Optimal. Leaf size=189 \[ -\frac {c}{7 a^2 x^7}+\frac {2 b c-a d}{5 a^3 x^5}-\frac {3 b^2 c-2 a b d+a^2 e}{3 a^4 x^3}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}+\frac {\sqrt {b} \left (9 b^3 c-7 a b^2 d+5 a^2 b e-3 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{11/2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1819, 1816,
211} \begin {gather*} \frac {2 b c-a d}{5 a^3 x^5}-\frac {c}{7 a^2 x^7}-\frac {a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac {\sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-3 a^3 f+5 a^2 b e-7 a b^2 d+9 b^3 c\right )}{2 a^{11/2}}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{a^5 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1816
Rule 1819
Rubi steps
\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )^2} \, dx &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}-\frac {\int \frac {-2 c+2 \left (\frac {b c}{a}-d\right ) x^2-\frac {2 \left (b^2 c-a b d+a^2 e\right ) x^4}{a^2}+\frac {2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{a^3}-\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^8}{a^4}}{x^8 \left (a+b x^2\right )} \, dx}{2 a}\\ &=\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}-\frac {\int \left (-\frac {2 c}{a x^8}-\frac {2 (-2 b c+a d)}{a^2 x^6}-\frac {2 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^4}-\frac {2 \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^4 x^2}+\frac {b \left (-9 b^3 c+7 a b^2 d-5 a^2 b e+3 a^3 f\right )}{a^4 \left (a+b x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac {c}{7 a^2 x^7}+\frac {2 b c-a d}{5 a^3 x^5}-\frac {3 b^2 c-2 a b d+a^2 e}{3 a^4 x^3}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}+\frac {\left (b \left (9 b^3 c-7 a b^2 d+5 a^2 b e-3 a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^5}\\ &=-\frac {c}{7 a^2 x^7}+\frac {2 b c-a d}{5 a^3 x^5}-\frac {3 b^2 c-2 a b d+a^2 e}{3 a^4 x^3}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}+\frac {b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}+\frac {\sqrt {b} \left (9 b^3 c-7 a b^2 d+5 a^2 b e-3 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{11/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 190, normalized size = 1.01 \begin {gather*} -\frac {c}{7 a^2 x^7}+\frac {2 b c-a d}{5 a^3 x^5}+\frac {-3 b^2 c+2 a b d-a^2 e}{3 a^4 x^3}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}-\frac {\sqrt {b} \left (-9 b^3 c+7 a b^2 d-5 a^2 b e+3 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 174, normalized size = 0.92
method | result | size |
default | \(-\frac {b \left (\frac {\left (\frac {1}{2} a^{3} f -\frac {1}{2} a^{2} b e +\frac {1}{2} a \,b^{2} d -\frac {1}{2} b^{3} c \right ) x}{b \,x^{2}+a}+\frac {\left (3 a^{3} f -5 a^{2} b e +7 a \,b^{2} d -9 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{5}}-\frac {c}{7 a^{2} x^{7}}-\frac {a d -2 b c}{5 a^{3} x^{5}}-\frac {a^{2} e -2 a b d +3 b^{2} c}{3 a^{4} x^{3}}-\frac {a^{3} f -2 a^{2} b e +3 a \,b^{2} d -4 b^{3} c}{a^{5} x}\) | \(174\) |
risch | \(\frac {-\frac {b \left (3 a^{3} f -5 a^{2} b e +7 a \,b^{2} d -9 b^{3} c \right ) x^{8}}{2 a^{5}}-\frac {\left (3 a^{3} f -5 a^{2} b e +7 a \,b^{2} d -9 b^{3} c \right ) x^{6}}{3 a^{4}}-\frac {\left (5 a^{2} e -7 a b d +9 b^{2} c \right ) x^{4}}{15 a^{3}}-\frac {\left (7 a d -9 b c \right ) x^{2}}{35 a^{2}}-\frac {c}{7 a}}{x^{7} \left (b \,x^{2}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{11} \textit {\_Z}^{2}+9 a^{6} b \,f^{2}-30 a^{5} b^{2} e f +42 a^{4} b^{3} d f +25 a^{4} b^{3} e^{2}-54 a^{3} b^{4} c f -70 a^{3} b^{4} d e +90 a^{2} b^{5} c e +49 a^{2} b^{5} d^{2}-126 a \,b^{6} c d +81 b^{7} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{11}+18 a^{6} b \,f^{2}-60 a^{5} b^{2} e f +84 a^{4} b^{3} d f +50 a^{4} b^{3} e^{2}-108 a^{3} b^{4} c f -140 a^{3} b^{4} d e +180 