Optimal. Leaf size=663 \[ -\frac{2 d^2 \left (\frac{d}{4 e}+x\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{\sqrt{256 a e^3+5 d^4} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )}+\frac{1}{3} \left (\frac{d}{4 e}+x\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}+\frac{\sqrt [4]{256 a e^3+5 d^4} \left (-3 d^2 \sqrt{256 a e^3+5 d^4}+256 a e^3+5 d^4\right ) \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{48 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac{d^2 \left (256 a e^3+5 d^4\right )^{3/4} \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{8 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \]
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Rubi [A] time = 0.810476, antiderivative size = 663, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {1106, 1091, 1197, 1103, 1195} \[ -\frac{2 d^2 \left (\frac{d}{4 e}+x\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{\sqrt{256 a e^3+5 d^4} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )}+\frac{1}{3} \left (\frac{d}{4 e}+x\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}+\frac{\sqrt [4]{256 a e^3+5 d^4} \left (-3 d^2 \sqrt{256 a e^3+5 d^4}+256 a e^3+5 d^4\right ) \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) F\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{48 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac{d^2 \left (256 a e^3+5 d^4\right )^{3/4} \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (256 a e^3+5 d^4\right ) \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right )^2}} \left (\frac{16 e^2 \left (\frac{d}{4 e}+x\right )^2}{\sqrt{256 a e^3+5 d^4}}+1\right ) E\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}+1\right )\right )}{8 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \]
Antiderivative was successfully verified.
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Rule 1106
Rule 1091
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx &=\operatorname{Subst}\left (\int \sqrt{\frac{1}{32} \left (\frac{5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4} \, dx,x,\frac{d}{4 e}+x\right )\\ &=\frac{1}{3} \left (\frac{d}{4 e}+x\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{\frac{1}{16} \left (\frac{5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2}{\sqrt{\frac{1}{32} \left (\frac{5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4}} \, dx,x,\frac{d}{4 e}+x\right )\\ &=\frac{1}{3} \left (\frac{d}{4 e}+x\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}+\frac{\left (d^2 \sqrt{5 d^4+256 a e^3}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{16 e^2 x^2}{\sqrt{5 d^4+256 a e^3}}}{\sqrt{\frac{1}{32} \left (\frac{5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4}} \, dx,x,\frac{d}{4 e}+x\right )}{16 e}+\frac{\left (5 d^4+256 a e^3-3 d^2 \sqrt{5 d^4+256 a e^3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{\frac{1}{32} \left (\frac{5 d^4}{e}+256 a e^2\right )-3 d^2 e x^2+8 e^3 x^4}} \, dx,x,\frac{d}{4 e}+x\right )}{48 e}\\ &=\frac{1}{3} \left (\frac{d}{4 e}+x\right ) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}-\frac{d^2 (d+4 e x) \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{2 e \sqrt{5 d^4+256 a e^3} \left (1+\frac{(d+4 e x)^2}{\sqrt{5 d^4+256 a e^3}}\right )}+\frac{d^2 \left (5 d^4+256 a e^3\right )^{3/4} \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (5 d^4+256 a e^3\right ) \left (1+\frac{(d+4 e x)^2}{\sqrt{5 d^4+256 a e^3}}\right )^2}} \left (1+\frac{(d+4 e x)^2}{\sqrt{5 d^4+256 a e^3}}\right ) E\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (1+\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}\right )\right )}{8 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac{\sqrt [4]{5 d^4+256 a e^3} \left (5 d^4+256 a e^3-3 d^2 \sqrt{5 d^4+256 a e^3}\right ) \sqrt{\frac{e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (5 d^4+256 a e^3\right ) \left (1+\frac{(d+4 e x)^2}{\sqrt{5 d^4+256 a e^3}}\right )^2}} \left (1+\frac{(d+4 e x)^2}{\sqrt{5 d^4+256 a e^3}}\right ) F\left (2 \tan ^{-1}\left (\frac{d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac{1}{2} \left (1+\frac{3 d^2}{\sqrt{5 d^4+256 a e^3}}\right )\right )}{48 \sqrt{2} e^2 \sqrt{8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}\\ \end{align*}
Mathematica [B] time = 6.12362, size = 7543, normalized size = 11.38 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.24, size = 7887, normalized size = 11.9 \begin{align*} \text{output 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 \sqrt{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}\,{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 (\sqrt{8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}}, 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 \sqrt{8 a e^{2} - d^{3} x + 8 d e^{2} x^{3} + 8 e^{3} x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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