Optimal. Leaf size=81 \[ \frac {b \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 c e \sqrt {b^2-4 a c}}+\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \]
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Rubi [A] time = 0.13, 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 = {1142, 1114, 634, 618, 206, 628} \[ \frac {b \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 c e \sqrt {b^2-4 a c}}+\frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 c e} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 1114
Rule 1142
Rubi steps
\begin {align*} \int \frac {(d+e x)^3}{a+b (d+e x)^2+c (d+e x)^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{a+b x^2+c x^4} \, dx,x,d+e x\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 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 c e}-\frac {b \operatorname {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 \operatorname {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 = 0.04, size = 77, normalized size = 0.95 \[ \frac {\log \left (a+b (d+e x)^2+c (d+e x)^4\right )-\frac {2 b \tan ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}}{4 c e} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 434, normalized size = 5.36 \[ \left [\frac {\sqrt {b^{2} - 4 \, a c} b \log \left (\frac {2 \, c^{2} e^{4} x^{4} + 8 \, c^{2} d e^{3} x^{3} + 2 \, c^{2} d^{4} + 2 \, {\left (6 \, c^{2} d^{2} + b c\right )} e^{2} x^{2} + 2 \, b c d^{2} + 4 \, {\left (2 \, c^{2} d^{3} + b c d\right )} e x + b^{2} - 2 \, a c + {\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a\right )}{4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e}, \frac {2 \, \sqrt {-b^{2} + 4 \, a c} b \arctan \left (-\frac {{\left (2 \, c e^{2} x^{2} + 4 \, c d e x + 2 \, c d^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left (b^{2} - 4 \, a c\right )} \log \left (c e^{4} x^{4} + 4 \, c d e^{3} x^{3} + c d^{4} + {\left (6 \, c d^{2} + b\right )} e^{2} x^{2} + b d^{2} + 2 \, {\left (2 \, c d^{3} + b d\right )} e x + a\right )}{4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 130, normalized size = 1.60 \[ -\frac {b \arctan \left (\frac {2 \, c d^{2} + 2 \, {\left (x^{2} e + 2 \, d x\right )} c e + b}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-1\right )}}{2 \, \sqrt {-b^{2} + 4 \, a c} c} + \frac {e^{\left (-1\right )} \log \left (c d^{4} + 2 \, {\left (x^{2} e + 2 \, d x\right )} c d^{2} e + {\left (x^{2} e + 2 \, d x\right )}^{2} c e^{2} + b d^{2} + {\left (x^{2} e + 2 \, d x\right )} b e + a\right )}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 151, normalized size = 1.86 \[ \frac {\left (\RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{3} e^{3}+3 \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{2} d \,e^{2}+3 \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right ) d^{2} e +d^{3}\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+x \right )}{2 e \left (2 c \,e^{3} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{3}+6 c d \,e^{2} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )^{2}+6 e c \,d^{2} \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+2 c \,d^{3}+b e \RootOf \left (\textit {\_Z}^{4} c \,e^{4}+4 \textit {\_Z}^{3} c d \,e^{3}+c \,d^{4}+b \,d^{2}+\left (6 c \,d^{2} e^{2}+b \,e^{2}\right ) \textit {\_Z}^{2}+\left (4 c \,d^{3} e +2 d e b \right ) \textit {\_Z} +a \right )+b d \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{3}}{{\left (e x + d\right )}^{4} c + {\left (e x + d\right )}^{2} b + a}\,{d x} \]
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 \[ \frac {4\,a\,c\,e\,\ln \left (c\,d^4+4\,c\,d^3\,e\,x+6\,c\,d^2\,e^2\,x^2+b\,d^2+4\,c\,d\,e^3\,x^3+2\,b\,d\,e\,x+c\,e^4\,x^4+b\,e^2\,x^2+a\right )}{16\,a\,c^2\,e^2-4\,b^2\,c\,e^2}-\frac {b^2\,e\,\ln \left (c\,d^4+4\,c\,d^3\,e\,x+6\,c\,d^2\,e^2\,x^2+b\,d^2+4\,c\,d\,e^3\,x^3+2\,b\,d\,e\,x+c\,e^4\,x^4+b\,e^2\,x^2+a\right )}{16\,a\,c^2\,e^2-4\,b^2\,c\,e^2}-\frac {b\,\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,d^2}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,e^2\,x^2}{\sqrt {4\,a\,c-b^2}}+\frac {4\,c\,d\,e\,x}{\sqrt {4\,a\,c-b^2}}\right )}{2\,c\,e\,\sqrt {4\,a\,c-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.64, size = 280, normalized size = 3.46 \[ \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a c e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + 2 a + 2 b^{2} e \left (- \frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + b d^{2}}{b e^{2}} \right )} + \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) \log {\left (\frac {2 d x}{e} + x^{2} + \frac {- 8 a c e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + 2 a + 2 b^{2} e \left (\frac {b \sqrt {- 4 a c + b^{2}}}{4 c e \left (4 a c - b^{2}\right )} + \frac {1}{4 c e}\right ) + b d^{2}}{b e^{2}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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