 a \sqrt{c+d x^4}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.018, size = 292, normalized size = 0.4 \begin{align*}{\frac{i}{b}\sqrt{c}\sqrt{1-{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}}\sqrt{1+{i{x}^{2}\sqrt{d}{\frac{1}{\sqrt{c}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{d}{\frac{1}{\sqrt{c}}}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}{\frac{1}{\sqrt{d}}}}-{\frac{a}{8\,{b}^{2}}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}b+a \right ) }{\frac{1}{{\it \_alpha}} \left ( -{{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}d{x}^{2}+2\,c}{2}{\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}{\frac{1}{\sqrt{d{x}^{4}+c}}}} \right ){\frac{1}{\sqrt{{\frac{-ad+bc}{b}}}}}}+2\,{\frac{{{\it \_alpha}}^{3}b}{a\sqrt{d{x}^{4}+c}}\sqrt{1-{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}\sqrt{1+{\frac{i\sqrt{d}{x}^{2}}{\sqrt{c}}}}{\it EllipticPi} \left ( x\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}},{\frac{i\sqrt{c}{{\it \_alpha}}^{2}b}{a\sqrt{d}}},{\sqrt{{\frac{-i\sqrt{d}}{\sqrt{c}}}}{\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ){\frac{1}{\sqrt{{\frac{i\sqrt{d}}{\sqrt{c}}}}}}} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x^{4} + c} x^{6}}{b d x^{8} +{\left (b c + a d\right )} x^{4} + a c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\left (a + b x^{4}\right ) \sqrt{c + d x^{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{{\left (b x^{4} + a\right )} \sqrt{d x^{4} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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