Optimal. Leaf size=235 \[ \frac {\sqrt [4]{5 d^4+256 a e^3} \sqrt {\frac {e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (5 d^4+256 a e^3\right ) \left (1+\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}\right )^2}} \left (1+\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}\right ) F\left (2 \tan ^{-1}\left (\frac {d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac {1}{2} \left (1+\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}\right )\right )}{\sqrt {2} e \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \]
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Rubi [A]
time = 0.13, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {1120, 1117}
\begin {gather*} \frac {\sqrt [4]{256 a e^3+5 d^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 ) F\left (2 \text {ArcTan}\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 )}{\sqrt {2} e \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1117
Rule 1120
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \, dx &=\text {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 )\\ &=\frac {\sqrt [4]{5 d^4+256 a e^3} \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 {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}\right )^2}} \left (1+\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^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 )}{\sqrt {2} e \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1065\) vs. \(2(235)=470\).
time = 11.55, size = 1065, normalized size = 4.53 \begin {gather*} -\frac {\left (-d+\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}-4 e x\right ) \left (d-\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}+4 e x\right ) \sqrt {-\frac {\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}} \left (d+\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}+4 e x\right )}{\left (\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}-\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}\right ) \left (-d+\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}-4 e x\right )}} \sqrt {\frac {3 d^2-2 \sqrt {d^4-64 a e^3}-\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}} \sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}+d \left (\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}-\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}\right )+4 e \left (\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}-\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}\right ) x}{\left (\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}+\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}\right ) \left (-d+\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}-4 e x\right )}} F\left (\sin ^{-1}\left (\sqrt {\frac {\left (\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}-\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}\right ) \left (d+\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}+4 e x\right )}{\left (\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}+\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}\right ) \left (-d+\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}-4 e x\right )}}\right )|\frac {\left (\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}+\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}\right )^2}{\left (\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}-\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}\right )^2}\right )}{2 e \left (\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}-\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}\right ) \sqrt {\frac {\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}} \left (-d+\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}-4 e x\right )}{\left (\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}+\sqrt {3 d^2+2 \sqrt {d^4-64 a e^3}}\right ) \left (-d+\sqrt {3 d^2-2 \sqrt {d^4-64 a e^3}}-4 e x\right )}} \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1703\) vs.
\(2(273)=546\).
time = 0.05, size = 1704, normalized size = 7.25
method | result | size |
default | \(\text {Expression too large to display}\) | \(1704\) |
elliptic | \(\text {Expression too large to display}\) | \(1704\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {1}{\sqrt {8 a e^{2} - d^{3} x + 8 d e^{2} x^{3} + 8 e^{3} x^{4}}}\, dx \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*} \text {could not integrate} \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 {1}{\sqrt {-d^3\,x+8\,d\,e^2\,x^3+8\,e^3\,x^4+8\,a\,e^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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