Optimal. Leaf size=128 \[ -\frac {\left (3 c d^2+4 b e^2\right ) x \sqrt {d-e x} \sqrt {d+e x}}{8 e^4}+\frac {c x^3 (-d+e x) \sqrt {d+e x}}{4 e^2 \sqrt {d-e x}}-\frac {\left (3 c d^4+4 b d^2 e^2+8 a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {d+e x}}\right )}{4 e^5} \]
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Rubi [A]
time = 0.06, antiderivative size = 179, normalized size of antiderivative = 1.40, number of steps
used = 5, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {534, 1173, 396,
223, 209} \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \text {ArcTan}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) \left (8 a e^4+4 b d^2 e^2+3 c d^4\right )}{8 e^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {x \left (d^2-e^2 x^2\right ) \left (4 b e^2+3 c d^2\right )}{8 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x^3 \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 396
Rule 534
Rule 1173
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{\sqrt {d-e x} \sqrt {d+e x}} \, dx &=\frac {\sqrt {d^2-e^2 x^2} \int \frac {a+b x^2+c x^4}{\sqrt {d^2-e^2 x^2}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {c x^3 \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\sqrt {d^2-e^2 x^2} \int \frac {-4 a e^2-\left (3 c d^2+4 b e^2\right ) x^2}{\sqrt {d^2-e^2 x^2}} \, dx}{4 e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (3 c d^2+4 b e^2\right ) x \left (d^2-e^2 x^2\right )}{8 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x^3 \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (\left (-8 a e^4+d^2 \left (-3 c d^2-4 b e^2\right )\right ) \sqrt {d^2-e^2 x^2}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{8 e^4 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (3 c d^2+4 b e^2\right ) x \left (d^2-e^2 x^2\right )}{8 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x^3 \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (\left (-8 a e^4+d^2 \left (-3 c d^2-4 b e^2\right )\right ) \sqrt {d^2-e^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\left (3 c d^2+4 b e^2\right ) x \left (d^2-e^2 x^2\right )}{8 e^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {c x^3 \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\left (3 c d^4+4 b d^2 e^2+8 a e^4\right ) \sqrt {d^2-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^5 \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 98, normalized size = 0.77 \begin {gather*} \frac {-e x \sqrt {d-e x} \sqrt {d+e x} \left (3 c d^2+4 b e^2+2 c e^2 x^2\right )+2 \left (3 c d^4+4 b d^2 e^2+8 a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{8 e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.14, size = 191, normalized size = 1.49
method | result | size |
risch | \(-\frac {x \left (2 c \,e^{2} x^{2}+4 e^{2} b +3 c \,d^{2}\right ) \sqrt {-e x +d}\, \sqrt {e x +d}}{8 e^{4}}+\frac {\left (\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) a}{\sqrt {e^{2}}}+\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) d^{2} b}{2 e^{2} \sqrt {e^{2}}}+\frac {3 \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) d^{4} c}{8 e^{4} \sqrt {e^{2}}}\right ) \sqrt {\left (e x +d \right ) \left (-e x +d \right )}}{\sqrt {e x +d}\, \sqrt {-e x +d}}\) | \(177\) |
default | \(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (2 \,\mathrm {csgn}\left (e \right ) c \,e^{3} x^{3} \sqrt {-e^{2} x^{2}+d^{2}}+4 \sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\left (e \right ) e^{3} b x +3 \sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\left (e \right ) e c \,d^{2} x -8 \arctan \left (\frac {\mathrm {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) a \,e^{4}-4 \arctan \left (\frac {\mathrm {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) b \,d^{2} e^{2}-3 \arctan \left (\frac {\mathrm {csgn}\left (e \right ) e x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right ) c \,d^{4}\right ) \mathrm {csgn}\left (e \right )}{8 e^{5} \sqrt {-e^{2} x^{2}+d^{2}}}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 107, normalized size = 0.84 \begin {gather*} \frac {3}{8} \, c d^{4} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} - \frac {1}{4} \, \sqrt {-x^{2} e^{2} + d^{2}} c x^{3} e^{\left (-2\right )} - \frac {3}{8} \, \sqrt {-x^{2} e^{2} + d^{2}} c d^{2} x e^{\left (-4\right )} + \frac {1}{2} \, b d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-3\right )} - \frac {1}{2} \, \sqrt {-x^{2} e^{2} + d^{2}} b x e^{\left (-2\right )} + a \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 100, normalized size = 0.78 \begin {gather*} -\frac {{\left (2 \, c e^{3} x^{3} + {\left (3 \, c d^{2} e + 4 \, b e^{3}\right )} x\right )} \sqrt {e x + d} \sqrt {-e x + d} + 2 \, {\left (3 \, c d^{4} + 4 \, b d^{2} e^{2} + 8 \, a e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{e x}\right )}{8 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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 [A]
time = 5.72, size = 116, normalized size = 0.91 \begin {gather*} \frac {1}{8} \, {\left ({\left (5 \, c d^{3} + 4 \, b d e^{2} - {\left (9 \, c d^{2} + 2 \, {\left ({\left (x e + d\right )} c - 3 \, c d\right )} {\left (x e + d\right )} + 4 \, b e^{2}\right )} {\left (x e + d\right )}\right )} \sqrt {x e + d} \sqrt {-x e + d} + 2 \, {\left (3 \, c d^{4} + 4 \, b d^{2} e^{2} + 8 \, a e^{4}\right )} \arcsin \left (\frac {\sqrt {2} \sqrt {x e + d}}{2 \, \sqrt {d}}\right )\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.86, size = 651, normalized size = 5.09 \begin {gather*} \frac {\frac {14\,b\,d^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^3}-\frac {14\,b\,d^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^5}+\frac {2\,b\,d^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^7}-\frac {2\,b\,d^2\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}{\sqrt {d-e\,x}-\sqrt {d}}}{e^3\,{\left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}+1\right )}^4}-\frac {4\,a\,\mathrm {atan}\left (\frac {e\,\left (\sqrt {d-e\,x}-\sqrt {d}\right )}{\sqrt {e^2}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )}{\sqrt {e^2}}-\frac {\frac {23\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^3}-\frac {333\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^5}+\frac {671\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^7}-\frac {671\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^9}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^9}+\frac {333\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{11}}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{11}}-\frac {23\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{13}}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{13}}-\frac {3\,c\,d^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{15}}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{15}}+\frac {3\,c\,d^4\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}{2\,\left (\sqrt {d-e\,x}-\sqrt {d}\right )}}{e^5\,{\left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}+1\right )}^8}+\frac {2\,b\,d^2\,\mathrm {atan}\left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )}{e^3}+\frac {3\,c\,d^4\,\mathrm {atan}\left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )}{2\,e^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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