Optimal. Leaf size=382 \[ \frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}-\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}} \]
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Rubi [A]
time = 0.26, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1868, 1869,
1890, 281, 211, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt {a} e+77 \sqrt {b} c\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (15 \sqrt {a} e+77 \sqrt {b} c\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {5 d \text {ArcTan}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 281
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 1868
Rule 1869
Rule 1890
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^4} \, dx &=-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}-\frac {\int \frac {-11 c-10 d x-9 e x^2}{\left (a+b x^4\right )^3} \, dx}{12 a}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {\int \frac {77 c+60 d x+45 e x^2}{\left (a+b x^4\right )^2} \, dx}{96 a^2}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}-\frac {\int \frac {-231 c-120 d x-45 e x^2}{a+b x^4} \, dx}{384 a^3}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}-\frac {\int \left (-\frac {120 d x}{a+b x^4}+\frac {-231 c-45 e x^2}{a+b x^4}\right ) \, dx}{384 a^3}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}-\frac {\int \frac {-231 c-45 e x^2}{a+b x^4} \, dx}{384 a^3}+\frac {(5 d) \int \frac {x}{a+b x^4} \, dx}{16 a^3}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {(5 d) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{32 a^3}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-15 e\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{256 a^3 b}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+15 e\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{256 a^3 b}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+15 e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^3 b}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+15 e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{512 a^3 b}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{512 \sqrt {2} a^{15/4} b^{3/4}}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx}{512 \sqrt {2} a^{15/4} b^{3/4}}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}-\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {5 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}-\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 379, normalized size = 0.99 \begin {gather*} \frac {\frac {8 a x (77 c+15 x (4 d+3 e x))}{a+b x^4}+\frac {32 a^2 x (11 c+x (10 d+9 e x))}{\left (a+b x^4\right )^2}-\frac {256 a^3 (a f-b x (c+x (d+e x)))}{b \left (a+b x^4\right )^3}-\frac {6 \sqrt [4]{a} \left (77 \sqrt {2} \sqrt {b} c+80 \sqrt [4]{a} \sqrt [4]{b} d+15 \sqrt {2} \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {6 \sqrt [4]{a} \left (77 \sqrt {2} \sqrt {b} c-80 \sqrt [4]{a} \sqrt [4]{b} d+15 \sqrt {2} \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {3 \sqrt {2} \left (-77 \sqrt [4]{a} \sqrt {b} c+15 a^{3/4} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}+\frac {3 \sqrt {2} \left (77 \sqrt [4]{a} \sqrt {b} c-15 a^{3/4} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}}{3072 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 340, normalized size = 0.89
method | result | size |
risch | \(\frac {\frac {15 e \,b^{2} x^{11}}{128 a^{3}}+\frac {5 d \,b^{2} x^{10}}{32 a^{3}}+\frac {77 c \,b^{2} x^{9}}{384 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {5 b d \,x^{6}}{12 a^{2}}+\frac {33 b c \,x^{5}}{64 a^{2}}+\frac {113 e \,x^{3}}{384 a}+\frac {11 d \,x^{2}}{32 a}+\frac {51 c x}{128 a}-\frac {f}{12 b}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (15 \textit {\_R}^{2} e +40 \textit {\_R} d +77 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{512 a^{3} b}\) | \(153\) |
default | \(\frac {\frac {15 e \,b^{2} x^{11}}{128 a^{3}}+\frac {5 d \,b^{2} x^{10}}{32 a^{3}}+\frac {77 c \,b^{2} x^{9}}{384 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {5 b d \,x^{6}}{12 a^{2}}+\frac {33 b c \,x^{5}}{64 a^{2}}+\frac {113 e \,x^{3}}{384 a}+\frac {11 d \,x^{2}}{32 a}+\frac {51 c x}{128 a}-\frac {f}{12 b}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\frac {77 c \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {20 d \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{\sqrt {a b}}+\frac {15 e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{128 a^{3}}\) | \(340\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 409, normalized size = 1.07 \begin {gather*} \frac {45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} + 126 \, a b^{2} x^{7} e + 160 \, a b^{2} d x^{6} + 198 \, a b^{2} c x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x - 32 \, a^{3} f}{384 \, {\left (a^{3} b^{4} x^{12} + 3 \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{4} + a^{6} b\right )}} + \frac {\frac {\sqrt {2} {\left (77 \, \sqrt {b} c - 15 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (77 \, \sqrt {b} c - 15 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e - 80 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e + 80 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{1024 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 16.51, size = 125011, normalized size = 327.25 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.66, size = 391, normalized size = 1.02 \begin {gather*} \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b^{3}} + \frac {45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} + 126 \, a b^{2} x^{7} e + 160 \, a b^{2} d x^{6} + 198 \, a b^{2} c x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x - 32 \, a^{3} f}{384 \, {\left (b x^{4} + a\right )}^{3} a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.25, size = 879, normalized size = 2.30 \begin {gather*} \left (\sum _{k=1}^4\ln \left (-\frac {b\,\left (3375\,a\,e^3-123200\,b\,c\,d^2+88935\,b\,c^2\,e-64000\,b\,d^3\,x+{\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )}^2\,a^7\,b^2\,c\,20185088-\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\,a^4\,b\,e^2\,x\,115200+92400\,b\,c\,d\,e\,x+\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\,a^3\,b^2\,c^2\,x\,3035648-{\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )}^2\,a^7\,b^2\,d\,x\,10485760+\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\,a^4\,b\,d\,e\,614400\right )}{a^9\,2097152}\right )\,\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\right )+\frac {\frac {11\,d\,x^2}{32\,a}-\frac {f}{12\,b}+\frac {113\,e\,x^3}{384\,a}+\frac {51\,c\,x}{128\,a}+\frac {77\,b^2\,c\,x^9}{384\,a^3}+\frac {5\,b^2\,d\,x^{10}}{32\,a^3}+\frac {15\,b^2\,e\,x^{11}}{128\,a^3}+\frac {33\,b\,c\,x^5}{64\,a^2}+\frac {5\,b\,d\,x^6}{12\,a^2}+\frac {21\,b\,e\,x^7}{64\,a^2}}{a^3+3\,a^2\,b\,x^4+3\,a\,b^2\,x^8+b^3\,x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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