Optimal. Leaf size=196 \[ \frac {3 b c-a d}{3 a^4 x^3}-\frac {c}{5 a^3 x^5}-\frac {a^2 e-3 a b d+6 b^2 c}{a^5 x}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-3 a^3 f+15 a^2 b e-35 a b^2 d+63 b^3 c\right )}{8 a^{11/2} \sqrt {b}}-\frac {x \left (-3 a^3 f+7 a^2 b e-11 a b^2 d+15 b^3 c\right )}{8 a^5 \left (a+b x^2\right )}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.35, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1805, 1802, 205} \begin {gather*} -\frac {x \left (7 a^2 b e-3 a^3 f-11 a b^2 d+15 b^3 c\right )}{8 a^5 \left (a+b x^2\right )}-\frac {x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (15 a^2 b e-3 a^3 f-35 a b^2 d+63 b^3 c\right )}{8 a^{11/2} \sqrt {b}}-\frac {a^2 e-3 a b d+6 b^2 c}{a^5 x}+\frac {3 b c-a d}{3 a^4 x^3}-\frac {c}{5 a^3 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 1802
Rule 1805
Rubi steps
\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^3} \, dx &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^4 \left (a+b x^2\right )^2}-\frac {\int \frac {-4 c+4 \left (\frac {b c}{a}-d\right ) x^2-\frac {4 \left (b^2 c-a b d+a^2 e\right ) x^4}{a^2}+\frac {3 \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 )^2} \, dx}{4 a}\\ &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^4 \left (a+b x^2\right )^2}-\frac {\left (15 b^3 c-11 a b^2 d+7 a^2 b e-3 a^3 f\right ) x}{8 a^5 \left (a+b x^2\right )}+\frac {\int \frac {8 c-8 \left (\frac {2 b c}{a}-d\right ) x^2+8 \left (\frac {3 b^2 c}{a^2}-\frac {2 b d}{a}+e\right ) x^4-\frac {\left (15 b^3 c-11 a b^2 d+7 a^2 b e-3 a^3 f\right ) x^6}{a^3}}{x^6 \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^4 \left (a+b x^2\right )^2}-\frac {\left (15 b^3 c-11 a b^2 d+7 a^2 b e-3 a^3 f\right ) x}{8 a^5 \left (a+b x^2\right )}+\frac {\int \left (\frac {8 c}{a x^6}+\frac {8 (-3 b c+a d)}{a^2 x^4}+\frac {8 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^2}+\frac {-63 b^3 c+35 a b^2 d-15 a^2 b e+3 a^3 f}{a^3 \left (a+b x^2\right )}\right ) \, dx}{8 a^2}\\ &=-\frac {c}{5 a^3 x^5}+\frac {3 b c-a d}{3 a^4 x^3}-\frac {6 b^2 c-3 a b d+a^2 e}{a^5 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^4 \left (a+b x^2\right )^2}-\frac {\left (15 b^3 c-11 a b^2 d+7 a^2 b e-3 a^3 f\right ) x}{8 a^5 \left (a+b x^2\right )}-\frac {\left (63 b^3 c-35 a b^2 d+15 a^2 b e-3 a^3 f\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^5}\\ &=-\frac {c}{5 a^3 x^5}+\frac {3 b c-a d}{3 a^4 x^3}-\frac {6 b^2 c-3 a b d+a^2 e}{a^5 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^4 \left (a+b x^2\right )^2}-\frac {\left (15 b^3 c-11 a b^2 d+7 a^2 b e-3 a^3 f\right ) x}{8 a^5 \left (a+b x^2\right )}-\frac {\left (63 b^3 c-35 a b^2 d+15 a^2 b e-3 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{11/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 196, normalized size = 1.00 \begin {gather*} \frac {3 b c-a d}{3 a^4 x^3}-\frac {c}{5 a^3 x^5}+\frac {a^2 (-e)+3 a b d-6 b^2 c}{a^5 x}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (3 a^3 f-15 a^2 b e+35 a b^2 d-63 b^3 c\right )}{8 a^{11/2} \sqrt {b}}+\frac {x \left (3 a^3 f-7 a^2 b e+11 a b^2 d-15 b^3 c\right )}{8 a^5 \left (a+b x^2\right )}+\frac {x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{4 a^4 \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.16, size = 628, normalized size = 3.20 \begin {gather*} \left [-\frac {30 \, {\left (63 \, a b^{5} c - 35 \, a^{2} b^{4} d + 15 \, a^{3} b^{3} e - 3 \, a^{4} b^{2} f\right )} x^{8} + 48 \, a^{5} b c + 50 \, {\left (63 \, a^{2} b^{4} c - 35 \, a^{3} b^{3} d + 15 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{6} + 16 \, {\left (63 \, a^{3} b^{3} c - 35 \, a^{4} b^{2} d + 15 \, a^{5} b e\right )} x^{4} - 16 \, {\left (9 \, a^{4} b^{2} c - 5 \, a^{5} b d\right )} x^{2} - 15 \, {\left ({\left (63 \, b^{5} c - 35 \, a b^{4} d + 15 \, a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{9} + 2 \, {\left (63 \, a b^{4} c - 35 \, a^{2} b^{3} d + 15 \, a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{7} + {\left (63 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 15 \, a^{4} b e - 3 \, a^{5} f\right )} x^{5}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{240 \, {\left (a^{6} b^{3} x^{9} + 2 \, a^{7} b^{2} x^{7} + a^{8} b x^{5}\right )}}, -\frac {15 \, {\left (63 \, a b^{5} c - 35 \, a^{2} b^{4} d + 15 \, a^{3} b^{3} e - 3 \, a^{4} b^{2} f\right )} x^{8} + 24 \, a^{5} b c + 25 \, {\left (63 \, a^{2} b^{4} c - 35 \, a^{3} b^{3} d + 15 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{6} + 