Optimal. Leaf size=167 \[ -\frac {a}{7 d^2 x^7}-\frac {b d-2 a e}{5 d^3 x^5}-\frac {c d^2-e (2 b d-3 a e)}{3 d^4 x^3}+\frac {e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}+\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {e^{3/2} \left (5 c d^2-e (7 b d-9 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{11/2}} \]
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Rubi [A]
time = 0.21, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1273, 1816,
211} \begin {gather*} \frac {e^{3/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (5 c d^2-e (7 b d-9 a e)\right )}{2 d^{11/2}}+\frac {e^2 x \left (a e^2-b d e+c d^2\right )}{2 d^5 \left (d+e x^2\right )}+\frac {e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}-\frac {c d^2-e (2 b d-3 a e)}{3 d^4 x^3}-\frac {b d-2 a e}{5 d^3 x^5}-\frac {a}{7 d^2 x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1273
Rule 1816
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{x^8 \left (d+e x^2\right )^2} \, dx &=\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {\int \frac {2 a d^4 e^2+2 d^3 e^2 (b d-a e) x^2+2 d^2 e^2 \left (c d^2-b d e+a e^2\right ) x^4-2 d e^3 \left (c d^2-b d e+a e^2\right ) x^6+e^4 \left (c d^2-b d e+a e^2\right ) x^8}{x^8 \left (d+e x^2\right )} \, dx}{2 d^5 e^2}\\ &=\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {\int \left (\frac {2 a d^3 e^2}{x^8}+\frac {2 d^2 e^2 (b d-2 a e)}{x^6}+\frac {2 d e^2 \left (c d^2-e (2 b d-3 a e)\right )}{x^4}+\frac {2 e^3 \left (-2 c d^2+e (3 b d-4 a e)\right )}{x^2}+\frac {e^4 \left (5 c d^2-e (7 b d-9 a e)\right )}{d+e x^2}\right ) \, dx}{2 d^5 e^2}\\ &=-\frac {a}{7 d^2 x^7}-\frac {b d-2 a e}{5 d^3 x^5}-\frac {c d^2-e (2 b d-3 a e)}{3 d^4 x^3}+\frac {e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}+\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {\left (e^2 \left (5 c d^2-e (7 b d-9 a e)\right )\right ) \int \frac {1}{d+e x^2} \, dx}{2 d^5}\\ &=-\frac {a}{7 d^2 x^7}-\frac {b d-2 a e}{5 d^3 x^5}-\frac {c d^2-e (2 b d-3 a e)}{3 d^4 x^3}+\frac {e \left (2 c d^2-e (3 b d-4 a e)\right )}{d^5 x}+\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {e^{3/2} \left (5 c d^2-e (7 b d-9 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 166, normalized size = 0.99 \begin {gather*} -\frac {a}{7 d^2 x^7}+\frac {-b d+2 a e}{5 d^3 x^5}+\frac {-c d^2+2 b d e-3 a e^2}{3 d^4 x^3}+\frac {e \left (2 c d^2-3 b d e+4 a e^2\right )}{d^5 x}+\frac {e^2 \left (c d^2-b d e+a e^2\right ) x}{2 d^5 \left (d+e x^2\right )}+\frac {e^{3/2} \left (5 c d^2-7 b d e+9 a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 149, normalized size = 0.89
method | result | size |
default | \(\frac {e^{2} \left (\frac {\left (\frac {1}{2} a \,e^{2}-\frac {1}{2} d e b +\frac {1}{2} c \,d^{2}\right ) x}{e \,x^{2}+d}+\frac {\left (9 a \,e^{2}-7 d e b +5 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \sqrt {d e}}\right )}{d^{5}}-\frac {a}{7 x^{7} d^{2}}-\frac {-2 a e +b d}{5 d^{3} x^{5}}-\frac {3 a \,e^{2}-2 d e b +c \,d^{2}}{3 d^{4} x^{3}}+\frac {e \left (4 a \,e^{2}-3 d e b +2 c \,d^{2}\right )}{d^{5} x}\) | \(149\) |
risch | \(\frac {\frac {e^{2} \left (9 a \,e^{2}-7 d e b +5 c \,d^{2}\right ) x^{8}}{2 d^{5}}+\frac {e \left (9 a \,e^{2}-7 d e b +5 c \,d^{2}\right ) x^{6}}{3 d^{4}}-\frac {\left (9 a \,e^{2}-7 d e b +5 c \,d^{2}\right ) x^{4}}{15 d^{3}}+\frac {\left (9 a e -7 b d \right ) x^{2}}{35 d^{2}}-\frac {a}{7 d}}{x^{7} \left (e \,x^{2}+d \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (d^{11} \textit {\_Z}^{2}+81 a^{2} e^{7}-126 a b d \,e^{6}+90 a c \,d^{2} e^{5}+49 b^{2} d^{2} e^{5}-70 b c \,d^{3} e^{4}+25 c^{2} d^{4} e^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} d^{11}+162 a^{2} e^{7}-252 a b d \,e^{6}+180 a c \,d^{2} e^{5}+98 b^{2} d^{2} e^{5}-140 b c \,d^{3} e^{4}+50 c^{2} d^{4} e^{3}\right ) x +\left (-9 a \,d^{6} e^{3}+7 b \,d^{7} e^{2}-5 c \,d^{8} e \right ) \textit {\_R} \right )\right )}{4}\) | \(294\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 165, normalized size = 0.99 \begin {gather*} \frac {105 \, {\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} x^{8} + 