Optimal. Leaf size=163 \[ \frac {x^3 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-7 a^3 f+5 a^2 b e-3 a b^2 d+b^3 c\right )}{2 \sqrt {a} b^{9/2}}-\frac {x \left (-7 a^3 f+5 a^2 b e-3 a b^2 d+b^3 c\right )}{2 a b^4}+\frac {x^3 (b e-2 a f)}{3 b^3}+\frac {f x^5}{5 b^2} \]
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Rubi [A] time = 0.23, antiderivative size = 163, 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 = {1804, 1585, 1261, 205} \[ \frac {x^3 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {x \left (5 a^2 b e-7 a^3 f-3 a b^2 d+b^3 c\right )}{2 a b^4}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (5 a^2 b e-7 a^3 f-3 a b^2 d+b^3 c\right )}{2 \sqrt {a} b^{9/2}}+\frac {x^3 (b e-2 a f)}{3 b^3}+\frac {f x^5}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 1261
Rule 1585
Rule 1804
Rubi steps
\begin {align*} \int \frac {x^2 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{2 a \left (a+b x^2\right )}-\frac {\int \frac {x \left (\left (b c-3 a d+\frac {3 a^2 e}{b}-\frac {3 a^3 f}{b^2}\right ) x-2 a \left (e-\frac {a f}{b}\right ) x^3-2 a f x^5\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{2 a \left (a+b x^2\right )}-\frac {\int \frac {x^2 \left (b c-3 a d+\frac {3 a^2 e}{b}-\frac {3 a^3 f}{b^2}-2 a \left (e-\frac {a f}{b}\right ) x^2-2 a f x^4\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{2 a \left (a+b x^2\right )}-\frac {\int \left (c-\frac {a \left (3 b^2 d-5 a b e+7 a^2 f\right )}{b^3}-\frac {2 a (b e-2 a f) x^2}{b^2}-\frac {2 a f x^4}{b}+\frac {-a b^3 c+3 a^2 b^2 d-5 a^3 b e+7 a^4 f}{b^3 \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=-\frac {\left (b^3 c-3 a b^2 d+5 a^2 b e-7 a^3 f\right ) x}{2 a b^4}+\frac {(b e-2 a f) x^3}{3 b^3}+\frac {f x^5}{5 b^2}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{2 a \left (a+b x^2\right )}+\frac {\left (b^3 c-3 a b^2 d+5 a^2 b e-7 a^3 f\right ) \int \frac {1}{a+b x^2} \, dx}{2 b^4}\\ &=-\frac {\left (b^3 c-3 a b^2 d+5 a^2 b e-7 a^3 f\right ) x}{2 a b^4}+\frac {(b e-2 a f) x^3}{3 b^3}+\frac {f x^5}{5 b^2}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{2 a \left (a+b x^2\right )}+\frac {\left (b^3 c-3 a b^2 d+5 a^2 b e-7 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 148, normalized size = 0.91 \[ \frac {x \left (3 a^2 f-2 a b e+b^2 d\right )}{b^4}-\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (7 a^3 f-5 a^2 b e+3 a b^2 d-b^3 c\right )}{2 \sqrt {a} b^{9/2}}-\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 b^4 \left (a+b x^2\right )}+\frac {x^3 (b e-2 a f)}{3 b^3}+\frac {f x^5}{5 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 418, normalized size = 2.56 \[ \left [\frac {12 \, a b^{4} f x^{7} + 4 \, {\left (5 \, a b^{4} e - 7 \, a^{2} b^{3} f\right )} x^{5} + 20 \, {\left (3 \, a b^{4} d - 5 \, a^{2} b^{3} e + 7 \, a^{3} b^{2} f\right )} x^{3} + 15 \, {\left (a b^{3} c - 3 \, a^{2} b^{2} d + 5 \, a^{3} b e - 7 \, a^{4} f + {\left (b^{4} c - 3 \, a b^{3} d + 5 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 30 \, {\left (a b^{4} c - 3 \, a^{2} b^{3} d + 5 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x}{60 \, {\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}, \frac {6 \, a b^{4} f x^{7} + 2 \, {\left (5 \, a b^{4} e - 