Optimal. Leaf size=73 \[ -\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {\sqrt {c} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {c d \log (x)}{a^2}+\frac {c d \log \left (a+c x^2\right )}{2 a^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {815, 649, 211,
266} \begin {gather*} -\frac {\sqrt {c} e \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}+\frac {c d \log \left (a+c x^2\right )}{2 a^2}-\frac {c d \log (x)}{a^2}-\frac {d}{2 a x^2}-\frac {e}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 266
Rule 649
Rule 815
Rubi steps
\begin {align*} \int \frac {d+e x}{x^3 \left (a+c x^2\right )} \, dx &=\int \left (\frac {d}{a x^3}+\frac {e}{a x^2}-\frac {c d}{a^2 x}-\frac {c (a e-c d x)}{a^2 \left (a+c x^2\right )}\right ) \, dx\\ &=-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {c d \log (x)}{a^2}-\frac {c \int \frac {a e-c d x}{a+c x^2} \, dx}{a^2}\\ &=-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {c d \log (x)}{a^2}+\frac {\left (c^2 d\right ) \int \frac {x}{a+c x^2} \, dx}{a^2}-\frac {(c e) \int \frac {1}{a+c x^2} \, dx}{a}\\ &=-\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {\sqrt {c} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {c d \log (x)}{a^2}+\frac {c d \log \left (a+c x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 73, normalized size = 1.00 \begin {gather*} -\frac {d}{2 a x^2}-\frac {e}{a x}-\frac {\sqrt {c} e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {c d \log (x)}{a^2}+\frac {c d \log \left (a+c x^2\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 64, normalized size = 0.88
method | result | size |
default | \(-\frac {d}{2 a \,x^{2}}-\frac {e}{a x}-\frac {c d \ln \left (x \right )}{a^{2}}-\frac {c \left (-\frac {d \ln \left (c \,x^{2}+a \right )}{2}+\frac {a e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}\right )}{a^{2}}\) | \(64\) |
risch | \(\frac {-\frac {x e}{a}-\frac {d}{2 a}}{x^{2}}-\frac {c d \ln \left (x \right )}{a^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{4} \textit {\_Z}^{2}-2 a^{2} c d \textit {\_Z} +a c \,e^{2}+c^{2} d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 a^{3} \textit {\_R}^{2}-3 \textit {\_R} a c d +2 e^{2} c \right ) x +a^{2} e \textit {\_R} +2 c d e \right )\right )}{2}\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 62, normalized size = 0.85 \begin {gather*} -\frac {c \arctan \left (\frac {c x}{\sqrt {a c}}\right ) e}{\sqrt {a c} a} + \frac {c d \log \left (c x^{2} + a\right )}{2 \, a^{2}} - \frac {c d \log \left (x\right )}{a^{2}} - \frac {2 \, x e + d}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 6.82, size = 157, normalized size = 2.15 \begin {gather*} \left [\frac {a x^{2} \sqrt {-\frac {c}{a}} e \log \left (\frac {c x^{2} - 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) + c d x^{2} \log \left (c x^{2} + a\right ) - 2 \, c d x^{2} \log \left (x\right ) - 2 \, a x e - a d}{2 \, a^{2} x^{2}}, -\frac {2 \, a x^{2} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) e - c d x^{2} \log \left (c x^{2} + a\right ) + 2 \, c d x^{2} \log \left (x\right ) + 2 \, a x e + a d}{2 \, a^{2} x^{2}}\right ] \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. 360 vs.
\(2 (65) = 130\).
time = 1.73, size = 360, normalized size = 4.93 \begin {gather*} \left (\frac {c d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) \log {\left (x + \frac {- 12 a^{4} d \left (\frac {c d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right )^{2} - 2 a^{3} e^{2} \left (\frac {c d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) - 6 a^{2} c d^{2} \left (\frac {c d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) - 2 a c d e^{2} + 6 c^{2} d^{3}}{a c e^{3} + 9 c^{2} d^{2} e} \right )} + \left (\frac {c d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) \log {\left (x + \frac {- 12 a^{4} d \left (\frac {c d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right )^{2} - 2 a^{3} e^{2} \left (\frac {c d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) - 6 a^{2} c d^{2} \left (\frac {c d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{2 a^{4}}\right ) - 2 a c d e^{2} + 6 c^{2} d^{3}}{a c e^{3} + 9 c^{2} d^{2} e} \right )} + \frac {- d - 2 e x}{2 a x^{2}} - \frac {c d \log {\left (x \right )}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.01, size = 66, normalized size = 0.90 \begin {gather*} -\frac {c \arctan \left (\frac {c x}{\sqrt {a c}}\right ) e}{\sqrt {a c} a} + \frac {c d \log \left (c x^{2} + a\right )}{2 \, a^{2}} - \frac {c d \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {2 \, a x e + a d}{2 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 154, normalized size = 2.11 \begin {gather*} \frac {\ln \left (a\,e\,\sqrt {-a^5\,c}+3\,a^3\,c\,d-a^3\,c\,e\,x+3\,c\,d\,x\,\sqrt {-a^5\,c}\right )\,\left (e\,\sqrt {-a^5\,c}+a^2\,c\,d\right )}{2\,a^4}-\frac {\ln \left (a\,e\,\sqrt {-a^5\,c}-3\,a^3\,c\,d+a^3\,c\,e\,x+3\,c\,d\,x\,\sqrt {-a^5\,c}\right )\,\left (e\,\sqrt {-a^5\,c}-a^2\,c\,d\right )}{2\,a^4}-\frac {\frac {d}{2\,a}+\frac {e\,x}{a}}{x^2}-\frac {c\,d\,\ln \left (x\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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