Optimal. Leaf size=47 \[ -\frac {2 \tanh ^{-1}\left (\frac {b+2 c e (d+e x)}{\sqrt {b} \sqrt {b+4 c d e}}\right )}{\sqrt {b} \sqrt {b+4 c d e}} \]
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Rubi [A]
time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2006, 632, 212}
\begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {b+2 c e (d+e x)}{\sqrt {b} \sqrt {b+4 c d e}}\right )}{\sqrt {b} \sqrt {b+4 c d e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 2006
Rubi steps
\begin {align*} \int \frac {1}{b x+c (d+e x)^2} \, dx &=\int \frac {1}{c d^2+(b+2 c d e) x+c e^2 x^2} \, dx\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{b (b+4 c d e)-x^2} \, dx,x,b+2 c d e+2 c e^2 x\right )\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {b+2 c e (d+e x)}{\sqrt {b} \sqrt {b+4 c d e}}\right )}{\sqrt {b} \sqrt {b+4 c d e}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {b+2 c e (d+e x)}{\sqrt {b} \sqrt {b+4 c d e}}\right )}{\sqrt {b} \sqrt {b+4 c d e}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 43, normalized size = 0.91
method | result | size |
default | \(-\frac {2 \arctanh \left (\frac {2 c \,e^{2} x +2 c d e +b}{\sqrt {4 b c d e +b^{2}}}\right )}{\sqrt {4 b c d e +b^{2}}}\) | \(43\) |
risch | \(\frac {\ln \left (-2 c \,e^{2} x -2 c d e +\sqrt {b \left (4 c d e +b \right )}-b \right )}{\sqrt {b \left (4 c d e +b \right )}}-\frac {\ln \left (2 c \,e^{2} x +2 c d e +\sqrt {b \left (4 c d e +b \right )}+b \right )}{\sqrt {b \left (4 c d e +b \right )}}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 71, normalized size = 1.51 \begin {gather*} \frac {\log \left (\frac {2 \, c x e^{2} + 2 \, c d e + b - \sqrt {{\left (4 \, c d e + b\right )} b}}{2 \, c x e^{2} + 2 \, c d e + b + \sqrt {{\left (4 \, c d e + b\right )} b}}\right )}{\sqrt {{\left (4 \, c d e + b\right )} b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 190, normalized size = 4.04 \begin {gather*} \left [\frac {\log \left (\frac {2 \, c^{2} e^{4} x^{2} + 2 \, c^{2} d^{2} e^{2} + 4 \, b c d e + b^{2} + 2 \, {\left (2 \, c^{2} d e^{3} + b c e^{2}\right )} x - \sqrt {4 \, b c d e + b^{2}} {\left (2 \, c e^{2} x + 2 \, c d e + b\right )}}{c e^{2} x^{2} + c d^{2} + {\left (2 \, c d e + b\right )} x}\right )}{\sqrt {4 \, b c d e + b^{2}}}, \frac {2 \, \sqrt {-4 \, b c d e - b^{2}} \arctan \left (\frac {\sqrt {-4 \, b c d e - b^{2}} {\left (2 \, c e^{2} x + 2 \, c d e + b\right )}}{4 \, b c d e + b^{2}}\right )}{4 \, b c d e + b^{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. 151 vs.
\(2 (49) = 98\).
time = 0.14, size = 151, normalized size = 3.21 \begin {gather*} \sqrt {\frac {1}{b \left (b + 4 c d e\right )}} \log {\left (x + \frac {- b^{2} \sqrt {\frac {1}{b \left (b + 4 c d e\right )}} - 4 b c d e \sqrt {\frac {1}{b \left (b + 4 c d e\right )}} + b + 2 c d e}{2 c e^{2}} \right )} - \sqrt {\frac {1}{b \left (b + 4 c d e\right )}} \log {\left (x + \frac {b^{2} \sqrt {\frac {1}{b \left (b + 4 c d e\right )}} + 4 b c d e \sqrt {\frac {1}{b \left (b + 4 c d e\right )}} + b + 2 c d e}{2 c e^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.12, size = 48, normalized size = 1.02 \begin {gather*} \frac {2 \, \arctan \left (\frac {2 \, c x e^{2} + 2 \, c d e + b}{\sqrt {-4 \, b c d e - b^{2}}}\right )}{\sqrt {-4 \, b c d e - b^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 42, normalized size = 0.89 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {2\,c\,x\,e^2+2\,c\,d\,e+b}{\sqrt {b}\,\sqrt {b+4\,c\,d\,e}}\right )}{\sqrt {b}\,\sqrt {b+4\,c\,d\,e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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