Optimal. Leaf size=115 \[ \frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) x}{4 d \left (d+e x^2\right )^2}-\frac {\left (5 c d^2-e (b d+3 a e)\right ) x}{8 d^2 e^2 \left (d+e x^2\right )}+\frac {\left (3 c d^2+e (b d+3 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{5/2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.03, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1171, 393, 211}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (e (3 a e+b d)+3 c d^2\right )}{8 d^{5/2} e^{5/2}}-\frac {x \left (5 c d^2-e (3 a e+b d)\right )}{8 d^2 e^2 \left (d+e x^2\right )}+\frac {x \left (a e^2-b d e+c d^2\right )}{4 d e^2 \left (d+e x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 393
Rule 1171
Rubi steps
\begin {align*} \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^3} \, dx &=\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) x}{4 d \left (d+e x^2\right )^2}-\frac {\int \frac {-3 a+\frac {d (c d-b e)}{e^2}-\frac {4 c d x^2}{e}}{\left (d+e x^2\right )^2} \, dx}{4 d}\\ &=\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) x}{4 d \left (d+e x^2\right )^2}-\frac {\left (5 c d^2-e (b d+3 a e)\right ) x}{8 d^2 e^2 \left (d+e x^2\right )}-\frac {\left (-\frac {4 c d^2}{e}+e \left (-3 a+\frac {d (c d-b e)}{e^2}\right )\right ) \int \frac {1}{d+e x^2} \, dx}{8 d^2 e}\\ &=\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) x}{4 d \left (d+e x^2\right )^2}-\frac {\left (5 c d^2-e (b d+3 a e)\right ) x}{8 d^2 e^2 \left (d+e x^2\right )}+\frac {\left (3 c d^2+e (b d+3 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 110, normalized size = 0.96 \begin {gather*} \frac {x \left (-c d^2 \left (3 d+5 e x^2\right )+e \left (b d \left (-d+e x^2\right )+a e \left (5 d+3 e x^2\right )\right )\right )}{8 d^2 e^2 \left (d+e x^2\right )^2}+\frac {\left (3 c d^2+e (b d+3 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 107, normalized size = 0.93
method | result | size |
default | \(\frac {\frac {\left (3 a \,e^{2}+d e b -5 c \,d^{2}\right ) x^{3}}{8 d^{2} e}+\frac {\left (5 a \,e^{2}-d e b -3 c \,d^{2}\right ) x}{8 d \,e^{2}}}{\left (e \,x^{2}+d \right )^{2}}+\frac {\left (3 a \,e^{2}+d e b +3 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 d^{2} e^{2} \sqrt {d e}}\) | \(107\) |
risch | \(\frac {\frac {\left (3 a \,e^{2}+d e b -5 c \,d^{2}\right ) x^{3}}{8 d^{2} e}+\frac {\left (5 a \,e^{2}-d e b -3 c \,d^{2}\right ) x}{8 d \,e^{2}}}{\left (e \,x^{2}+d \right )^{2}}-\frac {3 \ln \left (e x +\sqrt {-d e}\right ) a}{16 \sqrt {-d e}\, d^{2}}-\frac {\ln \left (e x +\sqrt {-d e}\right ) b}{16 \sqrt {-d e}\, e d}-\frac {3 \ln \left (e x +\sqrt {-d e}\right ) c}{16 \sqrt {-d e}\, e^{2}}+\frac {3 \ln \left (-e x +\sqrt {-d e}\right ) a}{16 \sqrt {-d e}\, d^{2}}+\frac {\ln \left (-e x +\sqrt {-d e}\right ) b}{16 \sqrt {-d e}\, e d}+\frac {3 \ln \left (-e x +\sqrt {-d e}\right ) c}{16 \sqrt {-d e}\, e^{2}}\) | \(215\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 110, normalized size = 0.96 \begin {gather*} \frac {{\left (3 \, c d^{2} + b d e + 3 \, a