Optimal. Leaf size=341 \[ -\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}}-\frac {\left (5 \sqrt {a} e+21 \sqrt {b} c\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (5 \sqrt {a} e+21 \sqrt {b} c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2} \]
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Rubi [A] time = 0.31, antiderivative size = 341, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1855, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628} \[ -\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{11/4} b^{3/4}}-\frac {\left (5 \sqrt {a} e+21 \sqrt {b} c\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (5 \sqrt {a} e+21 \sqrt {b} c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}+\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 275
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 1855
Rule 1876
Rubi steps
\begin {align*} \int \frac {c+d x+e x^2}{\left (a+b x^4\right )^3} \, dx &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}-\frac {\int \frac {-7 c-6 d x-5 e x^2}{\left (a+b x^4\right )^2} \, dx}{8 a}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {\int \frac {21 c+12 d x+5 e x^2}{a+b x^4} \, dx}{32 a^2}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {\int \left (\frac {12 d x}{a+b x^4}+\frac {21 c+5 e x^2}{a+b x^4}\right ) \, dx}{32 a^2}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {\int \frac {21 c+5 e x^2}{a+b x^4} \, dx}{32 a^2}+\frac {(3 d) \int \frac {x}{a+b x^4} \, dx}{8 a^2}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {(3 d) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{16 a^2}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \int \frac {\sqrt {a} \sqrt {b}-b x^2}{a+b x^4} \, dx}{64 a^2 b}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}+5 e\right ) \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx}{64 a^2 b}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 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}{128 \sqrt {2} a^{9/4} b^{3/4}}-\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 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}{128 \sqrt {2} a^{9/4} b^{3/4}}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}+5 e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^2 b}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}+5 e\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{128 a^2 b}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}-\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a+b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a+b x^4\right )}+\frac {3 d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}-\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} b^{3/4}}-\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{128 \sqrt {2} a^{9/4} b^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 337, normalized size = 0.99 \[ \frac {\frac {\sqrt {2} \left (5 a^{3/4} e-21 \sqrt [4]{a} \sqrt {b} c\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{b^{3/4}}+\frac {\sqrt {2} \left (21 \sqrt [4]{a} \sqrt {b} c-5 a^{3/4} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{b^{3/4}}+\frac {32 a^2 x (c+x (d+e x))}{\left (a+b x^4\right )^2}-\frac {2 \sqrt [4]{a} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt {2} \sqrt {a} e+21 \sqrt {2} \sqrt {b} c\right )}{b^{3/4}}+\frac {2 \sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (-24 \sqrt [4]{a} \sqrt [4]{b} d+5 \sqrt {2} \sqrt {a} e+21 \sqrt {2} \sqrt {b} c\right )}{b^{3/4}}+\frac {8 a x (7 c+x (6 d+5 e x))}{a+b x^4}}{256 a^3} \]
Antiderivative was successfully verified.
