Optimal. Leaf size=152 \[ -\frac {c}{5 a^2 x^5}+\frac {2 b c-a d}{3 a^3 x^3}-\frac {3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^4 \left (a+b x^2\right )}-\frac {\left (7 b^3 c-5 a b^2 d+3 a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2} \sqrt {b}} \]
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Rubi [A]
time = 0.15, antiderivative size = 152, 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}{3 a^3 x^3}-\frac {c}{5 a^2 x^5}-\frac {a^2 e-2 a b d+3 b^2 c}{a^4 x}-\frac {\text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+3 a^2 b e-5 a b^2 d+7 b^3 c\right )}{2 a^{9/2} \sqrt {b}}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^4 \left (a+b x^2\right )} \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^6 \left (a+b x^2\right )^2} \, dx &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^4 \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 {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{a^3}}{x^6 \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^4 \left (a+b x^2\right )}-\frac {\int \left (-\frac {2 c}{a x^6}-\frac {2 (-2 b c+a d)}{a^2 x^4}-\frac {2 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^2}+\frac {7 b^3 c-5 a b^2 d+3 a^2 b e-a^3 f}{a^3 \left (a+b x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac {c}{5 a^2 x^5}+\frac {2 b c-a d}{3 a^3 x^3}-\frac {3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^4 \left (a+b x^2\right )}-\frac {\left (7 b^3 c-5 a b^2 d+3 a^2 b e-a^3 f\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^4}\\ &=-\frac {c}{5 a^2 x^5}+\frac {2 b c-a d}{3 a^3 x^3}-\frac {3 b^2 c-2 a b d+a^2 e}{a^4 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^4 \left (a+b x^2\right )}-\frac {\left (7 b^3 c-5 a b^2 d+3 a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 151, normalized size = 0.99 \begin {gather*} -\frac {c}{5 a^2 x^5}+\frac {2 b c-a d}{3 a^3 x^3}+\frac {-3 b^2 c+2 a b d-a^2 e}{a^4 x}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{2 a^4 \left (a+b x^2\right )}+\frac {\left (-7 b^3 c+5 a b^2 d-3 a^2 b e+a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{9/2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 137, normalized size = 0.90
method | result | size |
default | \(\frac {\frac {\left (\frac {1}{2} a^{3} f -\frac {1}{2} a^{2} b e +\frac {1}{2} a \,b^{2} d -\frac {1}{2} b^{3} c \right ) x}{b \,x^{2}+a}+\frac {\left (a^{3} f -3 a^{2} b e +5 a \,b^{2} d -7 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{a^{4}}-\frac {c}{5 a^{2} x^{5}}-\frac {a d -2 b c}{3 a^{3} x^{3}}-\frac {a^{2} e -2 a b d +3 b^{2} c}{a^{4} x}\) | \(137\) |
risch | \(\frac {\frac {\left (a^{3} f -3 a^{2} b e +5 a \,b^{2} d -7 b^{3} c \right ) x^{6}}{2 a^{4}}-\frac {\left (3 a^{2} e -5 a b d +7 b^{2} c \right ) x^{4}}{3 a^{3}}-\frac {\left (5 a d -7 b c \right ) x^{2}}{15 a^{2}}-\frac {c}{5 a}}{x^{5} \left (b \,x^{2}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{9} \textit {\_Z}^{2} b +a^{6} f^{2}-6 a^{5} b e f +10 a^{4} b^{2} d f +9 a^{4} b^{2} e^{2}-14 a^{3} b^{3} c f -30 a^{3} b^{3} d e +42 a^{2} b^{4} c e +25 a^{2} b^{4} d^{2}-70 a c d \,b^{5}+49 c^{2} b^{6}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{9} b +2 a^{6} f^{2}-12 a^{5} b e f +20 a^{4} b^{2} d f +18 a^{4} b^{2} e^{2}-28 a^{3} b^{3} c f -60 a^{3} b^{3} d e +84 a^{2} b^{4} c e +50 a^{2} b^{4} d^{2}-140 a c d \,b^{5}+98 c^{2} b^{6}\right ) x +\left (-a^{8} f +3 a^{7} b e -5 a^{6} b^{2} d +7 a^{5} b^{3} c \right ) \textit {\_R} \right )\right )}{4}\) | \(351\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 