7 \, a^{2} b^{3} f\right )} x^{5} + 10 \, {\left (3 \, a b^{4} d - 5 \, a^{2} b^{3} e + 7 \, a^{3} b^{2} f\right )} x^{3} + 15 \, {\left (a b^{3} c - 3 \, a^{2} b^{2} d + 5 \, a^{3} b e - 7 \, a^{4} f + {\left (b^{4} c - 3 \, a b^{3} d + 5 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - 15 \, {\left (a b^{4} c - 3 \, a^{2} b^{3} d + 5 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x}{30 \, {\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 152, normalized size = 0.93 \[ \frac {{\left (b^{3} c - 3 \, a b^{2} d - 7 \, a^{3} f + 5 \, a^{2} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} - \frac {b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {3 \, b^{8} f x^{5} - 10 \, a b^{7} f x^{3} + 5 \, b^{8} x^{3} e + 15 \, b^{8} d x + 45 \, a^{2} b^{6} f x - 30 \, a b^{7} x e}{15 \, b^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 212, normalized size = 1.30 \[ \frac {f \,x^{5}}{5 b^{2}}-\frac {2 a f \,x^{3}}{3 b^{3}}+\frac {e \,x^{3}}{3 b^{2}}+\frac {a^{3} f x}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {7 a^{3} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{4}}-\frac {a^{2} e x}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {5 a^{2} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{3}}+\frac {a d x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {3 a d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}-\frac {c x}{2 \left (b \,x^{2}+a \right ) b}+\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b}+\frac {3 a^{2} f x}{b^{4}}-\frac {2 a e x}{b^{3}}+\frac {d x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 140, normalized size = 0.86 \[ -\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} + \frac {{\left (b^{3} c - 3 \, a b^{2} d + 5 \, a^{2} b e - 7 \, a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{4}} + \frac {3 \, b^{2} f x^{5} + 5 \, {\left (b^{2} e - 2 \, a b f\right )} x^{3} + 15 \, {\left (b^{2} d - 2 \, a b e + 3 \, a^{2} f\right )} x}{15 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 153, normalized size = 0.94 \[ x^3\,\left (\frac {e}{3\,b^2}-\frac {2\,a\,f}{3\,b^3}\right )-x\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )-\frac {x\,\left (-\frac {f\,a^3}{2}+\frac {e\,a^2\,b}{2}-\frac {d\,a\,b^2}{2}+\frac {c\,b^3}{2}\right )}{b^5\,x^2+a\,b^4}+\frac {f\,x^5}{5\,b^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-7\,f\,a^3+5\,e\,a^2\,b-3\,d\,a\,b^2+c\,b^3\right )}{2\,\sqrt {a}\,b^{9/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.04, size = 221, normalized size = 1.36 \[ x^{3} \left (- \frac {2 a f}{3 b^{3}} + \frac {e}{3 b^{2}}\right ) + x \left (\frac {3 a^{2} f}{b^{4}} - \frac {2 a e}{b^{3}} + \frac {d}{b^{2}}\right ) + \frac {x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{2 a b^{4} + 2 b^{5} x^{2}} + \frac {\sqrt {- \frac {1}{a b^{9}}} \left (7 a^{3} f - 5 a^{2} b e + 3 a b^{2} d - b^{3} c\right ) \log {\left (- a b^{4} \sqrt {- \frac {1}{a b^{9}}} + x \right )}}{4} - \frac {\sqrt {- \frac {1}{a b^{9}}} \left (7 a^{3} f - 5 a^{2} b e + 3 a b^{2} d - b^{3} c\right ) \log {\left (a b^{4} \sqrt {- \frac {1}{a b^{9}}} + x \right )}}{4} + \frac {f x^{5}}{5 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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