8 \, {\left (63 \, a^{3} b^{3} c - 35 \, a^{4} b^{2} d + 15 \, a^{5} b e\right )} x^{4} - 8 \, {\left (9 \, a^{4} b^{2} c - 5 \, a^{5} b d\right )} x^{2} + 15 \, {\left ({\left (63 \, b^{5} c - 35 \, a b^{4} d + 15 \, a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{9} + 2 \, {\left (63 \, a b^{4} c - 35 \, a^{2} b^{3} d + 15 \, a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{7} + {\left (63 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 15 \, a^{4} b e - 3 \, a^{5} f\right )} x^{5}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{120 \, {\left (a^{6} b^{3} x^{9} + 2 \, a^{7} b^{2} x^{7} + a^{8} b x^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 198, normalized size = 1.01 \begin {gather*} -\frac {{\left (63 \, b^{3} c - 35 \, a b^{2} d - 3 \, a^{3} f + 15 \, a^{2} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{5}} - \frac {15 \, b^{4} c x^{3} - 11 \, a b^{3} d x^{3} - 3 \, a^{3} b f x^{3} + 7 \, a^{2} b^{2} x^{3} e + 17 \, a b^{3} c x - 13 \, a^{2} b^{2} d x - 5 \, a^{4} f x + 9 \, a^{3} b x e}{8 \, {\left (b x^{2} + a\right )}^{2} a^{5}} - \frac {90 \, b^{2} c x^{4} - 45 \, a b d x^{4} + 15 \, a^{2} x^{4} e - 15 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{5} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 300, normalized size = 1.53 \begin {gather*} \frac {3 b f \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{2}}-\frac {7 b^{2} e \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{3}}+\frac {11 b^{3} d \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{4}}-\frac {15 b^{4} c \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{5}}+\frac {5 f x}{8 \left (b \,x^{2}+a \right )^{2} a}-\frac {9 b e x}{8 \left (b \,x^{2}+a \right )^{2} a^{2}}+\frac {13 b^{2} d x}{8 \left (b \,x^{2}+a \right )^{2} a^{3}}-\frac {17 b^{3} c x}{8 \left (b \,x^{2}+a \right )^{2} a^{4}}+\frac {3 f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{2}}-\frac {15 b e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{3}}+\frac {35 b^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{4}}-\frac {63 b^{3} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{5}}-\frac {e}{a^{3} x}+\frac {3 b d}{a^{4} x}-\frac {6 b^{2} c}{a^{5} x}-\frac {d}{3 a^{3} x^{3}}+\frac {b c}{a^{4} x^{3}}-\frac {c}{5 a^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 202, normalized size = 1.03 \begin {gather*} -\frac {15 \, {\left (63 \, b^{4} c - 35 \, a b^{3} d + 15 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 25 \, {\left (63 \, a b^{3} c - 35 \, a^{2} b^{2} d + 15 \, a^{3} b e - 3 \, a^{4} f\right )} x^{6} + 24 \, a^{4} c + 8 \, {\left (63 \, a^{2} b^{2} c - 35 \, a^{3} b d + 15 \, a^{4} e\right )} x^{4} - 8 \, {\left (9 \, a^{3} b c - 5 \, a^{4} d\right )} x^{2}}{120 \, {\left (a^{5} b^{2} x^{9} + 2 \, a^{6} b x^{7} + a^{7} x^{5}\right )}} - \frac {{\left (63 \, b^{3} c - 35 \, a b^{2} d + 15 \, a^{2} b e - 3 \, a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 192, normalized size = 0.98 \begin {gather*} -\frac {\frac {c}{5\,a}+\frac {5\,x^6\,\left (-3\,f\,a^3+15\,e\,a^2\,b-35\,d\,a\,b^2+63\,c\,b^3\right )}{24\,a^4}+\frac {x^2\,\left (5\,a\,d-9\,b\,c\right )}{15\,a^2}+\frac {x^4\,\left (15\,e\,a^2-35\,d\,a\,b+63\,c\,b^2\right )}{15\,a^3}+\frac {b\,x^8\,\left (-3\,f\,a^3+15\,e\,a^2\,b-35\,d\,a\,b^2+63\,c\,b^3\right )}{8\,a^5}}{a^2\,x^5+2\,a\,b\,x^7+b^2\,x^9}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-3\,f\,a^3+15\,e\,a^2\,b-35\,d\,a\,b^2+63\,c\,b^3\right )}{8\,a^{11/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 150.75, size = 284, normalized size = 1.45 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{11} b}} \left (3 a^{3} f - 15 a^{2} b e + 35 a b^{2} d - 63 b^{3} c\right ) \log {\left (- a^{6} \sqrt {- \frac {1}{a^{11} b}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{11} b}} \left (3 a^{3} f - 15 a^{2} b e + 35 a b^{2} d - 63 b^{3} c\right ) \log {\left (a^{6} \sqrt {- \frac {1}{a^{11} b}} + x \right )}}{16} + \frac {- 24 a^{4} c + x^{8} \left (45 a^{3} b f - 225 a^{2} b^{2} e + 525 a b^{3} d - 945 b^{4} c\right ) + x^{6} \left (75 a^{4} f - 375 a^{3} b e + 875 a^{2} b^{2} d - 1575 a b^{3} c\right ) + x^{4} \left (- 120 a^{4} e + 280 a^{3} b d - 504 a^{2} b^{2} c\right ) + x^{2} \left (- 40 a^{4} d + 72 a^{3} b c\right )}{120 a^{7} x^{5} + 240 a^{6} b x^{7} + 120 a^{5} b^{2} x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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