Optimal. Leaf size=207 \[ \frac {x^5 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-63 a^3 f+35 a^2 b e-15 a b^2 d+3 b^3 c\right )}{8 \sqrt {a} b^{11/2}}-\frac {x \left (-13 a^3 f+9 a^2 b e-5 a b^2 d+b^3 c\right )}{8 b^5 \left (a+b x^2\right )}-\frac {x \left (-25 a^3 f+13 a^2 b e-5 a b^2 d+b^3 c\right )}{4 a b^5}+\frac {x^3 (b e-3 a f)}{3 b^4}+\frac {f x^5}{5 b^3} \]
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Rubi [A] time = 0.33, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1804, 1585, 1257, 1810, 205} \begin {gather*} \frac {x^5 \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac {x \left (9 a^2 b e-13 a^3 f-5 a b^2 d+b^3 c\right )}{8 b^5 \left (a+b x^2\right )}-\frac {x \left (13 a^2 b e-25 a^3 f-5 a b^2 d+b^3 c\right )}{4 a b^5}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (35 a^2 b e-63 a^3 f-15 a b^2 d+3 b^3 c\right )}{8 \sqrt {a} b^{11/2}}+\frac {x^3 (b e-3 a f)}{3 b^4}+\frac {f x^5}{5 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 1257
Rule 1585
Rule 1804
Rule 1810
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 )^3} \, dx &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^3 \left (\left (b c-5 a d+\frac {5 a^2 e}{b}-\frac {5 a^3 f}{b^2}\right ) x-4 a \left (e-\frac {a f}{b}\right ) x^3-4 a f x^5\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{4 a \left (a+b x^2\right )^2}-\frac {\int \frac {x^4 \left (b c-5 a d+\frac {5 a^2 e}{b}-\frac {5 a^3 f}{b^2}-4 a \left (e-\frac {a f}{b}\right ) x^2-4 a f x^4\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{4 a \left (a+b x^2\right )^2}-\frac {\left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right ) x}{8 b^5 \left (a+b x^2\right )}+\frac {\int \frac {a \left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right )-2 b \left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right ) x^2+8 a b^2 (b e-2 a f) x^4+8 a b^3 f x^6}{a+b x^2} \, dx}{8 a b^5}\\ &=\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{4 a \left (a+b x^2\right )^2}-\frac {\left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right ) x}{8 b^5 \left (a+b x^2\right )}+\frac {\int \left (-2 \left (b^3 c-5 a b^2 d+13 a^2 b e-25 a^3 f\right )+8 a b (b e-3 a f) x^2+8 a b^2 f x^4+\frac {3 a b^3 c-15 a^2 b^2 d+35 a^3 b e-63 a^4 f}{a+b x^2}\right ) \, dx}{8 a b^5}\\ &=-\frac {\left (b^3 c-5 a b^2 d+13 a^2 b e-25 a^3 f\right ) x}{4 a b^5}+\frac {(b e-3 a f) x^3}{3 b^4}+\frac {f x^5}{5 b^3}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{4 a \left (a+b x^2\right )^2}-\frac {\left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right ) x}{8 b^5 \left (a+b x^2\right )}+\frac {\left (3 b^3 c-15 a b^2 d+35 a^2 b e-63 a^3 f\right ) \int \frac {1}{a+b x^2} \, dx}{8 b^5}\\ &=-\frac {\left (b^3 c-5 a b^2 d+13 a^2 b e-25 a^3 f\right ) x}{4 a b^5}+\frac {(b e-3 a f) x^3}{3 b^4}+\frac {f x^5}{5 b^3}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{4 a \left (a+b x^2\right )^2}-\frac {\left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right ) x}{8 b^5 \left (a+b x^2\right )}+\frac {\left (3 b^3 c-15 a b^2 d+35 a^2 b e-63 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 \sqrt {a} b^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 176, normalized size = 0.85 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-63 a^3 f+35 a^2 b e-15 a b^2 d+3 b^3 c\right )}{8 \sqrt {a} b^{11/2}}+\frac {x \left (945 a^4 f-525 a^3 b \left (e-3 f x^2\right )+a^2 b^2 \left (225 d-875 e x^2+504 f x^4\right )-a b^3 \left (45 c-375 d x^2+280 e x^4+72 f x^6\right )+b^4 x^2 \left (8 \left (15 d x^2+5 e x^4+3 f x^6\right )-75 c\right )\right )}{120 b^5 \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.94, size = 614, normalized size = 2.97 \begin {gather*} \left [\frac {48 \, a b^{5} f x^{9} + 16 \, {\left (5 \, a b^{5} e - 9 \, a^{2} b^{4} f\right )} x^{7} + 16 \, {\left (15 \, a b^{5} d - 35 \, a^{2} b^{4} e + 63 \, a^{3} b^{3} f\right )} x^{5} - 50 \, {\left (3 \, a b^{5} c - 15 \, a^{2} b^{4} d + 35 \, a^{3} b^{3} e - 63 \, a^{4} b^{2} f\right )} x^{3} + 15 \, {\left (3 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 35 \, a^{4} b e - 63 \, a^{5} f + {\left (3 \, b^{5} c - 15 \, a b^{4} d + 35 \, a^{2} b^{3} e - 63 \, a^{3} b^{2} f\right )} x^{4} + 2 \, {\left (3 \, a b^{4} c - 15 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 63 \, a^{4} 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 (3 \, a^{2} b^{4} c - 15 \, a^{3} b^{3} d + 35 \, a^{4} b^{2} e - 63 \, a^{5} b f\right )} x}{240 \, {\left (a b^{8} x^{4} + 2 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}}, \frac {24 \, a b^{5} f x^{9} + 8 \, {\left (5 \, a b^{5} e - 9 \, a^{2} b^{4} f\right )} x^{7} + 8 \, {\left (15 \, a b^{5} d - 35 \, a^{2} b^{4} e + 63 \, a^{3} b^{3} f\right )} x^{5} - 25 \, {\left (3 \, a b^{5} c - 15 \, a^{2} b^{4} d + 35 \, a^{3} b^{3} e - 63 \, a^{4} b^{2} f\right )} x^{3} + 15 \, {\left (3 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 35 \, a^{4} b e - 63 \, a^{5} f + {\left (3 \, b^{5} c - 15 \, a b^{4} d + 35 \, a^{2} b^{3} e - 63 \, a^{3} b^{2} f\right )} x^{4} + 2 \, {\left (3 \, a b^{4} c - 15 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 63 \, a^{4} b f\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - 15 \, {\left (3 \, a^{2} b^{4} c - 15 \, a^{3} b^{3} d + 35 \, a^{4} b^{2} e - 63 \, a^{5} b f\right )} x}{120 \, {\left (a b^{8} x^{4} + 2 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 200, normalized size = 0.97 \begin {gather*} \frac {{\left (3 \, b^{3} c - 15 \, a b^{2} d - 63 \, a^{3} f + 35 \, a^{2} b e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{5}} - \frac {5 \, b^{4} c x^{3} - 9 \, a b^{3} d x^{3} - 17 \, a^{3} b f x^{3} + 13 \, a^{2} b^{2} x^{3} e + 3 \, a b^{3} c x - 7 \, a^{2} b^{2} d x - 15 \, a^{4} f x + 11 \, a^{3} b x e}{8 \, {\left (b x^{2} + a\right )}^{2} b^{5}} + \frac {3 \, b^{12} f x^{5} - 15 \, a b^{11} f x^{3} + 5 \, b^{12} x^{3} e + 15 \, b^{12} d x + 90 \, a^{2} b^{10} f x - 45 \, a b^{11} x e}{15 \, b^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 294, normalized size = 1.42 \begin {gather*} \frac {17 a^{3} f \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}-\frac {13 a^{2} e \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}+\frac {9 a d \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{2}}-\frac {5 c \,x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b}+\frac {f \,x^{5}}{5 b^{3}}+\frac {15 a^{4} f x}{8 \left (b \,x^{2}+a \right )^{2} b^{5}}-\frac {11 a^{3} e x}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}+\frac {7 a^{2} d x}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}-\frac {3 a c x}{8 \left (b \,x^{2}+a \right )^{2} b^{2}}-\frac {a f \,x^{3}}{b^{4}}+\frac {e \,x^{3}}{3 b^{3}}-\frac {63 a^{3} f \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{5}}+\frac {35 a^{2} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{4}}-\frac {15 a d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{3}}+\frac {3 c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{2}}+\frac {6 a^{2} f x}{b^{5}}-\frac {3 a e x}{b^{4}}+\frac {d x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 193, normalized size = 0.93 \begin {gather*} -\frac {{\left (5 \, b^{4} c - 9 \, a b^{3} d + 13 \, a^{2} b^{2} e - 17 \, a^{3} b f\right )} x^{3} + {\left (3 \, a b^{3} c - 7 \, a^{2} b^{2} d + 11 \, a^{3} b e - 15 \, a^{4} f\right )} x}{8 \, {\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} + \frac {{\left (3 \, b^{3} c - 15 \, a b^{2} d + 35 \, a^{2} b e - 63 \, a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{5}} + \frac {3 \, b^{2} f x^{5} + 5 \, {\left (b^{2} e - 3 \, a b f\right )} x^{3} + 15 \, {\left (b^{2} d - 3 \, a b e + 6 \, a^{2} f\right )} x}{15 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.95, size = 206, normalized size = 1.00 \begin {gather*} x^3\,\left (\frac {e}{3\,b^3}-\frac {a\,f}{b^4}\right )-x\,\left (\frac {3\,a^2\,f}{b^5}-\frac {d}{b^3}+\frac {3\,a\,\left (\frac {e}{b^3}-\frac {3\,a\,f}{b^4}\right )}{b}\right )-\frac {x^3\,\left (-\frac {17\,f\,a^3\,b}{8}+\frac {13\,e\,a^2\,b^2}{8}-\frac {9\,d\,a\,b^3}{8}+\frac {5\,c\,b^4}{8}\right )-x\,\left (\frac {15\,f\,a^4}{8}-\frac {11\,e\,a^3\,b}{8}+\frac {7\,d\,a^2\,b^2}{8}-\frac {3\,c\,a\,b^3}{8}\right )}{a^2\,b^5+2\,a\,b^6\,x^2+b^7\,x^4}+\frac {f\,x^5}{5\,b^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-63\,f\,a^3+35\,e\,a^2\,b-15\,d\,a\,b^2+3\,c\,b^3\right )}{8\,\sqrt {a}\,b^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.84, size = 280, normalized size = 1.35 \begin {gather*} x^{3} \left (- \frac {a f}{b^{4}} + \frac {e}{3 b^{3}}\right ) + x \left (\frac {6 a^{2} f}{b^{5}} - \frac {3 a e}{b^{4}} + \frac {d}{b^{3}}\right ) + \frac {\sqrt {- \frac {1}{a b^{11}}} \left (63 a^{3} f - 35 a^{2} b e + 15 a b^{2} d - 3 b^{3} c\right ) \log {\left (- a b^{5} \sqrt {- \frac {1}{a b^{11}}} + x \right )}}{16} - \frac {\sqrt {- \frac {1}{a b^{11}}} \left (63 a^{3} f - 35 a^{2} b e + 15 a b^{2} d - 3 b^{3} c\right ) \log {\left (a b^{5} \sqrt {- \frac {1}{a b^{11}}} + x \right )}}{16} + \frac {x^{3} \left (17 a^{3} b f - 13 a^{2} b^{2} e + 9 a b^{3} d - 5 b^{4} c\right ) + x \left (15 a^{4} f - 11 a^{3} b e + 7 a^{2} b^{2} d - 3 a b^{3} c\right )}{8 a^{2} b^{5} + 16 a b^{6} x^{2} + 8 b^{7} x^{4}} + \frac {f x^{5}}{5 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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