∫ 1____________ (df+efx)3(a+b(d+ex)2+c(d+ex)4) dx [644]

Optimal. Leaf size=133 \[ -\frac {1}{2 a e f^3 (d+e x)^2}-\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c} e f^3}-\frac {b \log (d+e x)}{a^2 e f^3}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f^3} \]

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Rubi [A]
time = 0.13, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1156, 1128, 723, 814, 648, 632, 212, 642} \begin {gather*} -\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 e f^3 \sqrt {b^2-4 a c}}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f^3}-\frac {b \log (d+e x)}{a^2 e f^3}-\frac {1}{2 a e f^3 (d+e x)^2} \end {gather*}

\begin {align*} \int \frac {1}{(d f+e f x)^3 \left (a+b (d+e x)^2+c (d+e x)^4\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^3 \left (a+b x^2+c x^4\right )} \, dx,x,d+e x\right )}{e f^3}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x^2 \left (a+b x+c x^2\right )} \, dx,x,(d+e x)^2\right )}{2 e f^3}\\ &=-\frac {1}{2 a e f^3 (d+e x)^2}+\frac {\text {Subst}\left (\int \frac {-b-c x}{x \left (a+b x+c x^2\right )} \, dx,x,(d+e x)^2\right )}{2 a e f^3}\\ &=-\frac {1}{2 a e f^3 (d+e x)^2}+\frac {\text {Subst}\left (\int \left (-\frac {b}{a x}+\frac {b^2-a c+b c x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,(d+e x)^2\right )}{2 a e f^3}\\ &=-\frac {1}{2 a e f^3 (d+e x)^2}-\frac {b \log (d+e x)}{a^2 e f^3}+\frac {\text {Subst}\left (\int \frac {b^2-a c+b c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{2 a^2 e f^3}\\ &=-\frac {1}{2 a e f^3 (d+e x)^2}-\frac {b \log (d+e x)}{a^2 e f^3}+\frac {b \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^2 e f^3}+\frac {\left (b^2-2 a c\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,(d+e x)^2\right )}{4 a^2 e f^3}\\ &=-\frac {1}{2 a e f^3 (d+e x)^2}-\frac {b \log (d+e x)}{a^2 e f^3}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f^3}-\frac {\left (b^2-2 a c\right ) \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^2 e f^3}\\ &=-\frac {1}{2 a e f^3 (d+e x)^2}-\frac {\left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c (d+e x)^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c} e f^3}-\frac {b \log (d+e x)}{a^2 e f^3}+\frac {b \log \left (a+b (d+e x)^2+c (d+e x)^4\right )}{4 a^2 e f^3}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 157, normalized size = 1.18 \begin {gather*} \frac {-\frac {2 a}{(d+e x)^2}-4 b \log (d+e x)+\frac {\left (b^2-2 a c+b \sqrt {b^2-4 a c}\right ) \log \left (b-\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\sqrt {b^2-4 a c}}+\frac {\left (-b^2+2 a c+b \sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c (d+e x)^2\right )}{\sqrt {b^2-4 a c}}}{4 a^2 e f^3} \end {gather*}

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.24, size = 217, normalized size = 1.63


