Optimal. Leaf size=196 \[ -\frac {c}{5 a^3 x^5}+\frac {3 b c-a d}{3 a^4 x^3}-\frac {6 b^2 c-3 a b d+a^2 e}{a^5 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^4 \left (a+b x^2\right )^2}-\frac {\left (15 b^3 c-11 a b^2 d+7 a^2 b e-3 a^3 f\right ) x}{8 a^5 \left (a+b x^2\right )}-\frac {\left (63 b^3 c-35 a b^2 d+15 a^2 b e-3 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{11/2} \sqrt {b}} \]
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Rubi [A]
time = 0.24, 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 = {1819, 1816,
211} \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 {\text {ArcTan}\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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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 )^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 = 0.08, size = 196, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 176, normalized size = 0.90
method | result | size |
default | \(\frac {\frac {\left (\frac {3}{8} a^{3} b f -\frac {7}{8} a^{2} e \,b^{2}+\frac {11}{8} a d \,b^{3}-\frac {15}{8} c \,b^{4}\right ) x^{3}+\frac {a \left (5 a^{3} f -9 a^{2} b e +13 a \,b^{2} d -17 b^{3} c \right ) x}{8}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (3 a^{3} f -15 a^{2} b e +35 a \,b^{2} d -63 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}}{a^{5}}-\frac {c}{5 a^{3} x^{5}}-\frac {a d -3 b c}{3 a^{4} x^{3}}-\frac {a^{2} e -3 a b d +6 b^{2} c}{a^{5} x}\) | \(176\) |
risch | \(\frac {\frac {b \left (3 a^{3} f -15 a^{2} b e +35 a \,b^{2} d -63 b^{3} c \right ) x^{8}}{8 a^{5}}+\frac {5 \left (3 a^{3} f -15 a^{2} b e +35 a \,b^{2} d -63 b^{3} c \right ) x^{6}}{24 a^{4}}-\frac {\left (15 a^{2} e -35 a b d +63 b^{2} c \right ) x^{4}}{15 a^{3}}-\frac {\left (5 a d -9 b c \right ) x^{2}}{15 a^{2}}-\frac {c}{5 a}}{x^{5} \left (b \,x^{2}+a \right )^{2}}-\frac {3 \ln \left (-\sqrt {-a b}\, x +a \right ) f}{16 \sqrt {-a b}\, a^{2}}+\frac {15 \ln \left (-\sqrt {-a b}\, x +a \right ) b e}{16 \sqrt {-a b}\, a^{3}}-\frac {35 \ln \left (-\sqrt {-a b}\, x +a \right ) b^{2} d}{16 \sqrt {-a b}\, a^{4}}+\frac {63 \ln \left (-\sqrt {-a b}\, x +a \right ) b^{3} c}{16 \sqrt {-a b}\, a^{5}}+\frac {3 \ln \left (-\sqrt {-a b}\, x -a \right ) f}{16 \sqrt {-a b}\, a^{2}}-\frac {15 \ln \left (-\sqrt {-a b}\, x -a \right ) b e}{16 \sqrt {-a b}\, a^{3}}+\frac {35 \ln \left (-\sqrt {-a b}\, x -a \right ) b^{2} d}{16 \sqrt {-a b}\, a^{4}}-\frac {63 \ln \left (-\sqrt {-a b}\, x -a \right ) b^{3} c}{16 \sqrt {-a b}\, a^{5}}\) | \(350\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 206, normalized size = 1.05 \begin {gather*} -\frac {15 \, {\left (63 \, b^{4} c - 35 \, a b^{3} d - 3 \, a^{3} b f + 15 \, a^{2} b^{2} e\right )} x^{8} + 25 \, {\left (63 \, a b^{3} c - 35 \, a^{2} b^{2} d - 3 \, a^{4} f + 15 \, a^{3} b e\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 - 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 5.35, size = 668, normalized size = 3.41 \begin {gather*} \left [-\frac {30 \, {\left (63 \, a b^{5} c - 35 \, a^{2} b^{4} d - 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 - 3 \, a^{5} b f\right )} x^{6} + 112 \, {\left (9 \, a^{3} b^{3} c - 5 \, a^{4} b^{2} d\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 - 3 \, a^{3} b^{2} f\right )} x^{9} + 2 \, {\left (63 \, a b^{4} c - 35 \, a^{2} b^{3} d - 3 \, a^{4} b f\right )} x^{7} + {\left (63 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d - 3 \, a^{5} f\right )} x^{5} + 15 \, {\left (a^{2} b^{3} x^{9} + 2 \, a^{3} b^{2} x^{7} + a^{4} 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 (15 \, a^{3} b^{3} x^{8} + 25 \, a^{4} b^{2} x^{6} + 8 \, a^{5} b x^{4}\right )} e}{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 - 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 - 3 \, a^{5} b f\right )} x^{6} + 56 \, {\left (9 \, a^{3} b^{3} c - 5 \, a^{4} b^{2} d\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 - 3 \, a^{3} b^{2} f\right )} x^{9} + 2 \, {\left (63 \, a b^{4} c - 35 \, a^{2} b^{3} d - 3 \, a^{4} b f\right )} x^{7} + {\left (63 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d - 3 \, a^{5} f\right )} x^{5} + 15 \, {\left (a^{2} b^{3} x^{9} + 2 \, a^{3} b^{2} x^{7} + a^{4} b x^{5}\right )} e\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 15 \, {\left (15 \, a^{3} b^{3} x^{8} + 25 \, a^{4} b^{2} x^{6} + 8 \, a^{5} b x^{4}\right )} e}{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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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 = 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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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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