Optimal. Leaf size=168 \[ \frac {3 b c-a d}{a^4 x}-\frac {c}{3 a^3 x^3}+\frac {x \left (\frac {b^2 c}{a^2}-\frac {b d}{a}-\frac {a f}{b}+e\right )}{4 a \left (a+b x^2\right )^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f+3 a^2 b e-15 a b^2 d+35 b^3 c\right )}{8 a^{9/2} b^{3/2}}+\frac {x \left (a^3 f+3 a^2 b e-7 a b^2 d+11 b^3 c\right )}{8 a^4 b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.24, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1805, 1259, 1261, 205} \begin {gather*} \frac {x \left (3 a^2 b e+a^3 f-7 a b^2 d+11 b^3 c\right )}{8 a^4 b \left (a+b x^2\right )}+\frac {x \left (\frac {b^2 c}{a^2}-\frac {b d}{a}-\frac {a f}{b}+e\right )}{4 a \left (a+b x^2\right )^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (3 a^2 b e+a^3 f-15 a b^2 d+35 b^3 c\right )}{8 a^{9/2} b^{3/2}}+\frac {3 b c-a d}{a^4 x}-\frac {c}{3 a^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 1259
Rule 1261
Rule 1805
Rubi steps
\begin {align*} \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^3} \, dx &=\frac {\left (\frac {b^2 c}{a^2}-\frac {b d}{a}+e-\frac {a f}{b}\right ) x}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {-4 c+4 \left (\frac {b c}{a}-d\right ) x^2+\left (-\frac {3 b^2 c}{a^2}+\frac {3 b d}{a}-3 e-\frac {a f}{b}\right ) x^4}{x^4 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac {\left (\frac {b^2 c}{a^2}-\frac {b d}{a}+e-\frac {a f}{b}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac {\left (11 b^3 c-7 a b^2 d+3 a^2 b e+a^3 f\right ) x}{8 a^4 b \left (a+b x^2\right )}-\frac {\int \frac {-8 a^2 b^2 c+8 a b^2 (2 b c-a d) x^2-b \left (11 b^3 c-7 a b^2 d+3 a^2 b e+a^3 f\right ) x^4}{x^4 \left (a+b x^2\right )} \, dx}{8 a^4 b^2}\\ &=\frac {\left (\frac {b^2 c}{a^2}-\frac {b d}{a}+e-\frac {a f}{b}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac {\left (11 b^3 c-7 a b^2 d+3 a^2 b e+a^3 f\right ) x}{8 a^4 b \left (a+b x^2\right )}-\frac {\int \left (-\frac {8 a b^2 c}{x^4}+\frac {8 b^2 (3 b c-a d)}{x^2}-\frac {b \left (35 b^3 c-15 a b^2 d+3 a^2 b e+a^3 f\right )}{a+b x^2}\right ) \, dx}{8 a^4 b^2}\\ &=-\frac {c}{3 a^3 x^3}+\frac {3 b c-a d}{a^4 x}+\frac {\left (\frac {b^2 c}{a^2}-\frac {b d}{a}+e-\frac {a f}{b}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac {\left (11 b^3 c-7 a b^2 d+3 a^2 b e+a^3 f\right ) x}{8 a^4 b \left (a+b x^2\right )}+\frac {\left (35 b^3 c-15 a b^2 d+3 a^2 b e+a^3 f\right ) \int \frac {1}{a+b x^2} \, dx}{8 a^4 b}\\ &=-\frac {c}{3 a^3 x^3}+\frac {3 b c-a d}{a^4 x}+\frac {\left (\frac {b^2 c}{a^2}-\frac {b d}{a}+e-\frac {a f}{b}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac {\left (11 b^3 c-7 a b^2 d+3 a^2 b e+a^3 f\right ) x}{8 a^4 b \left (a+b x^2\right )}+\frac {\left (35 b^3 c-15 a b^2 d+3 a^2 b e+a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{9/2} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 169, normalized size = 1.01 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f+3 a^2 b e-15 a b^2 d+35 b^3 c\right )}{8 a^{9/2} b^{3/2}}+\frac {-3 a^4 f x^4+a^3 b \left (3 x^2 \left (-8 d+5 e x^2+f x^4\right )-8 c\right )+a^2 b^2 x^2 \left (56 c-75 d x^2+9 e x^4\right )+5 a b^3 x^4 \left (35 c-9 d x^2\right )+105 b^4 c x^6}{24 a^4 b x^3 \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^4 \left (a+b x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.87, size = 570, normalized size = 3.39 \begin {gather*} \left [-\frac {16 \, a^{4} b^{2} c - 6 \, {\left (35 \, a b^{5} c - 15 \, a^{2} b^{4} d + 3 \, a^{3} b^{3} e + a^{4} b^{2} f\right )} x^{6} - 2 \, {\left (175 \, a^{2} b^{4} c - 75 \, a^{3} b^{3} d + 15 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{4} - 16 \, {\left (7 \, a^{3} b^{3} c - 3 \, a^{4} b^{2} d\right )} x^{2} + 3 \, {\left ({\left (35 \, b^{5} c - 15 \, a b^{4} d + 3 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{7} + 2 \, {\left (35 \, a b^{4} c - 15 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e + a^{4} b f\right )} x^{5} + {\left (35 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 3 \, a^{4} b e + a^{5} f\right )} x^{3}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{48 \, {\left (a^{5} b^{4} x^{7} + 