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fricas [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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giac [A] time = 0.19, size = 336, normalized size = 0.99 \[ \frac {5 \, b x^{7} e + 6 \, b d x^{6} + 7 \, b c x^{5} + 9 \, a x^{3} e + 10 \, a d x^{2} + 11 \, a c x}{32 \, {\left (b x^{4} + a\right )}^{2} a^{2}} + \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (12 \, \sqrt {2} \sqrt {a b} b^{2} d + 21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} b^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} - \frac {\sqrt {2} {\left (21 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{3} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 396, normalized size = 1.16 \[ \frac {e \,x^{3}}{8 \left (b \,x^{4}+a \right )^{2} a}+\frac {d \,x^{2}}{8 \left (b \,x^{4}+a \right )^{2} a}+\frac {5 e \,x^{3}}{32 \left (b \,x^{4}+a \right ) a^{2}}+\frac {c x}{8 \left (b \,x^{4}+a \right )^{2} a}+\frac {3 d \,x^{2}}{16 \left (b \,x^{4}+a \right ) a^{2}}+\frac {7 c x}{32 \left (b \,x^{4}+a \right ) a^{2}}+\frac {3 d \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{16 \sqrt {a b}\, a^{2}}+\frac {5 \sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {5 \sqrt {2}\, e \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {5 \sqrt {2}\, e \ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{256 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{128 a^{3}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{128 a^{3}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{256 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.09, size = 336, normalized size = 0.99 \[ \frac {5 \, b e x^{7} + 6 \, b d x^{6} + 7 \, b c x^{5} + 9 \, a e x^{3} + 10 \, a d x^{2} + 11 \, a c x}{32 \, {\left (a^{2} b^{2} x^{8} + 2 \, a^{3} b x^{4} + a^{4}\right )}} + \frac {\frac {\sqrt {2} {\left (21 \, \sqrt {b} c - 5 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (21 \, \sqrt {b} c - 5 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e - 24 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e + 24 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{256 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.05, size = 826, normalized size = 2.42 \[ \frac {\frac {5\,d\,x^2}{16\,a}+\frac {9\,e\,x^3}{32\,a}+\frac {11\,c\,x}{32\,a}+\frac {7\,b\,c\,x^5}{32\,a^2}+\frac {3\,b\,d\,x^6}{16\,a^2}+\frac {5\,b\,e\,x^7}{32\,a^2}}{a^2+2\,a\,b\,x^4+b^2\,x^8}+\left (\sum _{k=1}^4\ln \left (-\frac {b\,\left (125\,a\,e^3-3024\,b\,c\,d^2+2205\,b\,c^2\,e-1728\,b\,d^3\,x+{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,c\,344064-\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\,a^3\,b\,e^2\,x\,3200+2520\,b\,c\,d\,e\,x+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\,a^2\,b^2\,c^2\,x\,56448-{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,d\,x\,196608+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\,a^3\,b\,d\,e\,15360\right )}{a^6\,32768}\right )\,\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4+6881280\,a^6\,b^2\,c\,e\,z^2+4718592\,a^6\,b^2\,d^2\,z^2-2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4+625\,a^2\,e^4+194481\,b^2\,c^4,z,k\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 40.86, size = 558, normalized size = 1.64 \[ \operatorname {RootSum} {\left (268435456 t^{4} a^{11} b^{3} + t^{2} \left (6881280 a^{6} b^{2} c e + 4718592 a^{6} b^{2} d^{2}\right ) + t \left (153600 a^{4} b d e^{2} - 2709504 a^{3} b^{2} c^{2} d\right ) + 625 a^{2} e^{4} + 22050 a b c^{2} e^{2} - 60480 a b c d^{2} e + 20736 a b d^{4} + 194481 b^{2} c^{4}, \left (t \mapsto t \log {\left (x + \frac {262144000 t^{3} a^{10} b^{2} e^{3} - 4624220160 t^{3} a^{9} b^{3} c^{2} e + 12683575296 t^{3} a^{9} b^{3} c d^{2} + 309657600 t^{2} a^{7} b^{2} c d e^{2} - 283115520 t^{2} a^{7} b^{2} d^{3} e + 1820786688 t^{2} a^{6} b^{3} c^{3} d + 5040000 t a^{5} b c e^{4} + 6912000 t a^{5} b d^{2} e^{3} - 118540800 t a^{4} b^{2} c^{3} e^{2} + 365783040 t a^{4} b^{2} c^{2} d^{2} e + 111476736 t a^{4} b^{2} c d^{4} + 522764928 t a^{3} b^{3} c^{5} + 112500 a^{3} d e^{5} + 4536000 a^{2} b c d^{3} e^{2} - 2488320 a^{2} b d^{5} e + 58344300 a b^{2} c^{4} d e - 80015040 a b^{2} c^{3} d^{3}}{15625 a^{3} e^{6} - 275625 a^{2} b c^{2} e^{4} + 3024000 a^{2} b c d^{2} e^{3} - 2073600 a^{2} b d^{4} e^{2} - 4862025 a b^{2} c^{4} e^{2} + 53343360 a b^{2} c^{3} d^{2} e - 36578304 a b^{2} c^{2} d^{4} + 85766121 b^{3} c^{6}} \right )} \right )\right )} + \frac {11 a c x + 10 a d x^{2} + 9 a e x^{3} + 7 b c x^{5} + 6 b d x^{6} + 5 b e x^{7}}{32 a^{4} + 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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