Optimal. Leaf size=189 \[ \frac {2 b c-a d}{5 a^3 x^5}-\frac {c}{7 a^2 x^7}-\frac {a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-3 a^3 f+5 a^2 b e-7 a b^2 d+9 b^3 c\right )}{2 a^{11/2}}+\frac {b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{a^5 x} \]
________________________________________________________________________________________
Rubi [A] time = 0.29, 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 = {1805, 1802, 205} \begin {gather*} \frac {b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac {2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{a^5 x}+\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (5 a^2 b e-3 a^3 f-7 a b^2 d+9 b^3 c\right )}{2 a^{11/2}}-\frac {a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac {2 b c-a d}{5 a^3 x^5}-\frac {c}{7 a^2 x^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 1802
Rule 1805
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 190, normalized size = 1.01 \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} \tan ^{-1}\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.
[In]
[Out]
________________________________________________________________________________________
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^8 \left (a+b x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.02, size = 488, normalized size = 2.58 \begin {gather*} \left [\frac {210 \, {\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 140 \, {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{6} - 60 \, a^{4} c - 28 \, {\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d + 5 \, a^{4} e\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 + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{9} + {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{7}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{420 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}, \frac {105 \, {\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 70 \, {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\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} + 105 \, {\left ({\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{9} + {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{7}\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{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.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.43, 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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 268, normalized size = 1.42 \begin {gather*} -\frac {b f x}{2 \left (b \,x^{2}+a \right ) a^{2}}-\frac {3 b f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{2}}+\frac {b^{2} e x}{2 \left (b \,x^{2}+a \right ) a^{3}}+\frac {5 b^{2} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{3}}-\frac {b^{3} d x}{2 \left (b \,x^{2}+a \right ) a^{4}}-\frac {7 b^{3} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{4}}+\frac {b^{4} c x}{2 \left (b \,x^{2}+a \right ) a^{5}}+\frac {9 b^{4} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a^{5}}-\frac {f}{a^{2} x}+\frac {2 b e}{a^{3} x}-\frac {3 b^{2} d}{a^{4} x}+\frac {4 b^{3} c}{a^{5} x}-\frac {e}{3 a^{2} x^{3}}+\frac {2 b d}{3 a^{3} x^{3}}-\frac {b^{2} c}{a^{4} x^{3}}-\frac {d}{5 a^{2} x^{5}}+\frac {2 b c}{5 a^{3} x^{5}}-\frac {c}{7 a^{2} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.05, size = 194, normalized size = 1.03 \begin {gather*} \frac {105 \, {\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 70 \, {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\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 + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 99.02, size = 394, normalized size = 2.08 \begin {gather*} \frac {\sqrt {- \frac {b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right ) \log {\left (- \frac {a^{6} \sqrt {- \frac {b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right )}{3 a^{3} b f - 5 a^{2} b^{2} e + 7 a b^{3} d - 9 b^{4} c} + x \right )}}{4} - \frac {\sqrt {- \frac {b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right ) \log {\left (\frac {a^{6} \sqrt {- \frac {b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right )}{3 a^{3} b f - 5 a^{2} b^{2} e + 7 a b^{3} d - 9 b^{4} c} + x \right )}}{4} + \frac {- 30 a^{4} c + x^{8} \left (- 315 a^{3} b f + 525 a^{2} b^{2} e - 735 a b^{3} d + 945 b^{4} c\right ) + x^{6} \left (- 210 a^{4} f + 350 a^{3} b e - 490 a^{2} b^{2} d + 630 a b^{3} c\right ) + x^{4} \left (- 70 a^{4} e + 98 a^{3} b d - 126 a^{2} b^{2} c\right ) + x^{2} \left (- 42 a^{4} d + 54 a^{3} b c\right )}{210 a^{6} x^{7} + 210 a^{5} b x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________