Optimal. Leaf size=94 \[ \frac {b \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a e \sqrt {b^2-4 a c}}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}+\frac {\log (d+e x)}{a e} \]
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Rubi [A] time = 0.13, 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 = {1142, 1114, 705, 29, 634, 618, 206, 628} \[ \frac {b \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a e \sqrt {b^2-4 a c}}-\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a e}+\frac {\log (d+e x)}{a e} \]
Antiderivative was successfully verified.
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Rule 29
Rule 206
Rule 618
Rule 628
Rule 634
Rule 705
Rule 1114
Rule 1142
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 {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \left (a+b x+c x^2\right )} \, dx,x,(d+e x)^2\right )}{2 e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,(d+e x)^2\right )}{2 a e}+\frac {\operatorname {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 {\operatorname {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 \operatorname {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 \operatorname {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 = 0.08, size = 128, normalized size = 1.36 \[ \frac {4 \sqrt {b^2-4 a c} \log (d+e x)-\left (\sqrt {b^2-4 a c}+b\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c (d+e x)^2\right )+\left (b-\sqrt {b^2-4 a c}\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c (d+e x)^2\right )}{4 a e \sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 468, normalized size = 4.98 \[ \left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \, {\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \, {\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c + {\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a\right ) + 4 \, {\left (b^{2} - 4 \, a c\right )} \log \left (e x + d\right )}{4 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (b^{2} - 4 \, a c\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a\right ) + 4 \, {\left (b^{2} - 4 \, a c\right )} \log \left (e x + d\right )}{4 \, {\left (a b^{2} - 4 \, a^{2} c\right )} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.15, size = 274, normalized size = 2.91 \[ -\frac {e^{\left (-1\right )} \log \left ({\left | c x^{4} e^{4} + 4 \, c d x^{3} e^{3} + 6 \, c d^{2} x^{2} e^{2} + 4 \, c d^{3} x e + c d^{4} + b x^{2} e^{2} + 2 \, b d x e + b d^{2} + a \right |}\right )}{4 \, a} + \frac {e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a} - \frac {{\left (\frac {a b c e^{3} \log \left ({\left | b x^{2} e^{2} + 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e + b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} + 2 \, a \right |}\right )}{\sqrt {b^{2} - 4 \, a c}} - \frac {a b c e^{3} \log \left ({\left | -b x^{2} e^{2} - 2 \, b d x e + \sqrt {b^{2} - 4 \, a c} x^{2} e^{2} + 2 \, \sqrt {b^{2} - 4 \, a c} d x e - b d^{2} + \sqrt {b^{2} - 4 \, a c} d^{2} - 2 \, a \right |}\right )}{\sqrt {b^{2} - 4 \, a c}}\right )} e^{\left (-4\right )}}{4 \, a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 184, normalized size = 1.96 \[ \frac {\ln \left (e x +d \right )}{a e}+\frac {\left (-c \,e^{3} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{3}-3 c d \,e^{2} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{2}-c \,d^{3}+\left (-3 c \,d^{2}-b \right ) \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right ) e -b d \right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+x \right )}{2 a e \left (2 c \,e^{3} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{3}+6 c d \,e^{2} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{2}+6 e c \,d^{2} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+2 c \,d^{3}+b e \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+b d \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.04, size = 320, normalized size = 3.40 \[ \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a^{2} c e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) + 2 a b^{2} e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) - 2 a c + b^{2} + b c d^{2}}{b c e^{2}} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a^{2} c e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) + 2 a b^{2} e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 a e \left (4 a c - b^{2}\right )} - \frac {1}{4 a e}\right ) - 2 a c + b^{2} + b c d^{2}}{b c e^{2}} \right )} + \frac {\log {\left (\frac {d}{e} + x \right )}}{a e} \]
Verification of antiderivative is not currently implemented for this CAS.
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