Optimal. Leaf size=94 \[ \frac {b \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c} e}+\frac {\log (d+e x)}{a e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e} \]
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Rubi [A]
time = 0.09, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1156, 1128,
719, 29, 648, 632, 212, 642} \begin {gather*} \frac {b \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a e \sqrt {b^2-4 a c}}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}+\frac {\log (d+e x)}{a e} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 212
Rule 632
Rule 642
Rule 648
Rule 719
Rule 1128
Rule 1156
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{e}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )} \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,(d+e x)^2\right )}{2 a e}+\frac {\text {Subst}\left (\int \frac {-b-c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 a e}\\ &=\frac {\log (d+e x)}{a e}-\frac {\text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a e}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a e}\\ &=\frac {\log (d+e x)}{a e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}+\frac {b \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c (d+e x)^2\right )}{2 a e}\\ &=\frac {b \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c} e}+\frac {\log (d+e x)}{a e}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 128, normalized size = 1.36 \begin {gather*} \frac {4 \sqrt {b^2-4 a c} \log (d+e x)-\left (b+\sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )+\left (b-\sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{4 a \sqrt {b^2-4 a c} e} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.18, size = 184, normalized size = 1.96
method | result | size |
risch | \(\frac {\ln \left (e x +d \right )}{a e}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (4 a^{2} c \,e^{2}-a \,b^{2} e^{2}\right ) \textit {\_Z}^{2}+\left (4 a c e -b^{2} e \right ) \textit {\_Z} +c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (10 e^{3} a c -3 b^{2} e^{3}\right ) \textit {\_R} +5 c \,e^{2}\right ) x^{2}+\left (\left (20 a c d \,e^{2}-6 b^{2} d \,e^{2}\right ) \textit {\_R} +10 c d e \right ) x +\left (10 a \,d^{2} e c -3 b^{2} d^{2} e -a b e \right ) \textit {\_R} +5 c \,d^{2}+2 b \right )\right )}{2}\) | \(155\) |
default | \(\frac {\ln \left (e x +d \right )}{a e}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (-e^{3} c \,\textit {\_R}^{3}-3 d \,e^{2} c \,\textit {\_R}^{2}+e \left (-3 c \,d^{2}-b \right ) \textit {\_R} -c \,d^{3}-b d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{2 a e}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 460, normalized size = 4.89 \begin {gather*} \left [\frac {{\left (\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} x^{4} e^{4} + 8 \, c^{2} d x^{3} e^{3} + 2 \, c^{2} d^{4} + 2 \, b c d^{2} + 2 \, {\left (6 \, c^{2} d^{2} + b c\right )} x^{2} e^{2} + 4 \, {\left (2 \, c^{2} d^{3} + b c d\right )} x e + b^{2} - 2 \, a c + {\left (2 \, c x^{2} e^{2} + 4 \, c d x e + 2 \, c d^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} x^{2} e^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} x e + a}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} x^{2} e^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} x e + a\right ) + 4 \, {\left (b^{2} - 4 \, a c\right )} \log \left (x e + d\right )\right )} e^{\left (-1\right )}}{4 \, {\left (a b^{2} - 4 \, a^{2} c\right )}}, \frac {{\left (2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {{\left (2 \, c x^{2} e^{2} + 4 \, c d x e + 2 \, c d^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} x^{2} e^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} x e + a\right ) + 4 \, {\left (b^{2} - 4 \, a c\right )} \log \left (x e + d\right )\right )} e^{\left (-1\right )}}{4 \, {\left (a b^{2} - 4 \, a^{2} c\right )}}\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. 320 vs.
\(2 (78) = 156\).
time = 13.55, size = 320, normalized size = 3.40 \begin {gather*} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a^{2} c e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) + 2 a b^{2} e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) - 2 a c + b^{2} + b c d^{2}}{b c e^{2}} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a^{2} c e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) + 2 a b^{2} e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) - 2 a c + b^{2} + b c d^{2}}{b c e^{2}} \right )} + \frac {\log {\left (\frac {d}{e} + x \right )}}{a e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 274 vs.
\(2 (87) = 174\).
time = 4.83, size = 274, normalized size = 2.91 \begin {gather*} -\frac {e^{\left (-1\right )} \log \left ({\left | c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a \right |}\right )}{4 \, a} + \frac {e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a} - \frac {{\left (\frac {a b c e^{3} \log \left ({\left | b x^{2} e^{2} + 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right )}{\sqrt {b^{2} - 4 \, a c}} - \frac {a b c e^{3} \log \left ({\left | -b x^{2} e^{2} - 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{\sqrt {b^{2} - 4 \, a c}}\right )} e^{\left (-4\right )}}{4 \, a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.50, size = 2173, normalized size = 23.12 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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