2 \, a^{6} b^{3} x^{5} + a^{7} b^{2} x^{3}\right )}}, -\frac {8 \, a^{4} b^{2} c - 3 \, {\left (35 \, a b^{5} c - 15 \, a^{2} b^{4} d + 3 \, a^{3} b^{3} e + a^{4} b^{2} f\right )} x^{6} - {\left (175 \, a^{2} b^{4} c - 75 \, a^{3} b^{3} d + 15 \, a^{4} b^{2} e - 3 \, a^{5} b f\right )} x^{4} - 8 \, {\left (7 \, a^{3} b^{3} c - 3 \, a^{4} b^{2} d\right )} x^{2} - 3 \, {\left ({\left (35 \, b^{5} c - 15 \, a b^{4} d + 3 \, a^{2} b^{3} e + a^{3} b^{2} f\right )} x^{7} + 2 \, {\left (35 \, a b^{4} c - 15 \, a^{2} b^{3} d + 3 \, a^{3} b^{2} e + a^{4} b f\right )} x^{5} + {\left (35 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 3 \, a^{4} b e + a^{5} f\right )} x^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{24 \, {\left (a^{5} b^{4} x^{7} + 2 \, a^{6} b^{3} x^{5} + a^{7} b^{2} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 170, normalized size = 1.01 \begin {gather*} \frac {{\left (35 \, b^{3} c - 15 \, a b^{2} d + a^{3} f + 3 \, a^{2} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4} b} + \frac {11 \, b^{4} c x^{3} - 7 \, a b^{3} d x^{3} + a^{3} b f x^{3} + 3 \, a^{2} b^{2} x^{3} e + 13 \, a b^{3} c x - 9 \, a^{2} b^{2} d x - a^{4} f x + 5 \, a^{3} b x e}{8 \, {\left (b x^{2} + a\right )}^{2} a^{4} b} + \frac {9 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 264, normalized size = 1.57 \begin {gather*} \frac {f \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a}+\frac {3 b e \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{2}}-\frac {7 b^{2} d \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{3}}+\frac {11 b^{3} c \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} a^{4}}+\frac {5 e x}{8 \left (b \,x^{2}+a \right )^{2} a}-\frac {9 b d x}{8 \left (b \,x^{2}+a \right )^{2} a^{2}}+\frac {13 b^{2} c x}{8 \left (b \,x^{2}+a \right )^{2} a^{3}}-\frac {f x}{8 \left (b \,x^{2}+a \right )^{2} b}+\frac {f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a b}+\frac {3 e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{2}}-\frac {15 b d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{3}}+\frac {35 b^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{4}}-\frac {d}{a^{3} x}+\frac {3 b c}{a^{4} x}-\frac {c}{3 a^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.07, size = 181, normalized size = 1.08 \begin {gather*} \frac {3 \, {\left (35 \, b^{4} c - 15 \, a b^{3} d + 3 \, a^{2} b^{2} e + a^{3} b f\right )} x^{6} - 8 \, a^{3} b c + {\left (175 \, a b^{3} c - 75 \, a^{2} b^{2} d + 15 \, a^{3} b e - 3 \, a^{4} f\right )} x^{4} + 8 \, {\left (7 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} x^{2}}{24 \, {\left (a^{4} b^{3} x^{7} + 2 \, a^{5} b^{2} x^{5} + a^{6} b x^{3}\right )}} + \frac {{\left (35 \, b^{3} c - 15 \, a b^{2} d + 3 \, a^{2} b e + a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 166, normalized size = 0.99 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (f\,a^3+3\,e\,a^2\,b-15\,d\,a\,b^2+35\,c\,b^3\right )}{8\,a^{9/2}\,b^{3/2}}-\frac {\frac {c}{3\,a}-\frac {x^6\,\left (f\,a^3+3\,e\,a^2\,b-15\,d\,a\,b^2+35\,c\,b^3\right )}{8\,a^4}+\frac {x^2\,\left (3\,a\,d-7\,b\,c\right )}{3\,a^2}-\frac {x^4\,\left (-3\,f\,a^3+15\,e\,a^2\,b-75\,d\,a\,b^2+175\,c\,b^3\right )}{24\,a^3\,b}}{a^2\,x^3+2\,a\,b\,x^5+b^2\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 70.49, size = 270, normalized size = 1.61 \begin {gather*} - \frac {\sqrt {- \frac {1}{a^{9} b^{3}}} \left (a^{3} f + 3 a^{2} b e - 15 a b^{2} d + 35 b^{3} c\right ) \log {\left (- a^{5} b \sqrt {- \frac {1}{a^{9} b^{3}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{a^{9} b^{3}}} \left (a^{3} f + 3 a^{2} b e - 15 a b^{2} d + 35 b^{3} c\right ) \log {\left (a^{5} b \sqrt {- \frac {1}{a^{9} b^{3}}} + x \right )}}{16} + \frac {- 8 a^{3} b c + x^{6} \left (3 a^{3} b f + 9 a^{2} b^{2} e - 45 a b^{3} d + 105 b^{4} c\right ) + x^{4} \left (- 3 a^{4} f + 15 a^{3} b e - 75 a^{2} b^{2} d + 175 a b^{3} c\right ) + x^{2} \left (- 24 a^{3} b d + 56 a^{2} b^{2} c\right )}{24 a^{6} b x^{3} + 48 a^{5} b^{2} x^{5} + 24 a^{4} b^{3} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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