Optimal. Leaf size=210 \[ \frac {x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac {a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}-\frac {a x^3 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^5}+\frac {x^5 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{5 b^4}-\frac {a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^{13/2}}+\frac {x^9 (b e-a f)}{9 b^2}+\frac {f x^{11}}{11 b} \]
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Rubi [A] time = 0.16, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1802, 205} \[ \frac {x^5 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{5 b^4}-\frac {a x^3 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^5}+\frac {a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^6}-\frac {a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^{13/2}}+\frac {x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac {x^9 (b e-a f)}{9 b^2}+\frac {f x^{11}}{11 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 1802
Rubi steps
\begin {align*} \int \frac {x^6 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx &=\int \left (\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )}{b^6}-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^4}{b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^6}{b^3}+\frac {(b e-a f) x^8}{b^2}+\frac {f x^{10}}{b}+\frac {-a^3 b^3 c+a^4 b^2 d-a^5 b e+a^6 f}{b^6 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^6}-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^7}{7 b^3}+\frac {(b e-a f) x^9}{9 b^2}+\frac {f x^{11}}{11 b}-\frac {\left (a^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{b^6}\\ &=\frac {a^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^6}-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^3}{3 b^5}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^5}{5 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^7}{7 b^3}+\frac {(b e-a f) x^9}{9 b^2}+\frac {f x^{11}}{11 b}-\frac {a^{5/2} \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{13/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 210, normalized size = 1.00 \[ \frac {x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}-\frac {a^2 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^6}+\frac {a x^3 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 b^5}+\frac {x^5 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{5 b^4}+\frac {a^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^{13/2}}+\frac {x^9 (b e-a f)}{9 b^2}+\frac {f x^{11}}{11 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 452, normalized size = 2.15 \[ \left [\frac {630 \, b^{5} f x^{11} + 770 \, {\left (b^{5} e - a b^{4} f\right )} x^{9} + 990 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{7} + 1386 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{5} - 2310 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} - 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6930 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{6930 \, b^{6}}, \frac {315 \, b^{5} f x^{11} + 385 \, {\left (b^{5} e - a b^{4} f\right )} x^{9} + 495 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{7} + 693 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{5} - 1155 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} - 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{3465 \, b^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 250, normalized size = 1.19 \[ -\frac {{\left (a^{3} b^{3} c - a^{4} b^{2} d - a^{6} f + a^{5} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{6}} + \frac {315 \, b^{10} f x^{11} - 385 \, a b^{9} f x^{9} + 385 \, b^{10} x^{9} e + 495 \, b^{10} d x^{7} + 495 \, a^{2} b^{8} f x^{7} - 495 \, a b^{9} x^{7} e + 693 \, b^{10} c x^{5} - 693 \, a b^{9} d x^{5} - 693 \, a^{3} b^{7} f x^{5} + 693 \, a^{2} b^{8} x^{5} e - 1155 \, a b^{9} c x^{3} + 1155 \, a^{2} b^{8} d x^{3} + 1155 \, a^{4} b^{6} f x^{3} - 1155 \, a^{3} b^{7} x^{3} e + 3465 \, a^{2} b^{8} c x - 3465 \, a^{3} b^{7} d x - 3465 \, a^{5} b^{5} f x + 3465 \, a^{4} b^{6} x e}{3465 \, b^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 278, normalized size = 1.32 \[ \frac {f \,x^{11}}{11 b}-\frac {a f \,x^{9}}{9 b^{2}}+\frac {e \,x^{9}}{9 b}+\frac {a^{2} f \,x^{7}}{7 b^{3}}-\frac {a e \,x^{7}}{7 b^{2}}+\frac {d \,x^{7}}{7 b}-\frac {a^{3} f \,x^{5}}{5 b^{4}}+\frac {a^{2} e \,x^{5}}{5 b^{3}}-\frac {a d \,x^{5}}{5 b^{2}}+\frac {c \,x^{5}}{5 b}+\frac {a^{4} f \,x^{3}}{3 b^{5}}-\frac {a^{3} e \,x^{3}}{3 b^{4}}+\frac {a^{2} d \,x^{3}}{3 b^{3}}-\frac {a c \,x^{3}}{3 b^{2}}+\frac {a^{6} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{6}}-\frac {a^{5} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{5}}+\frac {a^{4} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{4}}-\frac {a^{3} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}-\frac {a^{5} f x}{b^{6}}+\frac {a^{4} e x}{b^{5}}-\frac {a^{3} d x}{b^{4}}+\frac {a^{2} c x}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.90, size = 213, normalized size = 1.01 \[ -\frac {{\left (a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{6}} + \frac {315 \, b^{5} f x^{11} + 385 \, {\left (b^{5} e - a b^{4} f\right )} x^{9} + 495 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{7} + 693 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{5} - 1155 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{3} + 3465 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{3465 \, b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 289, normalized size = 1.38 \[ x^9\,\left (\frac {e}{9\,b}-\frac {a\,f}{9\,b^2}\right )+x^7\,\left (\frac {d}{7\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{7\,b}\right )+x^5\,\left (\frac {c}{5\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{5\,b}\right )+\frac {f\,x^{11}}{11\,b}+\frac {a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,x\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{f\,a^6-e\,a^5\,b+d\,a^4\,b^2-c\,a^3\,b^3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{b^{13/2}}-\frac {a\,x^3\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{3\,b}+\frac {a^2\,x\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.65, size = 384, normalized size = 1.83 \[ x^{9} \left (- \frac {a f}{9 b^{2}} + \frac {e}{9 b}\right ) + x^{7} \left (\frac {a^{2} f}{7 b^{3}} - \frac {a e}{7 b^{2}} + \frac {d}{7 b}\right ) + x^{5} \left (- \frac {a^{3} f}{5 b^{4}} + \frac {a^{2} e}{5 b^{3}} - \frac {a d}{5 b^{2}} + \frac {c}{5 b}\right ) + x^{3} \left (\frac {a^{4} f}{3 b^{5}} - \frac {a^{3} e}{3 b^{4}} + \frac {a^{2} d}{3 b^{3}} - \frac {a c}{3 b^{2}}\right ) + x \left (- \frac {a^{5} f}{b^{6}} + \frac {a^{4} e}{b^{5}} - \frac {a^{3} d}{b^{4}} + \frac {a^{2} c}{b^{3}}\right ) - \frac {\sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- \frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c} + x \right )}}{2} + \frac {\sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (\frac {b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c} + x \right )}}{2} + \frac {f x^{11}}{11 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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