method	result	size

default	\(\frac {-\frac {1}{2 a e \left (e x +d \right )^{2}}-\frac {b \ln \left (e x +d \right )}{e \,a^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (e^{4} c \,\textit {\_Z}^{4}+4 d \,e^{3} c \,\textit {\_Z}^{3}+\left (6 d^{2} e^{2} c +e^{2} b \right ) \textit {\_Z}^{2}+\left (4 d^{3} e c +2 d e b \right ) \textit {\_Z} +d^{4} c +d^{2} b +a \right )}{\sum }\frac {\left (b c \,e^{3} \textit {\_R}^{3}+3 b c d \,e^{2} \textit {\_R}^{2}+e \left (3 b c \,d^{2}-a c +b^{2}\right ) \textit {\_R} +b c \,d^{3}-a c d +b^{2} d \right ) \ln \left (x -\textit {\_R} \right )}{2 e^{3} c \,\textit {\_R}^{3}+6 d \,e^{2} c \,\textit {\_R}^{2}+6 c \,d^{2} e \textit {\_R} +2 c \,d^{3}+e b \textit {\_R} +b d}}{2 a^{2} e}}{f^{3}}\)	\(217\)
risch	\(-\frac {1}{2 a e \,f^{3} \left (e x +d \right )^{2}}-\frac {b \ln \left (e x +d \right )}{a^{2} e \,f^{3}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (4 a^{3} c \,e^{2} f^{6}-a^{2} b^{2} e^{2} f^{6}\right ) \textit {\_Z}^{2}+\left (-4 a b c e \,f^{3}+b^{3} e \,f^{3}\right ) \textit {\_Z} +c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (10 a^{3} c \,e^{4} f^{6}-3 a^{2} b^{2} e^{4} f^{6}\right ) \textit {\_R}^{2}-4 a b c \,e^{3} f^{3} \textit {\_R} +2 c^{2} e^{2}\right ) x^{2}+\left (\left (20 a^{3} c d \,e^{3} f^{6}-6 a^{2} b^{2} d \,e^{3} f^{6}\right ) \textit {\_R}^{2}-8 a b c d \,e^{2} f^{3} \textit {\_R} +4 c^{2} d e \right ) x +\left (10 a^{3} c \,d^{2} e^{2} f^{6}-3 a^{2} b^{2} d^{2} e^{2} f^{6}-a^{3} b \,e^{2} f^{6}\right ) \textit {\_R}^{2}+\left (-4 a b c \,d^{2} e \,f^{3}+a^{2} c e \,f^{3}-2 a \,b^{2} e \,f^{3}\right ) \textit {\_R} +2 c^{2} d^{2}+2 b c \right )\right )}{2}\)	\(310\)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (124) = 248\).
time = 0.49, size = 820, normalized size = 6.17 \begin {gather*} \left [-\frac {2 \, a b^{2} - 8 \, a^{2} c + {\left ({\left (b^{2} - 2 \, a c\right )} x^{2} e^{2} + 2 \, {\left (b^{2} - 2 \, a c\right )} d x e + {\left (b^{2} - 2 \, a c\right )} d^{2}\right )} \sqrt {b^{2} - 4 \, a c} \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 ({\left (b^{3} - 4 \, a b c\right )} x^{2} e^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d x e + {\left (b^{3} - 4 \, a b c\right )} d^{2}\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 ({\left (b^{3} - 4 \, a b c\right )} x^{2} e^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d x e + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \log \left (x e + d\right )}{4 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f^{3} x^{2} e^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d f^{3} x e^{2} + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} f^{3} e\right )}}, -\frac {2 \, a b^{2} - 8 \, a^{2} c + 2 \, {\left ({\left (b^{2} - 2 \, a c\right )} x^{2} e^{2} + 2 \, {\left (b^{2} - 2 \, a c\right )} d x e + {\left (b^{2} - 2 \, a c\right )} d^{2}\right )} \sqrt {-b^{2} + 4 \, a c} \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 ({\left (b^{3} - 4 \, a b c\right )} x^{2} e^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d x e + {\left (b^{3} - 4 \, a b c\right )} d^{2}\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 ({\left (b^{3} - 4 \, a b c\right )} x^{2} e^{2} + 2 \, {\left (b^{3} - 4 \, a b c\right )} d x e + {\left (b^{3} - 4 \, a b c\right )} d^{2}\right )} \log \left (x e + d\right )}{4 \, {\left ({\left (a^{2} b^{2} - 4 \, a^{3} c\right )} f^{3} x^{2} e^{3} + 2 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d f^{3} x e^{2} + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} d^{2} f^{3} e\right )}}\right ] \end {gather*}

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (124) = 248\).
time = 3.88, size = 348, normalized size = 2.62 \begin {gather*} \frac {b 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^{2} f^{3}} - \frac {b e^{\left (-1\right )} \log \left ({\left | x e + d \right |}\right )}{a^{2} f^{3}} - \frac {e^{\left (-1\right )}}{2 \, {\left (x e + d\right )}^{2} a f^{3}} + \frac {{\left (\frac {{\left (a^{2} b^{2} c f^{3} e^{3} - 2 \, a^{3} c^{2} f^{3} e^{3}\right )} \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 {{\left (a^{2} b^{2} c f^{3} e^{3} - 2 \, a^{3} c^{2} f^{3} e^{3}\right )} \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^{4} c f^{6}} \end {gather*}

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Mupad [B]
time = 6.98, size = 2500, normalized size = 18.80 \begin {gather*} \text {Too large to display} \end {gather*}

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3.7.44 \(\int \frac {1}{(d f+e f x)^3 (a+b (d+e x)^2+c (d+e x)^4)} \, dx\) [644]