Optimal. Leaf size=1146 \[ \text{result too large to display} \]
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Rubi [A] time = 1.56284, antiderivative size = 1146, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {471, 584, 305, 220, 1196, 490, 1217, 1707} \[ -\frac{\sqrt{d x^4+c} x^3}{4 (b c-a d) \left (b x^4+a\right )}+\frac{\sqrt{d} \sqrt{d x^4+c} x}{4 b (b c-a d) \left (\sqrt{d} x^2+\sqrt{c}\right )}+\frac{(3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^4+c}}\right )}{16 \sqrt [4]{-a} b^{5/4} (b c-a d)^{3/2}}+\frac{(3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{a d-b c} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{d x^4+c}}\right )}{16 \sqrt [4]{-a} b^{5/4} (a d-b c)^{3/2}}-\frac{\sqrt [4]{c} \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}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 b (b c-a d) \sqrt{d x^4+c}}+\frac{\sqrt [4]{c} \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 )}{8 b (b c-a d) \sqrt{d x^4+c}}-\frac{\left (\sqrt{c}-\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt [4]{d} (3 b c-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}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 b \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{\left (\sqrt{c}+\frac{\sqrt{-a} \sqrt{d}}{\sqrt{b}}\right ) \sqrt [4]{d} (3 b c-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}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{16 b \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{d x^4+c}}+\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 (3 b c-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}} \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 )}{32 \sqrt{-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 (3 b c-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}} \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 )}{32 \sqrt{-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{d x^4+c}} \]
Antiderivative was successfully verified.
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Rule 471
Rule 584
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 )^2 \sqrt{c+d x^4}} \, dx &=-\frac{x^3 \sqrt{c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}+\frac{\int \frac{x^2 \left (3 c+d x^4\right )}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx}{4 (b c-a d)}\\ &=-\frac{x^3 \sqrt{c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}+\frac{\int \left (\frac{d x^2}{b \sqrt{c+d x^4}}+\frac{(3 b c-a d) x^2}{b \left (a+b x^4\right ) \sqrt{c+d x^4}}\right ) \, dx}{4 (b c-a d)}\\ &=-\frac{x^3 \sqrt{c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}+\frac{d \int \frac{x^2}{\sqrt{c+d x^4}} \, dx}{4 b (b c-a d)}+\frac{(3 b c-a d) \int \frac{x^2}{\left (a+b x^4\right ) \sqrt{c+d x^4}} \, dx}{4 b (b c-a d)}\\ &=-\frac{x^3 \sqrt{c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}+\frac{\left (\sqrt{c} \sqrt{d}\right ) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{4 b (b c-a d)}-\frac{\left (\sqrt{c} \sqrt{d}\right ) \int \frac{1-\frac{\sqrt{d} x^2}{\sqrt{c}}}{\sqrt{c+d x^4}} \, dx}{4 b (b c-a d)}-\frac{(3 b c-a d) \int \frac{1}{\left (\sqrt{-a}-\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx}{8 b^{3/2} (b c-a d)}+\frac{(3 b c-a d) \int \frac{1}{\left (\sqrt{-a}+\sqrt{b} x^2\right ) \sqrt{c+d x^4}} \, dx}{8 b^{3/2} (b c-a d)}\\ &=\frac{\sqrt{d} x \sqrt{c+d x^4}}{4 b (b c-a d) \left (\sqrt{c}+\sqrt{d} x^2\right )}-\frac{x^3 \sqrt{c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}-\frac{\sqrt [4]{c} \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}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 b (b c-a d) \sqrt{c+d x^4}}+\frac{\sqrt [4]{c} \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 )}{8 b (b c-a d) \sqrt{c+d x^4}}-\frac{\left (\sqrt{c} \left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) (3 b c-a d)\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}{8 b (b c-a d) (b c+a d)}+\frac{\left (\sqrt{c} \left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) (3 b c-a d)\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}{8 b (b c-a d) (b c+a d)}-\frac{\left (\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt{d} (3 b c-a d)\right ) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{8 b^{3/2} (b c-a d) (b c+a d)}-\frac{\left (\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt{d} (3 b c-a d)\right ) \int \frac{1}{\sqrt{c+d x^4}} \, dx}{8 b^{3/2} (b c-a d) (b c+a d)}\\ &=\frac{\sqrt{d} x \sqrt{c+d x^4}}{4 b (b c-a d) \left (\sqrt{c}+\sqrt{d} x^2\right )}-\frac{x^3 \sqrt{c+d x^4}}{4 (b c-a d) \left (a+b x^4\right )}+\frac{(3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{16 \sqrt [4]{-a} b^{5/4} (b c-a d)^{3/2}}+\frac{(3 b c-a d) \tan ^{-1}\left (\frac{\sqrt{-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt{c+d x^4}}\right )}{16 \sqrt [4]{-a} b^{5/4} (-b c+a d)^{3/2}}-\frac{\sqrt [4]{c} \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}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} x}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{4 b (b c-a d) \sqrt{c+d x^4}}+\frac{\sqrt [4]{c} \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 )}{8 b (b c-a d) \sqrt{c+d x^4}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (3 b c-a 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 )}{16 b^{3/2} \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{c+d x^4}}-\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right ) \sqrt [4]{d} (3 b c-a 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 )}{16 b^{3/2} \sqrt [4]{c} (b c-a d) (b c+a d) \sqrt{c+d x^4}}+\frac{\left (\sqrt{b} \sqrt{c}+\sqrt{-a} \sqrt{d}\right )^2 (3 b c-a d) \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 )}{32 \sqrt{-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{c+d x^4}}-\frac{\left (\sqrt{b} \sqrt{c}-\sqrt{-a} \sqrt{d}\right )^2 (3 b c-a d) \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 )}{32 \sqrt{-a} b^{3/2} \sqrt [4]{c} \sqrt [4]{d} (b c-a d) (b c+a d) \sqrt{c+d x^4}}\\ \end{align*}
Mathematica [C] time = 0.122553, size = 162, normalized size = 0.14 \[ \frac{d x^7 \left (a+b x^4\right ) \sqrt{\frac{d x^4}{c}+1} F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )+7 c x^3 \left (a+b x^4\right ) \sqrt{\frac{d x^4}{c}+1} F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\frac{d x^4}{c},-\frac{b x^4}{a}\right )-7 a x^3 \left (c+d x^4\right )}{28 a \left (a+b x^4\right ) \sqrt{c+d x^4} (b c-a d)} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.026, size = 556, normalized size = 0.5 \begin{align*} \text{result too large to display} \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 )}^{2} \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 )}^{2} \sqrt{d x^{4} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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