Optimal. Leaf size=152 \[ -\frac {\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 d x^8}+\frac {b c d^2 \left (c^2 d-4 e\right )}{40 x^5}+\frac {b \left (c^2 d-e\right )^4 \tan ^{-1}(c x)}{8 d}-\frac {b c d \left (c^4 d^2-4 c^2 d e+6 e^2\right )}{24 x^3}+\frac {b c \left (c^2 d-2 e\right ) \left (c^4 d^2-2 c^2 d e+2 e^2\right )}{8 x}-\frac {b c d^3}{56 x^7} \]
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Rubi [A] time = 0.20, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {264, 4976, 12, 461, 203} \[ -\frac {\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 d x^8}-\frac {b c d \left (c^4 d^2-4 c^2 d e+6 e^2\right )}{24 x^3}+\frac {b c \left (c^2 d-2 e\right ) \left (c^4 d^2-2 c^2 d e+2 e^2\right )}{8 x}+\frac {b c d^2 \left (c^2 d-4 e\right )}{40 x^5}+\frac {b \left (c^2 d-e\right )^4 \tan ^{-1}(c x)}{8 d}-\frac {b c d^3}{56 x^7} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 264
Rule 461
Rule 4976
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{x^9} \, dx &=-\frac {\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 d x^8}-(b c) \int \frac {\left (d+e x^2\right )^4}{8 x^8 \left (-d-c^2 d x^2\right )} \, dx\\ &=-\frac {\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 d x^8}-\frac {1}{8} (b c) \int \frac {\left (d+e x^2\right )^4}{x^8 \left (-d-c^2 d x^2\right )} \, dx\\ &=-\frac {\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 d x^8}-\frac {1}{8} (b c) \int \left (-\frac {d^3}{x^8}+\frac {d^2 \left (c^2 d-4 e\right )}{x^6}-\frac {d \left (c^4 d^2-4 c^2 d e+6 e^2\right )}{x^4}+\frac {\left (c^2 d-2 e\right ) \left (c^4 d^2-2 c^2 d e+2 e^2\right )}{x^2}-\frac {\left (c^2 d-e\right )^4}{d \left (1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {b c d^3}{56 x^7}+\frac {b c d^2 \left (c^2 d-4 e\right )}{40 x^5}-\frac {b c d \left (c^4 d^2-4 c^2 d e+6 e^2\right )}{24 x^3}+\frac {b c \left (c^2 d-2 e\right ) \left (c^4 d^2-2 c^2 d e+2 e^2\right )}{8 x}-\frac {\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 d x^8}+\frac {\left (b c \left (c^2 d-e\right )^4\right ) \int \frac {1}{1+c^2 x^2} \, dx}{8 d}\\ &=-\frac {b c d^3}{56 x^7}+\frac {b c d^2 \left (c^2 d-4 e\right )}{40 x^5}-\frac {b c d \left (c^4 d^2-4 c^2 d e+6 e^2\right )}{24 x^3}+\frac {b c \left (c^2 d-2 e\right ) \left (c^4 d^2-2 c^2 d e+2 e^2\right )}{8 x}+\frac {b \left (c^2 d-e\right )^4 \tan ^{-1}(c x)}{8 d}-\frac {\left (d+e x^2\right )^4 \left (a+b \tan ^{-1}(c x)\right )}{8 d x^8}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 154, normalized size = 1.01 \[ -\frac {35 \left (\left (d^3+4 d^2 e x^2+6 d e^2 x^4+4 e^3 x^6\right ) \left (a+b \tan ^{-1}(c x)\right )+2 b c d e^2 x^5 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-c^2 x^2\right )+4 b c e^3 x^7 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-c^2 x^2\right )\right )+5 b c d^3 x \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};-c^2 x^2\right )+28 b c d^2 e x^3 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-c^2 x^2\right )}{280 x^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 228, normalized size = 1.50 \[ -\frac {420 \, a e^{3} x^{6} + 630 \, a d e^{2} x^{4} - 105 \, {\left (b c^{7} d^{3} - 4 \, b c^{5} d^{2} e + 6 \, b c^{3} d e^{2} - 4 \, b c e^{3}\right )} x^{7} + 15 \, b c d^{3} x + 420 \, a d^{2} e x^{2} + 35 \, {\left (b c^{5} d^{3} - 4 \, b c^{3} d^{2} e + 6 \, b c d e^{2}\right )} x^{5} + 105 \, a d^{3} - 21 \, {\left (b c^{3} d^{3} - 4 \, b c d^{2} e\right )} x^{3} + 105 \, {\left (4 \, b e^{3} x^{6} - {\left (b c^{8} d^{3} - 4 \, b c^{6} d^{2} e + 6 \, b c^{4} d e^{2} - 4 \, b c^{2} e^{3}\right )} x^{8} + 6 \, b d e^{2} x^{4} + 4 \, b d^{2} e x^{2} + b d^{3}\right )} \arctan \left (c x\right )}{840 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 