Optimal. Leaf size=126 \[ -\frac {a \sqrt {d-e x} \sqrt {d+e x}}{4 d^2 x^4}-\frac {\left (4 b d^2+3 a e^2\right ) \sqrt {d-e x} \sqrt {d+e x}}{8 d^4 x^2}-\frac {\left (8 c d^4+4 b d^2 e^2+3 a e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d-e x} \sqrt {d+e x}}{d}\right )}{8 d^5} \]
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Rubi [A]
time = 0.19, antiderivative size = 182, normalized size of antiderivative = 1.44, number of steps
used = 6, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {534, 1265, 911,
1171, 393, 214} \begin {gather*} -\frac {\sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \left (3 a e^4+4 b d^2 e^2+8 c d^4\right )}{8 d^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (3 a e^2+4 b d^2\right )}{8 d^4 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{4 d^2 x^4 \sqrt {d-e x} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 393
Rule 534
Rule 911
Rule 1171
Rule 1265
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{x^5 \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}{x^5 \sqrt {d^2-e^2 x^2}} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {\sqrt {d^2-e^2 x^2} \text {Subst}\left (\int \frac {a+b x+c x^2}{x^3 \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {\sqrt {d^2-e^2 x^2} \text {Subst}\left (\int \frac {\frac {c d^4+b d^2 e^2+a e^4}{e^4}-\frac {\left (2 c d^2+b e^2\right ) x^2}{e^4}+\frac {c x^4}{e^4}}{\left (\frac {d^2}{e^2}-\frac {x^2}{e^2}\right )^3} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{e^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{4 d^2 x^4 \sqrt {d-e x} \sqrt {d+e x}}+\frac {\sqrt {d^2-e^2 x^2} \text {Subst}\left (\int \frac {-3 a-\frac {4 \left (c d^4+b d^2 e^2\right )}{e^4}+\frac {4 c d^2 x^2}{e^4}}{\left (\frac {d^2}{e^2}-\frac {x^2}{e^2}\right )^2} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{4 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{4 d^2 x^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (4 b d^2+3 a e^2\right ) \left (d^2-e^2 x^2\right )}{8 d^4 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (\left (4 b+\frac {8 c d^2}{e^2}+\frac {3 a e^2}{d^2}\right ) \sqrt {d^2-e^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{8 d^2 \sqrt {d-e x} \sqrt {d+e x}}\\ &=-\frac {a \left (d^2-e^2 x^2\right )}{4 d^2 x^4 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (4 b d^2+3 a e^2\right ) \left (d^2-e^2 x^2\right )}{8 d^4 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (8 c d^4+4 b d^2 e^2+3 a e^4\right ) \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^5 \sqrt {d-e x} \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 102, normalized size = 0.81 \begin {gather*} -\frac {\frac {d \sqrt {d-e x} \sqrt {d+e x} \left (2 a d^2+4 b d^2 x^2+3 a e^2 x^2\right )}{x^4}+2 \left (8 c d^4+4 b d^2 e^2+3 a e^4\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d-e x}}\right )}{8 d^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 = 0.16, size = 222, normalized size = 1.76
method | result | size |
risch | \(-\frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \left (3 a \,e^{2} x^{2}+4 b \,d^{2} x^{2}+2 a \,d^{2}\right )}{8 d^{4} x^{4}}+\frac {\left (-\frac {3 \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right ) e^{4} a}{8 d^{4} \sqrt {d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right ) e^{2} b}{2 d^{2} \sqrt {d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right ) c}{\sqrt {d^{2}}}\right ) \sqrt {\left (e x +d \right ) \left (-e x +d \right )}}{\sqrt {e x +d}\, \sqrt {-e x +d}}\) | \(213\) |
default | \(-\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (3 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\left (d \right )+d \right )}{x}\right ) a \,e^{4} x^{4}+4 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\left (d \right )+d \right )}{x}\right ) b \,d^{2} e^{2} x^{4}+8 \ln \left (\frac {2 d \left (\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\left (d \right )+d \right )}{x}\right ) c \,d^{4} x^{4}+3 \,\mathrm {csgn}\left (d \right ) a d \,e^{2} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}+4 \,\mathrm {csgn}\left (d \right ) b \,d^{3} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}+2 \,\mathrm {csgn}\left (d \right ) a \,d^{3} \sqrt {-e^{2} x^{2}+d^{2}}\right ) \mathrm {csgn}\left (d \right )}{8 d^{5} \sqrt {-e^{2} x^{2}+d^{2}}\, x^{4}}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 184, normalized size = 1.46 \begin {gather*} -\frac {c \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d} - \frac {b e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{3}} - \frac {3 \, a e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-x^{2} e^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{5}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}} b}{2 \, d^{2} x^{2}} - \frac {3 \, \sqrt {-x^{2} e^{2} + d^{2}} a e^{2}}{8 \, d^{4} x^{2}} - \frac {\sqrt {-x^{2} e^{2} + d^{2}} a}{4 \, d^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 102, normalized size = 0.81 \begin {gather*} \frac {{\left (8 \, c d^{4} + 4 \, b d^{2} e^{2} + 3 \, a e^{4}\right )} x^{4} \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) - {\left (2 \, a d^{3} + {\left (4 \, b d^{3} + 3 \, a d e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{8 \, d^{5} x^{4}} \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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.82, size = 932, normalized size = 7.40 \begin {gather*} \frac {\frac {a\,e^4}{4}+\frac {6\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-\frac {53\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}-\frac {87\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^6}+\frac {657\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}{4\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^8}-\frac {121\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{10}}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{10}}}{\frac {256\,d^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}-\frac {1024\,d^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^6}+\frac {1536\,d^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^8}-\frac {1024\,d^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{10}}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{10}}+\frac {256\,d^5\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^{12}}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^{12}}}-\frac {\frac {b\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-\frac {b\,e^2}{2}+\frac {15\,b\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{2\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}}{\frac {16\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-\frac {32\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}+\frac {16\,d^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^6}}+\frac {c\,\left (\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )-\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )\right )}{d}-\frac {3\,a\,e^4\,\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )}{8\,d^5}-\frac {b\,e^2\,\ln \left (\frac {\sqrt {d+e\,x}-\sqrt {d}}{\sqrt {d-e\,x}-\sqrt {d}}\right )}{2\,d^3}+\frac {3\,a\,e^4\,\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )}{8\,d^5}+\frac {b\,e^2\,\ln \left (\frac {{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}-1\right )}{2\,d^3}+\frac {7\,a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{256\,d^5\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2}+\frac {a\,e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}{1024\,d^5\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^4}+\frac {b\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}{32\,d^3\,{\left (\sqrt {d-e\,x}-\sqrt {d}\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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