Optimal. Leaf size=81 \[ \frac {b \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c} e}+\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \]
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Rubi [A]
time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1156, 1128,
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 c e \sqrt {b^2-4 a c}}+\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 1128
Rule 1156
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx &=\frac {\text {Subst}\left (\int \frac {x^3}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{e}\\ &=\frac {\text {Subst}\left (\int \frac {x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 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 c e}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 c e}\\ &=\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c 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 c e}\\ &=\frac {b \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 c \sqrt {b^2-4 a c} e}+\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 77, normalized size = 0.95 \begin {gather*} \frac {-\frac {2 b \tan ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 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 = 151, normalized size = 1.86
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{3} e^{3}+3 \textit {\_R}^{2} d \,e^{2}+3 \textit {\_R} \,d^{2} e +d^{3}\right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{2 e}\) | \(151\) |
risch | \(\frac {\ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) a}{\left (4 a c -b^{2}\right ) e}-\frac {\ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) b^{2}}{4 c \left (4 a c -b^{2}\right ) e}+\frac {\ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e +2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}+\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}+2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{4 c \left (4 a c -b^{2}\right ) e}+\frac {\ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) a}{\left (4 a c -b^{2}\right ) e}-\frac {\ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) b^{2}}{4 c \left (4 a c -b^{2}\right ) e}-\frac {\ln \left (\left (-4 a b c \,e^{2}+b^{3} e^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,e^{2}\right ) x^{2}+\left (-8 a b c d e +2 b^{3} d e -2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b d e \right ) x -4 a b c \,d^{2}+b^{3} d^{2}-\sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, b \,d^{2}-2 \sqrt {-b^{2} \left (4 a c -b^{2}\right )}\, a \right ) \sqrt {-b^{2} \left (4 a c -b^{2}\right )}}{4 c \left (4 a c -b^{2}\right ) e}\) | \(1002\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 149 vs.
\(2 (74) = 148\).
time = 0.36, size = 424, normalized size = 5.23 \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 )\right )} e^{\left (-1\right )}}{4 \, {\left (b^{2} c - 4 \, a c^{2}\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 )\right )} e^{\left (-1\right )}}{4 \, {\left (b^{2} c - 4 \, a c^{2}\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. 280 vs.
\(2 (68) = 136\).
time = 0.92, size = 280, normalized size = 3.46 \begin {gather*} \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a c e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + 2 a + 2 b^{2} e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + b d^{2}}{b e^{2}} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a c e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + 2 a + 2 b^{2} e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + b d^{2}}{b e^{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.94, size = 130, normalized size = 1.60 \begin {gather*} -\frac {b \arctan \left (\frac {2 \, c d^{2} + 2 \, {\left (x^{2} e + 2 \, d x\right )} c e + b}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-1\right )}}{2 \, \sqrt {-b^{2} + 4 \, a c} c} + \frac {e^{\left (-1\right )} \log \left (c d^{4} + 2 \, {\left (x^{2} e + 2 \, d x\right )} c d^{2} e + {\left (x^{2} e + 2 \, d x\right )}^{2} c e^{2} + b d^{2} + {\left (x^{2} e + 2 \, d x\right )} b e + a\right )}{4 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.76, size = 278, normalized size = 3.43 \begin {gather*} \frac {4\,a\,c\,e\,\ln \left (c\,d^4+4\,c\,d^3\,e\,x+6\,c\,d^2\,e^2\,x^2+b\,d^2+4\,c\,d\,e^3\,x^3+2\,b\,d\,e\,x+c\,e^4\,x^4+b\,e^2\,x^2+a\right )}{16\,a\,c^2\,e^2-4\,b^2\,c\,e^2}-\frac {b^2\,e\,\ln \left (c\,d^4+4\,c\,d^3\,e\,x+6\,c\,d^2\,e^2\,x^2+b\,d^2+4\,c\,d\,e^3\,x^3+2\,b\,d\,e\,x+c\,e^4\,x^4+b\,e^2\,x^2+a\right )}{16\,a\,c^2\,e^2-4\,b^2\,c\,e^2}-\frac {b\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,d^2}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,e^2\,x^2}{\sqrt {4\,a\,c-b^2}}+\frac {4\,c\,d\,e\,x}{\sqrt {4\,a\,c-b^2}}\right )}{2\,c\,e\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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