Optimal. Leaf size=126 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {d-e x} \sqrt {d+e x}}{d}\right ) \left (3 a e^4+4 b d^2 e^2+8 c d^4\right )}{8 d^5}-\frac {\sqrt {d-e x} \sqrt {d+e x} \left (3 a e^2+4 b d^2\right )}{8 d^4 x^2}-\frac {a \sqrt {d-e x} \sqrt {d+e x}}{4 d^2 x^4} \]
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Rubi [A] time = 0.28, 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 = {520, 1251, 897, 1157, 385, 208} \[ -\frac {\sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \left (3 a e^4+4 b d^2 e^2+8 c d^4\right )}{8 d^5 \sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (d^2-e^2 x^2\right ) \left (3 a e^2+4 b d^2\right )}{8 d^4 x^2 \sqrt {d-e x} \sqrt {d+e x}}-\frac {a \left (d^2-e^2 x^2\right )}{4 d^2 x^4 \sqrt {d-e x} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 385
Rule 520
Rule 897
Rule 1157
Rule 1251
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} \operatorname {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} \operatorname {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} \operatorname {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 ) \operatorname {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 = 0.16, size = 134, normalized size = 1.06 \[ \frac {-\left (x^4 \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \left (3 a e^4+4 b d^2 e^2+8 c d^4\right )\right )-d \left (d^2-e^2 x^2\right ) \left (2 a d^2+3 a e^2 x^2+4 b d^2 x^2\right )}{8 d^5 x^4 \sqrt {d-e x} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 102, normalized size = 0.81 \[ \frac {{\left (8 \, c d^{4} + 4 \, b d^{2} e^{2} + 3 \, a e^{4}\right )} x^{4} \log \left (\frac {\sqrt {e x + d} \sqrt {-e x + d} - d}{x}\right ) - {\left (2 \, a d^{3} + {\left (4 \, b d^{3} + 3 \, a d e^{2}\right )} x^{2}\right )} \sqrt {e x + d} \sqrt {-e x + d}}{8 \, d^{5} x^{4}} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 222, normalized size = 1.76 \[ -\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \left (3 a \,e^{4} x^{4} \ln \left (\frac {2 \left (d +\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\relax (d )\right ) d}{x}\right )+4 b \,d^{2} e^{2} x^{4} \ln \left (\frac {2 \left (d +\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\relax (d )\right ) d}{x}\right )+8 c \,d^{4} x^{4} \ln \left (\frac {2 \left (d +\sqrt {-e^{2} x^{2}+d^{2}}\, \mathrm {csgn}\relax (d )\right ) d}{x}\right )+3 \sqrt {-e^{2} x^{2}+d^{2}}\, a d \,e^{2} x^{2} \mathrm {csgn}\relax (d )+4 \sqrt {-e^{2} x^{2}+d^{2}}\, b \,d^{3} x^{2} \mathrm {csgn}\relax (d )+2 \sqrt {-e^{2} x^{2}+d^{2}}\, a \,d^{3} \mathrm {csgn}\relax (d )\right ) \mathrm {csgn}\relax (d )}{8 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 193, normalized size = 1.53 \[ -\frac {c \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{d} - \frac {b e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{3}} - \frac {3 \, a e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{5}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} b}{2 \, d^{2} x^{2}} - \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} a e^{2}}{8 \, d^{4} x^{2}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} a}{4 \, d^{2} x^{4}} \]
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 \[ \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} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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