Optimal. Leaf size=202 \[ \frac {x^5 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right )}{2 b^{11/2}}+\frac {x \left (-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right )}{2 b^5}-\frac {x^3 \left (-9 a^3 f+7 a^2 b e-5 a b^2 d+3 b^3 c\right )}{6 a b^4}+\frac {x^5 (b e-2 a f)}{5 b^3}+\frac {f x^7}{7 b^2} \]
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Rubi [A] time = 0.23, antiderivative size = 202, 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} \[ \frac {x^5 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac {x^3 \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{6 a b^4}+\frac {x \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{2 b^5}-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (7 a^2 b e-9 a^3 f-5 a b^2 d+3 b^3 c\right )}{2 b^{11/2}}+\frac {x^5 (b e-2 a f)}{5 b^3}+\frac {f x^7}{7 b^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 1261
Rule 1585
Rule 1804
Rubi steps
\begin {align*} \int \frac {x^4 \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^5}{2 a \left (a+b x^2\right )}-\frac {\int \frac {x^3 \left (\left (3 b c-5 a d+\frac {5 a^2 e}{b}-\frac {5 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^5}{2 a \left (a+b x^2\right )}-\frac {\int \frac {x^4 \left (3 b c-5 a d+\frac {5 a^2 e}{b}-\frac {5 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^5}{2 a \left (a+b x^2\right )}-\frac {\int \left (-\frac {a \left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right )}{b^4}+\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x^2}{b^3}-\frac {2 a (b e-2 a f) x^4}{b^2}-\frac {2 a f x^6}{b}+\frac {3 a^2 b^3 c-5 a^3 b^2 d+7 a^4 b e-9 a^5 f}{b^4 \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x}{2 b^5}-\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x^3}{6 a b^4}+\frac {(b e-2 a f) x^5}{5 b^3}+\frac {f x^7}{7 b^2}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac {\left (a \left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right )\right ) \int \frac {1}{a+b x^2} \, dx}{2 b^5}\\ &=\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x}{2 b^5}-\frac {\left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) x^3}{6 a b^4}+\frac {(b e-2 a f) x^5}{5 b^3}+\frac {f x^7}{7 b^2}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{2 a \left (a+b x^2\right )}-\frac {\sqrt {a} \left (3 b^3 c-5 a b^2 d+7 a^2 b e-9 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 187, normalized size = 0.93 \[ \frac {x^3 \left (3 a^2 f-2 a b e+b^2 d\right )}{3 b^4}+\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (9 a^3 f-7 a^2 b e+5 a b^2 d-3 b^3 c\right )}{2 b^{11/2}}+\frac {x \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{b^5}+\frac {x \left (a^4 (-f)+a^3 b e-a^2 b^2 d+a b^3 c\right )}{2 b^5 \left (a+b x^2\right )}+\frac {x^5 (b e-2 a f)}{5 b^3}+\frac {f x^7}{7 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 478, normalized size = 2.37 \[ \left [\frac {60 \, b^{4} f x^{9} + 12 \, {\left (7 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 28 \, {\left (5 \, b^{4} d - 7 \, a b^{3} e + 9 \, a^{2} b^{2} f\right )} x^{5} + 140 \, {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{3} - 105 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f + {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 210 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} x}{420 \, {\left (b^{6} x^{2} + a b^{5}\right )}}, \frac {30 \, b^{4} f x^{9} + 6 \, {\left (7 \, b^{4} e - 9 \, a b^{3} f\right )} x^{7} + 14 \, {\left (5 \, b^{4} d - 7 \, a b^{3} e + 9 \, a^{2} b^{2} f\right )} x^{5} + 70 \, {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{3} - 105 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f + {\left (3 \, b^{4} c - 5 \, a b^{3} d + 7 \, a^{2} b^{2} e - 9 \, a^{3} b f\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 105 \, {\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} x}{210 \, {\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 201, normalized size = 1.00 \[ -\frac {{\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d - 9 \, a^{4} f + 7 \, a^{3} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {a b^{3} c x - a^{2} b^{2} d x - a^{4} f x + a^{3} b x e}{2 \, {\left (b x^{2} + a\right )} b^{5}} + \frac {15 \, b^{12} f x^{7} - 42 \, a b^{11} f x^{5} + 21 \, b^{12} x^{5} e + 35 \, b^{12} d x^{3} + 105 \, a^{2} b^{10} f x^{3} - 70 \, a b^{11} x^{3} e + 105 \, b^{12} c x - 210 \, a b^{11} d x - 420 \, a^{3} b^{9} f x + 315 \, a^{2} b^{10} x e}{105 \, b^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 258, normalized size = 1.28 \[ \frac {f \,x^{7}}{7 b^{2}}-\frac {2 a f \,x^{5}}{5 b^{3}}+\frac {e \,x^{5}}{5 b^{2}}+\frac {a^{2} f \,x^{3}}{b^{4}}-\frac {2 a e \,x^{3}}{3 b^{3}}+\frac {d \,x^{3}}{3 b^{2}}-\frac {a^{4} f x}{2 \left (b \,x^{2}+a \right ) b^{5}}+\frac {9 a^{4} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{5}}+\frac {a^{3} e x}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {7 a^{3} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{4}}-\frac {a^{2} d x}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {5 a^{2} d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{3}}+\frac {a c x}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {3 a c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{2}}-\frac {4 a^{3} f x}{b^{5}}+\frac {3 a^{2} e x}{b^{4}}-\frac {2 a d x}{b^{3}}+\frac {c x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 183, normalized size = 0.91 \[ \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} - \frac {{\left (3 \, a b^{3} c - 5 \, a^{2} b^{2} d + 7 \, a^{3} b e - 9 \, a^{4} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {15 \, b^{3} f x^{7} + 21 \, {\left (b^{3} e - 2 \, a b^{2} f\right )} x^{5} + 35 \, {\left (b^{3} d - 2 \, a b^{2} e + 3 \, a^{2} b f\right )} x^{3} + 105 \, {\left (b^{3} c - 2 \, a b^{2} d + 3 \, a^{2} b e - 4 \, a^{3} f\right )} x}{105 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 288, normalized size = 1.43 \[ x^5\,\left (\frac {e}{5\,b^2}-\frac {2\,a\,f}{5\,b^3}\right )+x\,\left (\frac {c}{b^2}-\frac {a^2\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b^2}+\frac {2\,a\,\left (\frac {a^2\,f}{b^4}-\frac {d}{b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{b}\right )}{b}\right )-x^3\,\left (\frac {a^2\,f}{3\,b^4}-\frac {d}{3\,b^2}+\frac {2\,a\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )}{3\,b}\right )-\frac {x\,\left (\frac {f\,a^4}{2}-\frac {e\,a^3\,b}{2}+\frac {d\,a^2\,b^2}{2}-\frac {c\,a\,b^3}{2}\right )}{b^6\,x^2+a\,b^5}+\frac {f\,x^7}{7\,b^2}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,\left (-9\,f\,a^3+7\,e\,a^2\,b-5\,d\,a\,b^2+3\,c\,b^3\right )}{9\,f\,a^4-7\,e\,a^3\,b+5\,d\,a^2\,b^2-3\,c\,a\,b^3}\right )\,\left (-9\,f\,a^3+7\,e\,a^2\,b-5\,d\,a\,b^2+3\,c\,b^3\right )}{2\,b^{11/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.76, size = 257, normalized size = 1.27 \[ x^{5} \left (- \frac {2 a f}{5 b^{3}} + \frac {e}{5 b^{2}}\right ) + x^{3} \left (\frac {a^{2} f}{b^{4}} - \frac {2 a e}{3 b^{3}} + \frac {d}{3 b^{2}}\right ) + x \left (- \frac {4 a^{3} f}{b^{5}} + \frac {3 a^{2} e}{b^{4}} - \frac {2 a d}{b^{3}} + \frac {c}{b^{2}}\right ) + \frac {x \left (- a^{4} f + a^{3} b e - a^{2} b^{2} d + a b^{3} c\right )}{2 a b^{5} + 2 b^{6} x^{2}} - \frac {\sqrt {- \frac {a}{b^{11}}} \left (9 a^{3} f - 7 a^{2} b e + 5 a b^{2} d - 3 b^{3} c\right ) \log {\left (- b^{5} \sqrt {- \frac {a}{b^{11}}} + x \right )}}{4} + \frac {\sqrt {- \frac {a}{b^{11}}} \left (9 a^{3} f - 7 a^{2} b e + 5 a b^{2} d - 3 b^{3} c\right ) \log {\left (b^{5} \sqrt {- \frac {a}{b^{11}}} + x \right )}}{4} + \frac {f x^{7}}{7 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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