Optimal. Leaf size=128 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {d+e x}}\right ) \left (8 a e^4+4 b d^2 e^2+3 c d^4\right )}{4 e^5}-\frac {x \sqrt {d-e x} \sqrt {d+e x} \left (4 b e^2+3 c d^2\right )}{8 e^4}+\frac {c x^3 (e x-d) \sqrt {d+e x}}{4 e^2 \sqrt {d-e x}} \]
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Rubi [A] time = 0.09, 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 = {520, 1159, 388, 217, 203} \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \tan ^{-1}\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 203
Rule 217
Rule 388
Rule 520
Rule 1159
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 ) \operatorname {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 = 0.56, size = 157, normalized size = 1.23 \begin {gather*} -\frac {16 \tan ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {d+e x}}\right ) \left (a e^4+b d^2 e^2+c d^4\right )+e x \sqrt {d-e x} \sqrt {d+e x} \left (4 b e^2+3 c d^2+2 c e^2 x^2\right )-\frac {2 d^{5/2} \sqrt {\frac {e x}{d}+1} \left (4 b e^2+5 c d^2\right ) \sin ^{-1}\left (\frac {\sqrt {d-e x}}{\sqrt {2} \sqrt {d}}\right )}{\sqrt {d+e x}}}{8 e^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 207, normalized size = 1.62 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right ) \left (8 a e^4+4 b d^2 e^2+3 c d^4\right )}{4 e^5}-\frac {d^2 \sqrt {d+e x} \left (\frac {d+e x}{d-e x}-1\right ) \left (\frac {4 b e^2 (d+e x)^2}{(d-e x)^2}+\frac {8 b e^2 (d+e x)}{d-e x}+4 b e^2+\frac {5 c d^2 (d+e x)^2}{(d-e x)^2}+\frac {2 c d^2 (d+e x)}{d-e x}+5 c d^2\right )}{4 e^5 \sqrt {d-e x} \left (\frac {d+e x}{d-e x}+1\right )^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.25, 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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giac [A] time = 0.59, size = 135, normalized size = 1.05 \begin {gather*} \frac {1}{8} \, {\left (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 ) e^{\left (-4\right )} - {\left ({\left (2 \, {\left ({\left (x e + d\right )} c e^{\left (-4\right )} - 3 \, c d e^{\left (-4\right )}\right )} {\left (x e + d\right )} + {\left (9 \, c d^{2} e^{16} + 4 \, b e^{18}\right )} e^{\left (-20\right )}\right )} {\left (x e + d\right )} - {\left (5 \, c d^{3} e^{16} + 4 \, b d e^{18}\right )} e^{\left (-20\right )}\right )} \sqrt {x e + d} \sqrt {-x e + d}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 191, normalized size = 1.49 \begin {gather*} -\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (2 \sqrt {-e^{2} x^{2}+d^{2}}\, c \,e^{3} x^{3} \mathrm {csgn}\relax (e )-8 a \,e^{4} \arctan \left (\frac {e x \,\mathrm {csgn}\relax (e )}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )-4 b \,d^{2} e^{2} \arctan \left (\frac {e x \,\mathrm {csgn}\relax (e )}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )+4 \sqrt {-e^{2} x^{2}+d^{2}}\, b \,e^{3} x \,\mathrm {csgn}\relax (e )-3 c \,d^{4} \arctan \left (\frac {e x \,\mathrm {csgn}\relax (e )}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )+3 \sqrt {-e^{2} x^{2}+d^{2}}\, c \,d^{2} e x \,\mathrm {csgn}\relax (e )\right ) \mathrm {csgn}\relax (e )}{8 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 113, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {-e^{2} x^{2} + d^{2}} c x^{3}}{4 \, e^{2}} + \frac {3 \, c d^{4} \arcsin \left (\frac {e x}{d}\right )}{8 \, e^{5}} + \frac {b d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{3}} + \frac {a \arcsin \left (\frac {e x}{d}\right )}{e} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} c d^{2} x}{8 \, e^{4}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b x}{2 \, e^{2}} \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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sympy [F(-1)] 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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