Optimal. Leaf size=152 \[ -\frac {b c d^3}{56 x^7}+\frac {b c d^2 \left (c^2 d-4 e\right )}{40 x^5}-\frac {b c d \left (c^4 d^2-4 c^2 d e+6 e^2\right )}{24 x^3}+\frac {b c \left (c^2 d-2 e\right ) \left (c^4 d^2-2 c^2 d e+2 e^2\right )}{8 x}+\frac {b \left (c^2 d-e\right )^4 \text {ArcTan}(c x)}{8 d}-\frac {\left (d+e x^2\right )^4 (a+b \text {ArcTan}(c x))}{8 d x^8} \]
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Rubi [A]
time = 0.14, 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 = {270, 5096, 12,
472, 209} \begin {gather*} -\frac {\left (d+e x^2\right )^4 (a+b \text {ArcTan}(c x))}{8 d x^8}+\frac {b \text {ArcTan}(c x) \left (c^2 d-e\right )^4}{8 d}+\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 c d^3}{56 x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 270
Rule 472
Rule 5096
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 [A]
time = 3.01, size = 204, normalized size = 1.34 \begin {gather*} \frac {-105 a \left (d^3+4 d^2 e x^2+6 d e^2 x^4+4 e^3 x^6\right )+b c x \left (-420 e^3 x^6+210 d e^2 x^4 \left (-1+3 c^2 x^2\right )-28 d^2 e x^2 \left (3-5 c^2 x^2+15 c^4 x^4\right )+d^3 \left (-15+21 c^2 x^2-35 c^4 x^4+105 c^6 x^6\right )\right )+105 b \left (-4 e^3 x^6 \left (1+c^2 x^2\right )+6 d e^2 x^4 \left (-1+c^4 x^4\right )-4 d^2 e x^2 \left (1+c^6 x^6\right )+d^3 \left (-1+c^8 x^8\right )\right ) \text {ArcTan}(c x)}{840 x^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(299\) vs.
\(2(140)=280\).
time = 0.43, size = 300, normalized size = 1.97
method | result | size |
derivativedivides | \(c^{8} \left (\frac {a \left (-\frac {3 d \,e^{2}}{4 c^{2} x^{4}}-\frac {d^{3}}{8 c^{2} x^{8}}-\frac {d^{2} e}{2 c^{2} x^{6}}-\frac {e^{3}}{2 c^{2} x^{2}}\right )}{c^{6}}-\frac {3 b \arctan \left (c x \right ) d \,e^{2}}{4 c^{8} x^{4}}-\frac {b \arctan \left (c x \right ) d^{3}}{8 c^{8} x^{8}}-\frac {b \arctan \left (c x \right ) d^{2} e}{2 c^{8} x^{6}}-\frac {b \arctan \left (c x \right ) e^{3}}{2 c^{8} x^{2}}+\frac {b \,d^{3} \arctan \left (c x \right )}{8}-\frac {b \,d^{2} e \arctan \left (c x \right )}{2 c^{2}}+\frac {3 b d \,e^{2} \arctan \left (c x \right )}{4 c^{4}}-\frac {b \,e^{3} \arctan \left (c x \right )}{2 c^{6}}+\frac {b \,d^{3}}{8 c x}-\frac {b \,d^{2} e}{2 c^{3} x}+\frac {3 b d \,e^{2}}{4 c^{5} x}-\frac {b \,e^{3}}{2 c^{7} x}+\frac {b \,d^{3}}{40 c^{5} x^{5}}-\frac {b \,d^{2} e}{10 c^{7} x^{5}}-\frac {b \,d^{3}}{56 c^{7} x^{7}}-\frac {b \,d^{3}}{24 c^{3} x^{3}}+\frac {b \,d^{2} e}{6 c^{5} x^{3}}-\frac {b d \,e^{2}}{4 c^{7} x^{3}}\right )\) | \(300\) |