a^{2} b^{5} c e +98 a^{2} b^{5} d^{2}-252 a \,b^{6} c d +162 b^{7} c^{2}\right ) x +\left (3 a^{9} f -5 a^{8} b e +7 a^{7} b^{2} d -9 a^{6} b^{3} c \right ) \textit {\_R} \right )\right )}{4}\) | \(393\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 198, normalized size = 1.05 \begin {gather*} \frac {105 \, {\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f + 5 \, a^{2} b^{2} e\right )} x^{8} + 70 \, {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d - 3 \, a^{4} f + 5 \, a^{3} b e\right )} x^{6} - 30 \, a^{4} c - 14 \, {\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d + 5 \, a^{4} e\right )} x^{4} + 6 \, {\left (9 \, a^{3} b c - 7 \, a^{4} d\right )} x^{2}}{210 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}} + \frac {{\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f + 5 \, a^{2} b^{2} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.62, size = 524, normalized size = 2.77 \begin {gather*} \left [\frac {210 \, {\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f\right )} x^{8} + 140 \, {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d - 3 \, a^{4} f\right )} x^{6} - 60 \, a^{4} c - 28 \, {\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d\right )} x^{4} + 12 \, {\left (9 \, a^{3} b c - 7 \, a^{4} d\right )} x^{2} + 105 \, {\left ({\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f\right )} x^{9} + {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d - 3 \, a^{4} f\right )} x^{7} + 5 \, {\left (a^{2} b^{2} x^{9} + a^{3} b x^{7}\right )} e\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 70 \, {\left (15 \, a^{2} b^{2} x^{8} + 10 \, a^{3} b x^{6} - 2 \, a^{4} x^{4}\right )} e}{420 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}, \frac {105 \, {\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f\right )} x^{8} + 70 \, {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d - 3 \, a^{4} f\right )} x^{6} - 30 \, a^{4} c - 14 \, {\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d\right )} x^{4} + 6 \, {\left (9 \, a^{3} b c - 7 \, a^{4} d\right )} x^{2} + 105 \, {\left ({\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f\right )} x^{9} + {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d - 3 \, a^{4} f\right )} x^{7} + 5 \, {\left (a^{2} b^{2} x^{9} + a^{3} b x^{7}\right )} e\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 35 \, {\left (15 \, a^{2} b^{2} x^{8} + 10 \, a^{3} b x^{6} - 2 \, a^{4} x^{4}\right )} e}{210 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}\right ] \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 = 1.41, size = 201, normalized size = 1.06 \begin {gather*} \frac {{\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f + 5 \, a^{2} b^{2} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{5}} + \frac {b^{4} c x - a b^{3} d x - a^{3} b f x + a^{2} b^{2} x e}{2 \, {\left (b x^{2} + a\right )} a^{5}} + \frac {420 \, b^{3} c x^{6} - 315 \, a b^{2} d x^{6} - 105 \, a^{3} f x^{6} + 210 \, a^{2} b x^{6} e - 105 \, a b^{2} c x^{4} + 70 \, a^{2} b d x^{4} - 35 \, a^{3} x^{4} e + 42 \, a^{2} b c x^{2} - 21 \, a^{3} d x^{2} - 15 \, a^{3} c}{105 \, a^{5} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.99, size = 181, normalized size = 0.96 \begin {gather*} \frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-3\,f\,a^3+5\,e\,a^2\,b-7\,d\,a\,b^2+9\,c\,b^3\right )}{2\,a^{11/2}}-\frac {\frac {c}{7\,a}-\frac {x^6\,\left (-3\,f\,a^3+5\,e\,a^2\,b-7\,d\,a\,b^2+9\,c\,b^3\right )}{3\,a^4}+\frac {x^2\,\left (7\,a\,d-9\,b\,c\right )}{35\,a^2}+\frac {x^4\,\left (5\,e\,a^2-7\,d\,a\,b+9\,c\,b^2\right )}{15\,a^3}-\frac {b\,x^8\,\left (-3\,f\,a^3+5\,e\,a^2\,b-7\,d\,a\,b^2+9\,c\,b^3\right )}{2\,a^5}}{b\,x^9+a\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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