Optimal. Leaf size=111 \[ -\frac {\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{6 d x^6}+\frac {b c d \left (c^2 d-3 e\right )}{18 x^3}-\frac {b \left (c^2 d-e\right )^3 \tan ^{-1}(c x)}{6 d}-\frac {b c \left (c^4 d^2-3 c^2 d e+3 e^2\right )}{6 x}-\frac {b c d^2}{30 x^5} \]
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Rubi [A] time = 0.15, antiderivative size = 111, 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} \[ -\frac {\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{6 d x^6}-\frac {b c \left (c^4 d^2-3 c^2 d e+3 e^2\right )}{6 x}+\frac {b c d \left (c^2 d-3 e\right )}{18 x^3}-\frac {b \left (c^2 d-e\right )^3 \tan ^{-1}(c x)}{6 d}-\frac {b c d^2}{30 x^5} \]
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 )^2 \left (a+b \tan ^{-1}(c x)\right )}{x^7} \, dx &=-\frac {\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{6 d x^6}-(b c) \int \frac {\left (d+e x^2\right )^3}{6 x^6 \left (-d-c^2 d x^2\right )} \, dx\\ &=-\frac {\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{6 d x^6}-\frac {1}{6} (b c) \int \frac {\left (d+e x^2\right )^3}{x^6 \left (-d-c^2 d x^2\right )} \, dx\\ &=-\frac {\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{6 d x^6}-\frac {1}{6} (b c) \int \left (-\frac {d^2}{x^6}+\frac {d \left (c^2 d-3 e\right )}{x^4}+\frac {-c^4 d^2+3 c^2 d e-3 e^2}{x^2}+\frac {\left (c^2 d-e\right )^3}{d \left (1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {b c d^2}{30 x^5}+\frac {b c d \left (c^2 d-3 e\right )}{18 x^3}-\frac {b c \left (c^4 d^2-3 c^2 d e+3 e^2\right )}{6 x}-\frac {\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{6 d x^6}-\frac {\left (b c \left (c^2 d-e\right )^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{6 d}\\ &=-\frac {b c d^2}{30 x^5}+\frac {b c d \left (c^2 d-3 e\right )}{18 x^3}-\frac {b c \left (c^4 d^2-3 c^2 d e+3 e^2\right )}{6 x}-\frac {b \left (c^2 d-e\right )^3 \tan ^{-1}(c x)}{6 d}-\frac {\left (d+e x^2\right )^3 \left (a+b \tan ^{-1}(c x)\right )}{6 d x^6}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 112, normalized size = 1.01 \[ -\frac {5 \left (\left (d^2+3 d e x^2+3 e^2 x^4\right ) \left (a+b \tan ^{-1}(c x)\right )+b c d e x^3 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-c^2 x^2\right )+3 b c e^2 x^5 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-c^2 x^2\right )\right )+b c d^2 x \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-c^2 x^2\right )}{30 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 145, normalized size = 1.31 \[ -\frac {45 \, a e^{2} x^{4} + 15 \, {\left (b c^{5} d^{2} - 3 \, b c^{3} d e + 3 \, b c e^{2}\right )} x^{5} + 3 \, b c d^{2} x + 45 \, a d e x^{2} - 5 \, {\left (b c^{3} d^{2} - 3 \, b c d e\right )} x^{3} + 15 \, a d^{2} + 15 \, {\left (3 \, b e^{2} x^{4} + {\left (b c^{6} d^{2} - 3 \, b c^{4} d e + 3 \, b c^{2} e^{2}\right )} x^{6} + 3 \, b d e x^{2} + b d^{2}\right )} \arctan \left (c x\right )}{90 \, x^{6}} \]