e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {5}{2}\right )}}{8 \, d^{\frac {5}{2}}} - \frac {{\left (5 \, c d^{2} e - b d e^{2} - 3 \, a e^{3}\right )} x^{3} + {\left (3 \, c d^{3} + b d^{2} e - 5 \, a d e^{2}\right )} x}{8 \, {\left (d^{2} x^{4} e^{4} + 2 \, d^{3} x^{2} e^{3} + d^{4} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 389, normalized size = 3.38 \begin {gather*} \left [-\frac {6 \, c d^{4} x e - 6 \, a d x^{3} e^{4} + {\left (3 \, a x^{4} e^{4} + 3 \, c d^{4} + {\left (b d x^{4} + 6 \, a d x^{2}\right )} e^{3} + {\left (3 \, c d^{2} x^{4} + 2 \, b d^{2} x^{2} + 3 \, a d^{2}\right )} e^{2} + {\left (6 \, c d^{3} x^{2} + b d^{3}\right )} e\right )} \sqrt {-d e} \log \left (\frac {x^{2} e - 2 \, \sqrt {-d e} x - d}{x^{2} e + d}\right ) - 2 \, {\left (b d^{2} x^{3} + 5 \, a d^{2} x\right )} e^{3} + 2 \, {\left (5 \, c d^{3} x^{3} + b d^{3} x\right )} e^{2}}{16 \, {\left (d^{3} x^{4} e^{5} + 2 \, d^{4} x^{2} e^{4} + d^{5} e^{3}\right )}}, -\frac {3 \, c d^{4} x e - 3 \, a d x^{3} e^{4} - {\left (3 \, a x^{4} e^{4} + 3 \, c d^{4} + {\left (b d x^{4} + 6 \, a d x^{2}\right )} e^{3} + {\left (3 \, c d^{2} x^{4} + 2 \, b d^{2} x^{2} + 3 \, a d^{2}\right )} e^{2} + {\left (6 \, c d^{3} x^{2} + b d^{3}\right )} e\right )} \sqrt {d} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\frac {1}{2}} - {\left (b d^{2} x^{3} + 5 \, a d^{2} x\right )} e^{3} + {\left (5 \, c d^{3} x^{3} + b d^{3} x\right )} e^{2}}{8 \, {\left (d^{3} x^{4} e^{5} + 2 \, d^{4} x^{2} e^{4} + d^{5} e^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.78, size = 196, normalized size = 1.70 \begin {gather*} - \frac {\sqrt {- \frac {1}{d^{5} e^{5}}} \cdot \left (3 a e^{2} + b d e + 3 c d^{2}\right ) \log {\left (- d^{3} e^{2} \sqrt {- \frac {1}{d^{5} e^{5}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{d^{5} e^{5}}} \cdot \left (3 a e^{2} + b d e + 3 c d^{2}\right ) \log {\left (d^{3} e^{2} \sqrt {- \frac {1}{d^{5} e^{5}}} + x \right )}}{16} + \frac {x^{3} \cdot \left (3 a e^{3} + b d e^{2} - 5 c d^{2} e\right ) + x \left (5 a d e^{2} - b d^{2} e - 3 c d^{3}\right )}{8 d^{4} e^{2} + 16 d^{3} e^{3} x^{2} + 8 d^{2} e^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.23, size = 101, normalized size = 0.88 \begin {gather*} \frac {{\left (3 \, c d^{2} + b d e + 3 \, a e^{2}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {5}{2}\right )}}{8 \, d^{\frac {5}{2}}} - \frac {{\left (5 \, c d^{2} x^{3} e - b d x^{3} e^{2} + 3 \, c d^{3} x - 3 \, a x^{3} e^{3} + b d^{2} x e - 5 \, a d x e^{2}\right )} e^{\left (-2\right )}}{8 \, {\left (x^{2} e + d\right )}^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 112, normalized size = 0.97 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (3\,c\,d^2+b\,d\,e+3\,a\,e^2\right )}{8\,d^{5/2}\,e^{5/2}}-\frac {\frac {x\,\left (3\,c\,d^2+b\,d\,e-5\,a\,e^2\right )}{8\,d\,e^2}-\frac {x^3\,\left (-5\,c\,d^2+b\,d\,e+3\,a\,e^2\right )}{8\,d^2\,e}}{d^2+2\,d\,e\,x^2+e^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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