70 \, {\left (5 \, c d^{3} e - 7 \, b d^{2} e^{2} + 9 \, a d e^{3}\right )} x^{6} - 30 \, a d^{4} - 14 \, {\left (5 \, c d^{4} - 7 \, b d^{3} e + 9 \, a d^{2} e^{2}\right )} x^{4} - 6 \, {\left (7 \, b d^{4} - 9 \, a d^{3} e\right )} x^{2}}{210 \, {\left (d^{5} x^{9} e + d^{6} x^{7}\right )}} + \frac {{\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, d^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 458, normalized size = 2.74 \begin {gather*} \left [\frac {1890 \, a x^{8} e^{4} - 140 \, c d^{4} x^{4} - 84 \, b d^{4} x^{2} - 60 \, a d^{4} + 105 \, {\left (5 \, c d^{3} x^{7} e + 9 \, a x^{9} e^{4} - {\left (7 \, b d x^{9} - 9 \, a d x^{7}\right )} e^{3} + {\left (5 \, c d^{2} x^{9} - 7 \, b d^{2} x^{7}\right )} e^{2}\right )} \sqrt {-\frac {e}{d}} \log \left (\frac {x^{2} e + 2 \, d x \sqrt {-\frac {e}{d}} - d}{x^{2} e + d}\right ) - 210 \, {\left (7 \, b d x^{8} - 6 \, a d x^{6}\right )} e^{3} + 14 \, {\left (75 \, c d^{2} x^{8} - 70 \, b d^{2} x^{6} - 18 \, a d^{2} x^{4}\right )} e^{2} + 4 \, {\left (175 \, c d^{3} x^{6} + 49 \, b d^{3} x^{4} + 27 \, a d^{3} x^{2}\right )} e}{420 \, {\left (d^{5} x^{9} e + d^{6} x^{7}\right )}}, \frac {945 \, a x^{8} e^{4} - 70 \, c d^{4} x^{4} - 42 \, b d^{4} x^{2} - 30 \, a d^{4} + \frac {105 \, {\left (5 \, c d^{3} x^{7} e + 9 \, a x^{9} e^{4} - {\left (7 \, b d x^{9} - 9 \, a d x^{7}\right )} e^{3} + {\left (5 \, c d^{2} x^{9} - 7 \, b d^{2} x^{7}\right )} e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}}}{\sqrt {d}} - 105 \, {\left (7 \, b d x^{8} - 6 \, a d x^{6}\right )} e^{3} + 7 \, {\left (75 \, c d^{2} x^{8} - 70 \, b d^{2} x^{6} - 18 \, a d^{2} x^{4}\right )} e^{2} + 2 \, {\left (175 \, c d^{3} x^{6} + 49 \, b d^{3} x^{4} + 27 \, a d^{3} x^{2}\right )} e}{210 \, {\left (d^{5} x^{9} e + d^{6} x^{7}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs.
\(2 (156) = 312\).
time = 1.35, size = 328, normalized size = 1.96 \begin {gather*} - \frac {\sqrt {- \frac {e^{3}}{d^{11}}} \cdot \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right ) \log {\left (- \frac {d^{6} \sqrt {- \frac {e^{3}}{d^{11}}} \cdot \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right )}{9 a e^{4} - 7 b d e^{3} + 5 c d^{2} e^{2}} + x \right )}}{4} + \frac {\sqrt {- \frac {e^{3}}{d^{11}}} \cdot \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right ) \log {\left (\frac {d^{6} \sqrt {- \frac {e^{3}}{d^{11}}} \cdot \left (9 a e^{2} - 7 b d e + 5 c d^{2}\right )}{9 a e^{4} - 7 b d e^{3} + 5 c d^{2} e^{2}} + x \right )}}{4} + \frac {- 30 a d^{4} + x^{8} \cdot \left (945 a e^{4} - 735 b d e^{3} + 525 c d^{2} e^{2}\right ) + x^{6} \cdot \left (630 a d e^{3} - 490 b d^{2} e^{2} + 350 c d^{3} e\right ) + x^{4} \left (- 126 a d^{2} e^{2} + 98 b d^{3} e - 70 c d^{4}\right ) + x^{2} \cdot \left (54 a d^{3} e - 42 b d^{4}\right )}{210 d^{6} x^{7} + 210 d^{5} e x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.41, size = 164, normalized size = 0.98 \begin {gather*} \frac {{\left (5 \, c d^{2} e^{2} - 7 \, b d e^{3} + 9 \, a e^{4}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {1}{2}\right )}}{2 \, d^{\frac {11}{2}}} + \frac {c d^{2} x e^{2} - b d x e^{3} + a x e^{4}}{2 \, {\left (x^{2} e + d\right )} d^{5}} + \frac {210 \, c d^{2} x^{6} e - 315 \, b d x^{6} e^{2} - 35 \, c d^{3} x^{4} + 420 \, a x^{6} e^{3} + 70 \, b d^{2} x^{4} e - 105 \, a d x^{4} e^{2} - 21 \, b d^{3} x^{2} + 42 \, a d^{2} x^{2} e - 15 \, a d^{3}}{105 \, d^{5} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 156, normalized size = 0.93 \begin {gather*} \frac {\frac {x^2\,\left (9\,a\,e-7\,b\,d\right )}{35\,d^2}-\frac {a}{7\,d}-\frac {x^4\,\left (5\,c\,d^2-7\,b\,d\,e+9\,a\,e^2\right )}{15\,d^3}+\frac {e\,x^6\,\left (5\,c\,d^2-7\,b\,d\,e+9\,a\,e^2\right )}{3\,d^4}+\frac {e^2\,x^8\,\left (5\,c\,d^2-7\,b\,d\,e+9\,a\,e^2\right )}{2\,d^5}}{e\,x^9+d\,x^7}+\frac {e^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (5\,c\,d^2-7\,b\,d\,e+9\,a\,e^2\right )}{2\,d^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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