154, normalized size = 1.01 \begin {gather*} -\frac {15 \, {\left (7 \, b^{3} c - 5 \, a b^{2} d - a^{3} f + 3 \, a^{2} b e\right )} x^{6} + 10 \, {\left (7 \, a b^{2} c - 5 \, a^{2} b d + 3 \, a^{3} e\right )} x^{4} + 6 \, a^{3} c - 2 \, {\left (7 \, a^{2} b c - 5 \, a^{3} d\right )} x^{2}}{30 \, {\left (a^{4} b x^{7} + a^{5} x^{5}\right )}} - \frac {{\left (7 \, b^{3} c - 5 \, a b^{2} d - a^{3} f + 3 \, a^{2} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.66, size = 470, normalized size = 3.09 \begin {gather*} \left [-\frac {30 \, {\left (7 \, a b^{4} c - 5 \, a^{2} b^{3} d - a^{4} b f\right )} x^{6} + 12 \, a^{4} b c + 20 \, {\left (7 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d\right )} x^{4} - 4 \, {\left (7 \, a^{3} b^{2} c - 5 \, a^{4} b d\right )} x^{2} + 15 \, {\left ({\left (7 \, b^{4} c - 5 \, a b^{3} d - a^{3} b f\right )} x^{7} + {\left (7 \, a b^{3} c - 5 \, a^{2} b^{2} d - a^{4} f\right )} x^{5} + 3 \, {\left (a^{2} b^{2} x^{7} + a^{3} b x^{5}\right )} e\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 30 \, {\left (3 \, a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4}\right )} e}{60 \, {\left (a^{5} b^{2} x^{7} + a^{6} b x^{5}\right )}}, -\frac {15 \, {\left (7 \, a b^{4} c - 5 \, a^{2} b^{3} d - a^{4} b f\right )} x^{6} + 6 \, a^{4} b c + 10 \, {\left (7 \, a^{2} b^{3} c - 5 \, a^{3} b^{2} d\right )} x^{4} - 2 \, {\left (7 \, a^{3} b^{2} c - 5 \, a^{4} b d\right )} x^{2} + 15 \, {\left ({\left (7 \, b^{4} c - 5 \, a b^{3} d - a^{3} b f\right )} x^{7} + {\left (7 \, a b^{3} c - 5 \, a^{2} b^{2} d - a^{4} f\right )} x^{5} + 3 \, {\left (a^{2} b^{2} x^{7} + a^{3} b x^{5}\right )} e\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 15 \, {\left (3 \, a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4}\right )} e}{30 \, {\left (a^{5} b^{2} x^{7} + a^{6} b x^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 20.56, size = 226, normalized size = 1.49 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{9} b}} \left (a^{3} f - 3 a^{2} b e + 5 a b^{2} d - 7 b^{3} c\right ) \log {\left (- a^{5} \sqrt {- \frac {1}{a^{9} b}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{9} b}} \left (a^{3} f - 3 a^{2} b e + 5 a b^{2} d - 7 b^{3} c\right ) \log {\left (a^{5} \sqrt {- \frac {1}{a^{9} b}} + x \right )}}{4} + \frac {- 6 a^{3} c + x^{6} \cdot \left (15 a^{3} f - 45 a^{2} b e + 75 a b^{2} d - 105 b^{3} c\right ) + x^{4} \left (- 30 a^{3} e + 50 a^{2} b d - 70 a b^{2} c\right ) + x^{2} \left (- 10 a^{3} d + 14 a^{2} b c\right )}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.94, size = 151, normalized size = 0.99 \begin {gather*} -\frac {{\left (7 \, b^{3} c - 5 \, a b^{2} d - a^{3} f + 3 \, a^{2} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{4}} - \frac {b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{2 \, {\left (b x^{2} + a\right )} a^{4}} - \frac {45 \, b^{2} c x^{4} - 30 \, a b d x^{4} + 15 \, a^{2} x^{4} e - 10 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{4} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.00, size = 145, normalized size = 0.95 \begin {gather*} -\frac {\frac {c}{5\,a}+\frac {x^6\,\left (-f\,a^3+3\,e\,a^2\,b-5\,d\,a\,b^2+7\,c\,b^3\right )}{2\,a^4}+\frac {x^2\,\left (5\,a\,d-7\,b\,c\right )}{15\,a^2}+\frac {x^4\,\left (3\,e\,a^2-5\,d\,a\,b+7\,c\,b^2\right )}{3\,a^3}}{b\,x^7+a\,x^5}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-f\,a^3+3\,e\,a^2\,b-5\,d\,a\,b^2+7\,c\,b^3\right )}{2\,a^{9/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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