Optimal. Leaf size=136 \[ \frac {x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^{9/2}}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^4}+\frac {x^5 (b e-a f)}{5 b^2}+\frac {f x^7}{7 b} \]
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Rubi [A] time = 0.11, antiderivative size = 136, 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} \[ \frac {x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^4}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^{9/2}}+\frac {x^3 \left (a^2 f-a b e+b^2 d\right )}{3 b^3}+\frac {x^5 (b e-a f)}{5 b^2}+\frac {f x^7}{7 b} \]
Antiderivative was successfully verified.
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Rule 205
Rule 1802
Rubi steps
\begin {align*} \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{a+b x^2} \, dx &=\int \left (\frac {b^3 c-a b^2 d+a^2 b e-a^3 f}{b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^2}{b^3}+\frac {(b e-a f) x^4}{b^2}+\frac {f x^6}{b}+\frac {-a b^3 c+a^2 b^2 d-a^3 b e+a^4 f}{b^4 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^3}{3 b^3}+\frac {(b e-a f) x^5}{5 b^2}+\frac {f x^7}{7 b}-\frac {\left (a \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^4}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^3}{3 b^3}+\frac {(b e-a f) x^5}{5 b^2}+\frac {f x^7}{7 b}-\frac {\sqrt {a} \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^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 128, normalized size = 0.94 \[ \frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^{9/2}}+\frac {x \left (-105 a^3 f+35 a^2 b \left (3 e+f x^2\right )-7 a b^2 \left (15 d+5 e x^2+3 f x^4\right )+b^3 \left (105 c+35 d x^2+21 e x^4+15 f x^6\right )\right )}{105 b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 286, normalized size = 2.10 \[ \left [\frac {30 \, b^{3} f x^{7} + 42 \, {\left (b^{3} e - a b^{2} f\right )} x^{5} + 70 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{3} - 105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 210 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{210 \, b^{4}}, \frac {15 \, b^{3} f x^{7} + 21 \, {\left (b^{3} e - a b^{2} f\right )} x^{5} + 35 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{3} - 105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{105 \, b^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 152, normalized size = 1.12 \[ -\frac {{\left (a b^{3} c - a^{2} b^{2} d - a^{4} f + a^{3} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{6} f x^{7} - 21 \, a b^{5} f x^{5} + 21 \, b^{6} x^{5} e + 35 \, b^{6} d x^{3} + 35 \, a^{2} b^{4} f x^{3} - 35 \, a b^{5} x^{3} e + 105 \, b^{6} c x - 105 \, a b^{5} d x - 105 \, a^{3} b^{3} f x + 105 \, a^{2} b^{4} x e}{105 \, b^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 182, normalized size = 1.34 \[ \frac {f \,x^{7}}{7 b}-\frac {a f \,x^{5}}{5 b^{2}}+\frac {e \,x^{5}}{5 b}+\frac {a^{2} f \,x^{3}}{3 b^{3}}-\frac {a e \,x^{3}}{3 b^{2}}+\frac {d \,x^{3}}{3 b}+\frac {a^{4} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{4}}-\frac {a^{3} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {a^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}-\frac {a c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, b}-\frac {a^{3} f x}{b^{4}}+\frac {a^{2} e x}{b^{3}}-\frac {a d x}{b^{2}}+\frac {c x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 133, normalized size = 0.98 \[ -\frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{4}} + \frac {15 \, b^{3} f x^{7} + 21 \, {\left (b^{3} e - a b^{2} f\right )} x^{5} + 35 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{3} + 105 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{105 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 193, normalized size = 1.42 \[ x^5\,\left (\frac {e}{5\,b}-\frac {a\,f}{5\,b^2}\right )+x^3\,\left (\frac {d}{3\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{3\,b}\right )+x\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )+\frac {f\,x^7}{7\,b}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{f\,a^4-e\,a^3\,b+d\,a^2\,b^2-c\,a\,b^3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{b^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.13, size = 185, normalized size = 1.36 \[ x^{5} \left (- \frac {a f}{5 b^{2}} + \frac {e}{5 b}\right ) + x^{3} \left (\frac {a^{2} f}{3 b^{3}} - \frac {a e}{3 b^{2}} + \frac {d}{3 b}\right ) + x \left (- \frac {a^{3} f}{b^{4}} + \frac {a^{2} e}{b^{3}} - \frac {a d}{b^{2}} + \frac {c}{b}\right ) - \frac {\sqrt {- \frac {a}{b^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- b^{4} \sqrt {- \frac {a}{b^{9}}} + x \right )}}{2} + \frac {\sqrt {- \frac {a}{b^{9}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (b^{4} \sqrt {- \frac {a}{b^{9}}} + x \right )}}{2} + \frac {f x^{7}}{7 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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