Optimal. Leaf size=105 \[ -\frac {d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac {b c \left (2 c^2 d-3 e\right )}{36 x^3}-\frac {1}{12} b c^4 \left (2 c^2 d-3 e\right ) \tan ^{-1}(c x)-\frac {b c^3 \left (2 c^2 d-3 e\right )}{12 x}-\frac {b c d}{30 x^5} \]
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Rubi [A] time = 0.11, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 4976, 12, 453, 325, 203} \[ -\frac {d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac {b c \left (2 c^2 d-3 e\right )}{36 x^3}-\frac {b c^3 \left (2 c^2 d-3 e\right )}{12 x}-\frac {1}{12} b c^4 \left (2 c^2 d-3 e\right ) \tan ^{-1}(c x)-\frac {b c d}{30 x^5} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 203
Rule 325
Rule 453
Rule 4976
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \tan ^{-1}(c x)\right )}{x^7} \, dx &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-(b c) \int \frac {-2 d-3 e x^2}{12 x^6 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {1}{12} (b c) \int \frac {-2 d-3 e x^2}{x^6 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {b c d}{30 x^5}-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {1}{12} \left (b c \left (2 c^2 d-3 e\right )\right ) \int \frac {1}{x^4 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {b c d}{30 x^5}+\frac {b c \left (2 c^2 d-3 e\right )}{36 x^3}-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}+\frac {1}{12} \left (b c^3 \left (2 c^2 d-3 e\right )\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac {b c d}{30 x^5}+\frac {b c \left (2 c^2 d-3 e\right )}{36 x^3}-\frac {b c^3 \left (2 c^2 d-3 e\right )}{12 x}-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}-\frac {1}{12} \left (b c^5 \left (2 c^2 d-3 e\right )\right ) \int \frac {1}{1+c^2 x^2} \, dx\\ &=-\frac {b c d}{30 x^5}+\frac {b c \left (2 c^2 d-3 e\right )}{36 x^3}-\frac {b c^3 \left (2 c^2 d-3 e\right )}{12 x}-\frac {1}{12} b c^4 \left (2 c^2 d-3 e\right ) \tan ^{-1}(c x)-\frac {d \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{4 x^4}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 97, normalized size = 0.92 \[ -\frac {a d}{6 x^6}-\frac {a e}{4 x^4}-\frac {b c d \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-c^2 x^2\right )}{30 x^5}-\frac {b c e \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-c^2 x^2\right )}{12 x^3}-\frac {b d \tan ^{-1}(c x)}{6 x^6}-\frac {b e \tan ^{-1}(c x)}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 98, normalized size = 0.93 \[ -\frac {15 \, {\left (2 \, b c^{5} d - 3 \, b c^{3} e\right )} x^{5} + 6 \, b c d x + 45 \, a e x^{2} - 5 \, {\left (2 \, b c^{3} d - 3 \, b c e\right )} x^{3} + 30 \, a d + 15 \, {\left ({\left (2 \, b c^{6} d - 3 \, b c^{4} e\right )} x^{6} + 3 \, b e x^{2} + 2 \, b d\right )} \arctan \left (c x\right )}{180 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 106, normalized size = 1.01 \[ -\frac {a e}{4 x^{4}}-\frac {a d}{6 x^{6}}-\frac {b \arctan \left (c x \right ) e}{4 x^{4}}-\frac {b \arctan \left (c x \right ) d}{6 x^{6}}-\frac {c^{5} b d}{6 x}+\frac {b \,c^{3} e}{4 x}+\frac {c^{3} b d}{18 x^{3}}-\frac {c b e}{12 x^{3}}-\frac {b c d}{30 x^{5}}-\frac {c^{6} b \arctan \left (c x \right ) d}{6}+\frac {b \,c^{4} e \arctan \left (c x \right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 103, normalized size = 0.98 \[ -\frac {1}{90} \, {\left ({\left (15 \, c^{5} \arctan \left (c x\right ) + \frac {15 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 3}{x^{5}}\right )} c + \frac {15 \, \arctan \left (c x\right )}{x^{6}}\right )} b d + \frac {1}{12} \, {\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac {3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac {3 \, \arctan \left (c x\right )}{x^{4}}\right )} b e - \frac {a e}{4 \, x^{4}} - \frac {a d}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.59, size = 130, normalized size = 1.24 \[ \frac {b\,c^4\,\mathrm {atan}\left (\frac {b\,c^2\,x\,\left (3\,e-2\,c^2\,d\right )}{3\,b\,c\,e-2\,b\,c^3\,d}\right )\,\left (3\,e-2\,c^2\,d\right )}{12}-\frac {\mathrm {atan}\left (c\,x\right )\,\left (\frac {b\,e\,x^2}{4}+\frac {b\,d}{6}\right )}{x^6}-\frac {x^3\,\left (b\,c\,e-\frac {2\,b\,c^3\,d}{3}\right )+2\,a\,d-c^2\,x^5\,\left (3\,b\,c\,e-2\,b\,c^3\,d\right )+3\,a\,e\,x^2+\frac {2\,b\,c\,d\,x}{5}}{12\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.60, size = 122, normalized size = 1.16 \[ - \frac {a d}{6 x^{6}} - \frac {a e}{4 x^{4}} - \frac {b c^{6} d \operatorname {atan}{\left (c x \right )}}{6} - \frac {b c^{5} d}{6 x} + \frac {b c^{4} e \operatorname {atan}{\left (c x \right )}}{4} + \frac {b c^{3} d}{18 x^{3}} + \frac {b c^{3} e}{4 x} - \frac {b c d}{30 x^{5}} - \frac {b c e}{12 x^{3}} - \frac {b d \operatorname {atan}{\left (c x \right )}}{6 x^{6}} - \frac {b e \operatorname {atan}{\left (c x \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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