default | \(c^{8} \left (\frac {a \left (-\frac {3 d \,e^{2}}{4 c^{2} x^{4}}-\frac {d^{3}}{8 c^{2} x^{8}}-\frac {d^{2} e}{2 c^{2} x^{6}}-\frac {e^{3}}{2 c^{2} x^{2}}\right )}{c^{6}}-\frac {3 b \arctan \left (c x \right ) d \,e^{2}}{4 c^{8} x^{4}}-\frac {b \arctan \left (c x \right ) d^{3}}{8 c^{8} x^{8}}-\frac {b \arctan \left (c x \right ) d^{2} e}{2 c^{8} x^{6}}-\frac {b \arctan \left (c x \right ) e^{3}}{2 c^{8} x^{2}}+\frac {b \,d^{3} \arctan \left (c x \right )}{8}-\frac {b \,d^{2} e \arctan \left (c x \right )}{2 c^{2}}+\frac {3 b d \,e^{2} \arctan \left (c x \right )}{4 c^{4}}-\frac {b \,e^{3} \arctan \left (c x \right )}{2 c^{6}}+\frac {b \,d^{3}}{8 c x}-\frac {b \,d^{2} e}{2 c^{3} x}+\frac {3 b d \,e^{2}}{4 c^{5} x}-\frac {b \,e^{3}}{2 c^{7} x}+\frac {b \,d^{3}}{40 c^{5} x^{5}}-\frac {b \,d^{2} e}{10 c^{7} x^{5}}-\frac {b \,d^{3}}{56 c^{7} x^{7}}-\frac {b \,d^{3}}{24 c^{3} x^{3}}+\frac {b \,d^{2} e}{6 c^{5} x^{3}}-\frac {b d \,e^{2}}{4 c^{7} x^{3}}\right )\) | \(300\) |
risch | \(\frac {i b \left (4 e^{3} x^{6}+6 e^{2} d \,x^{4}+4 d^{2} e \,x^{2}+d^{3}\right ) \ln \left (i c x +1\right )}{16 x^{8}}-\frac {840 b c \,e^{3} x^{7}+168 b c \,d^{2} e \,x^{3}+420 b c d \,e^{2} x^{5}-280 b \,c^{3} d^{2} e \,x^{5}-1260 b \,c^{3} d \,e^{2} x^{7}+70 x^{5} c^{5} d^{3} b +420 i b \,e^{3} x^{6} \ln \left (-i c x +1\right )+420 i b \,d^{2} e \,x^{2} \ln \left (-i c x +1\right )+630 i b d \,e^{2} x^{4} \ln \left (-i c x +1\right )+105 i b \,d^{3} \ln \left (-i c x +1\right )+30 b c \,d^{3} x +840 a \,e^{3} x^{6}-42 b \,d^{3} c^{3} x^{3}-210 x^{7} c^{7} d^{3} b +840 x^{7} c^{5} e \,d^{2} b +1260 a d \,e^{2} x^{4}+840 a \,d^{2} e \,x^{2}+210 d^{3} a -105 i \ln \left (-c x -i\right ) b \,c^{8} d^{3} x^{8}+105 i \ln \left (-c x +i\right ) b \,c^{8} d^{3} x^{8}+420 i \ln \left (-c x -i\right ) b \,c^{2} e^{3} x^{8}-420 i \ln \left (-c x +i\right ) b \,c^{2} e^{3} x^{8}+420 i \ln \left (-c x -i\right ) b \,c^{6} d^{2} e \,x^{8}-420 i \ln \left (-c x +i\right ) b \,c^{6} d^{2} e \,x^{8}-630 i \ln \left (-c x -i\right ) b \,c^{4} d \,e^{2} x^{8}+630 i \ln \left (-c x +i\right ) b \,c^{4} d \,e^{2} x^{8}}{1680 x^{8}}\) | \(446\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 216, normalized size = 1.42 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.38, size = 237, normalized size = 1.56 \begin {gather*} \frac {105 \, b c^{7} d^{3} x^{7} - 35 \, b c^{5} d^{3} x^{5} + 21 \, b c^{3} d^{3} x^{3} - 15 \, b c d^{3} x - 105 \, a d^{3} + 105 \, {\left (b c^{8} d^{3} x^{8} - b d^{3} - 4 \, {\left (b c^{2} x^{8} + b x^{6}\right )} e^{3} + 6 \, {\left (b c^{4} d x^{8} - b d x^{4}\right )} e^{2} - 4 \, {\left (b c^{6} d^{2} x^{8} + b d^{2} x^{2}\right )} e\right )} \arctan \left (c x\right ) - 420 \, {\left (b c x^{7} + a x^{6}\right )} e^{3} + 210 \, {\left (3 \, b c^{3} d x^{7} - b c d x^{5} - 3 \, a d x^{4}\right )} e^{2} - 28 \, {\left (15 \, b c^{5} d^{2} x^{7} - 5 \, b c^{3} d^{2} x^{5} + 3 \, b c d^{2} x^{3} + 15 \, a d^{2} x^{2}\right )} e}{840 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 309 vs.
\(2 (139) = 278\).
time = 0.66, size = 309, normalized size = 2.03 \begin {gather*} - \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}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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