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 = 168, normalized size = 1.51 \[ -\frac {a e d}{2 x^{4}}-\frac {a \,d^{2}}{6 x^{6}}-\frac {a \,e^{2}}{2 x^{2}}-\frac {b \arctan \left (c x \right ) e d}{2 x^{4}}-\frac {b \arctan \left (c x \right ) d^{2}}{6 x^{6}}-\frac {b \arctan \left (c x \right ) e^{2}}{2 x^{2}}-\frac {c^{5} b \,d^{2}}{6 x}+\frac {c^{3} b e d}{2 x}-\frac {c b \,e^{2}}{2 x}+\frac {c^{3} b \,d^{2}}{18 x^{3}}-\frac {c b e d}{6 x^{3}}-\frac {b c \,d^{2}}{30 x^{5}}-\frac {c^{6} b \arctan \left (c x \right ) d^{2}}{6}+\frac {c^{4} b \arctan \left (c x \right ) e d}{2}-\frac {c^{2} b \arctan \left (c x \right ) e^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 145, normalized size = 1.31 \[ -\frac {1}{90} \, {\left ({\left (15 \, c^{5} \arctan \left (c x\right ) + \frac {15 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 3}{x^{5}}\right )} c + \frac {15 \, \arctan \left (c x\right )}{x^{6}}\right )} b d^{2} + \frac {1}{6} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b d e - \frac {1}{2} \, {\left ({\left (c \arctan \left (c x\right ) + \frac {1}{x}\right )} c + \frac {\arctan \left (c x\right )}{x^{2}}\right )} b e^{2} - \frac {a e^{2}}{2 \, x^{2}} - \frac {a d e}{2 \, x^{4}} - \frac {a d^{2}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 256, normalized size = 2.31 \[ -\frac {\frac {a\,d^2}{6}+\frac {b\,d^2\,\mathrm {atan}\left (c\,x\right )}{6}-\frac {a\,c^4\,e^2\,x^8}{2}+\frac {a\,e\,x^4\,\left (d\,c^2+e\right )}{2}+\frac {b\,c\,x^5\,\left (2\,c^4\,d^2-6\,c^2\,d\,e+9\,e^2\right )}{18}+\frac {b\,c\,d^2\,x}{30}+\frac {a\,d\,x^2\,\left (d\,c^2+3\,e\right )}{6}+\frac {b\,c^3\,x^7\,\left (c^4\,d^2-3\,c^2\,d\,e+3\,e^2\right )}{6}+\frac {b\,c\,d\,x^3\,\left (15\,e-2\,c^2\,d\right )}{90}+\frac {b\,d\,x^2\,\mathrm {atan}\left (c\,x\right )\,\left (d\,c^2+3\,e\right )}{6}+\frac {b\,c^2\,e^2\,x^6\,\mathrm {atan}\left (c\,x\right )}{2}+\frac {b\,e\,x^4\,\mathrm {atan}\left (c\,x\right )\,\left (d\,c^2+e\right )}{2}}{c^2\,x^8+x^6}-\frac {\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {c^2}}\right )\,{\left (c^2\right )}^{5/2}\,\left (b\,c^4\,d^2-3\,b\,c^2\,d\,e+3\,b\,e^2\right )}{6\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.87, size = 192, normalized size = 1.73 \[ - \frac {a d^{2}}{6 x^{6}} - \frac {a d e}{2 x^{4}} - \frac {a e^{2}}{2 x^{2}} - \frac {b c^{6} d^{2} \operatorname {atan}{\left (c x \right )}}{6} - \frac {b c^{5} d^{2}}{6 x} + \frac {b c^{4} d e \operatorname {atan}{\left (c x \right )}}{2} + \frac {b c^{3} d^{2}}{18 x^{3}} + \frac {b c^{3} d e}{2 x} - \frac {b c^{2} e^{2} \operatorname {atan}{\left (c x \right )}}{2} - \frac {b c d^{2}}{30 x^{5}} - \frac {b c d e}{6 x^{3}} - \frac {b c e^{2}}{2 x} - \frac {b d^{2} \operatorname {atan}{\left (c x \right )}}{6 x^{6}} - \frac {b d e \operatorname {atan}{\left (c x \right )}}{2 x^{4}} - \frac {b e^{2} \operatorname {atan}{\left (c x \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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