265, normalized size = 1.74 \[ -\frac {a \,d^{3}}{8 x^{8}}-\frac {3 a d \,e^{2}}{4 x^{4}}-\frac {a \,d^{2} e}{2 x^{6}}-\frac {a \,e^{3}}{2 x^{2}}-\frac {b \arctan \left (c x \right ) d^{3}}{8 x^{8}}-\frac {3 b \arctan \left (c x \right ) d \,e^{2}}{4 x^{4}}-\frac {b \arctan \left (c x \right ) d^{2} e}{2 x^{6}}-\frac {b \arctan \left (c x \right ) e^{3}}{2 x^{2}}+\frac {c^{7} b \,d^{3}}{8 x}-\frac {c^{5} b \,d^{2} e}{2 x}+\frac {3 c^{3} b d \,e^{2}}{4 x}-\frac {c b \,e^{3}}{2 x}+\frac {c^{3} b \,d^{3}}{40 x^{5}}-\frac {c b \,d^{2} e}{10 x^{5}}-\frac {b c \,d^{3}}{56 x^{7}}-\frac {c^{5} b \,d^{3}}{24 x^{3}}+\frac {c^{3} b \,d^{2} e}{6 x^{3}}-\frac {c b d \,e^{2}}{4 x^{3}}+\frac {c^{8} b \arctan \left (c x \right ) d^{3}}{8}-\frac {c^{6} b \arctan \left (c x \right ) d^{2} e}{2}+\frac {3 c^{4} b \arctan \left (c x \right ) d \,e^{2}}{4}-\frac {c^{2} b \arctan \left (c x \right ) e^{3}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 218, normalized size = 1.43 \[ \frac {1}{840} \, {\left ({\left (105 \, c^{7} \arctan \left (c x\right ) + \frac {105 \, c^{6} x^{6} - 35 \, c^{4} x^{4} + 21 \, c^{2} x^{2} - 15}{x^{7}}\right )} c - \frac {105 \, \arctan \left (c x\right )}{x^{8}}\right )} b d^{3} - \frac {1}{30} \, {\left ({\left (15 \, c^{5} \arctan \left (c x\right ) + \frac {15 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 3}{x^{5}}\right )} c + \frac {15 \, \arctan \left (c x\right )}{x^{6}}\right )} b d^{2} e + \frac {1}{4} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b d e^{2} - \frac {1}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b e^{3} - \frac {a e^{3}}{2 \, x^{2}} - \frac {3 \, a d e^{2}}{4 \, x^{4}} - \frac {a d^{2} e}{2 \, x^{6}} - \frac {a d^{3}}{8 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 301, normalized size = 1.98 \[ \frac {b\,c^2\,\mathrm {atan}\left (\frac {b\,c^2\,x\,\left (2\,e-c^2\,d\right )\,\left (c^4\,d^2-2\,c^2\,d\,e+2\,e^2\right )}{b\,c^7\,d^3-4\,b\,c^5\,d^2\,e+6\,b\,c^3\,d\,e^2-4\,b\,c\,e^3}\right )\,\left (2\,e-c^2\,d\right )\,\left (c^4\,d^2-2\,c^2\,d\,e+2\,e^2\right )}{8}-\frac {\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,d^3}{8}+\frac {b\,d^2\,e\,x^2}{2}+\frac {3\,b\,d\,e^2\,x^4}{4}+\frac {b\,e^3\,x^6}{2}\right )}{x^8}-\frac {a\,d^3-x^3\,\left (\frac {b\,c^3\,d^3}{5}-\frac {4\,b\,c\,d^2\,e}{5}\right )-x^7\,\left (b\,c^7\,d^3-4\,b\,c^5\,d^2\,e+6\,b\,c^3\,d\,e^2-4\,b\,c\,e^3\right )+x^5\,\left (\frac {b\,c^5\,d^3}{3}-\frac {4\,b\,c^3\,d^2\,e}{3}+2\,b\,c\,d\,e^2\right )+4\,a\,e^3\,x^6+\frac {b\,c\,d^3\,x}{7}+4\,a\,d^2\,e\,x^2+6\,a\,d\,e^2\,x^4}{8\,x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.16, size = 309, normalized size = 2.03 \[ - \frac {a d^{3}}{8 x^{8}} - \frac {a d^{2} e}{2 x^{6}} - \frac {3 a d e^{2}}{4 x^{4}} - \frac {a e^{3}}{2 x^{2}} + \frac {b c^{8} d^{3} \operatorname {atan}{\left (c x \right )}}{8} + \frac {b c^{7} d^{3}}{8 x} - \frac {b c^{6} d^{2} e \operatorname {atan}{\left (c x \right )}}{2} - \frac {b c^{5} d^{3}}{24 x^{3}} - \frac {b c^{5} d^{2} e}{2 x} + \frac {3 b c^{4} d e^{2} \operatorname {atan}{\left (c x \right )}}{4} + \frac {b c^{3} d^{3}}{40 x^{5}} + \frac {b c^{3} d^{2} e}{6 x^{3}} + \frac {3 b c^{3} d e^{2}}{4 x} - \frac {b c^{2} e^{3} \operatorname {atan}{\left (c x \right )}}{2} - \frac {b c d^{3}}{56 x^{7}} - \frac {b c d^{2} e}{10 x^{5}} - \frac {b c d e^{2}}{4 x^{3}} - \frac {b c e^{3}}{2 x} - \frac {b d^{3} \operatorname {atan}{\left (c x \right )}}{8 x^{8}} - \frac {b d^{2} e \operatorname {atan}{\left (c x \right )}}{2 x^{6}} - \frac {3 b d e^{2} \operatorname {atan}{\left (c x \right )}}{4 x^{4}} - \frac {b e^{3} \operatorname {atan}{\left (c x \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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