Optimal. Leaf size=266 \[ -\frac {21 d \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 d \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}-\frac {21 d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {3 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}+\frac {x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2} \]
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Rubi [A] time = 0.25, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {1855, 1876, 211, 1165, 628, 1162, 617, 204, 275, 205} \[ \frac {x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}-\frac {21 d \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 d \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}-\frac {21 d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {3 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}+\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 205
Rule 211
Rule 275
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1855
Rule 1876
Rubi steps
\begin {align*} \int \frac {d+e x}{\left (a+c x^4\right )^3} \, dx &=\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}-\frac {\int \frac {-7 d-6 e x}{\left (a+c x^4\right )^2} \, dx}{8 a}\\ &=\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac {x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac {\int \frac {21 d+12 e x}{a+c x^4} \, dx}{32 a^2}\\ &=\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac {x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac {\int \left (\frac {21 d}{a+c x^4}+\frac {12 e x}{a+c x^4}\right ) \, dx}{32 a^2}\\ &=\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac {x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac {(21 d) \int \frac {1}{a+c x^4} \, dx}{32 a^2}+\frac {(3 e) \int \frac {x}{a+c x^4} \, dx}{8 a^2}\\ &=\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac {x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac {(21 d) \int \frac {\sqrt {a}-\sqrt {c} x^2}{a+c x^4} \, dx}{64 a^{5/2}}+\frac {(21 d) \int \frac {\sqrt {a}+\sqrt {c} x^2}{a+c x^4} \, dx}{64 a^{5/2}}+\frac {(3 e) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{16 a^2}\\ &=\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac {x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac {3 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}+\frac {(21 d) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^{5/2} \sqrt {c}}+\frac {(21 d) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{128 a^{5/2} \sqrt {c}}-\frac {(21 d) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}-\frac {(21 d) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}\\ &=\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac {x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac {3 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {21 d \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 d \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {(21 d) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}-\frac {(21 d) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}\\ &=\frac {x (d+e x)}{8 a \left (a+c x^4\right )^2}+\frac {x (7 d+6 e x)}{32 a^2 \left (a+c x^4\right )}+\frac {3 e \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {c}}-\frac {21 d \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 d \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} \sqrt [4]{c}}-\frac {21 d \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}+\frac {21 d \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{128 \sqrt {2} a^{11/4} \sqrt [4]{c}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 249, normalized size = 0.94 \[ \frac {\frac {32 a^{7/4} x (d+e x)}{\left (a+c x^4\right )^2}+\frac {8 a^{3/4} x (7 d+6 e x)}{a+c x^4}-\frac {6 \left (8 \sqrt [4]{a} e+7 \sqrt {2} \sqrt [4]{c} d\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt {c}}+\frac {6 \left (7 \sqrt {2} \sqrt [4]{c} d-8 \sqrt [4]{a} e\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt {c}}-\frac {21 \sqrt {2} d \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{\sqrt [4]{c}}+\frac {21 \sqrt {2} d \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{\sqrt [4]{c}}}{256 a^{11/4}} \]
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 = 0.32, size = 260, normalized size = 0.98 \[ \frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} d \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c} - \frac {21 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} d \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c} + \frac {3 \, \sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a c} c e + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} c d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{2}} + \frac {3 \, \sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a c} c e + 7 \, \left (a c^{3}\right )^{\frac {1}{4}} c d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{2}} + \frac {6 \, c x^{6} e + 7 \, c d x^{5} + 10 \, a x^{2} e + 11 \, a d x}{32 \, {\left (c x^{4} + a\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 222, normalized size = 0.83 \[ \frac {e \,x^{2}}{8 \left (c \,x^{4}+a \right )^{2} a}+\frac {d x}{8 \left (c \,x^{4}+a \right )^{2} a}+\frac {3 e \,x^{2}}{16 \left (c \,x^{4}+a \right ) a^{2}}+\frac {7 d x}{32 \left (c \,x^{4}+a \right ) a^{2}}+\frac {3 e \arctan \left (\sqrt {\frac {c}{a}}\, x^{2}\right )}{16 \sqrt {a c}\, a^{2}}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )}{128 a^{3}}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )}{128 a^{3}}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, d \ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{c}}}\right )}{256 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.20, size = 269, normalized size = 1.01 \[ \frac {6 \, c e x^{6} + 7 \, c d x^{5} + 10 \, a e x^{2} + 11 \, a d x}{32 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + \frac {3 \, {\left (\frac {7 \, \sqrt {2} d \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}} - \frac {7 \, \sqrt {2} d \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {1}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} d - 8 \, \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {1}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} d + 8 \, \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {1}{4}}}\right )}}{256 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 315, normalized size = 1.18 \[ \frac {\frac {5\,e\,x^2}{16\,a}+\frac {11\,d\,x}{32\,a}+\frac {7\,c\,d\,x^5}{32\,a^2}+\frac {3\,c\,e\,x^6}{16\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\left (\sum _{k=1}^4\ln \left (\frac {c^2\,\left (63\,d\,e^2+36\,e^3\,x-{\mathrm {root}\left (268435456\,a^{11}\,c^2\,z^4+4718592\,a^6\,c\,e^2\,z^2-2709504\,a^3\,c\,d^2\,e\,z+194481\,c\,d^4+20736\,a\,e^4,z,k\right )}^2\,a^5\,c\,d\,7168-\mathrm {root}\left (268435456\,a^{11}\,c^2\,z^4+4718592\,a^6\,c\,e^2\,z^2-2709504\,a^3\,c\,d^2\,e\,z+194481\,c\,d^4+20736\,a\,e^4,z,k\right )\,a^2\,c\,d^2\,x\,1176+{\mathrm {root}\left (268435456\,a^{11}\,c^2\,z^4+4718592\,a^6\,c\,e^2\,z^2-2709504\,a^3\,c\,d^2\,e\,z+194481\,c\,d^4+20736\,a\,e^4,z,k\right )}^2\,a^5\,c\,e\,x\,4096\right )\,3}{a^6\,2048}\right )\,\mathrm {root}\left (268435456\,a^{11}\,c^2\,z^4+4718592\,a^6\,c\,e^2\,z^2-2709504\,a^3\,c\,d^2\,e\,z+194481\,c\,d^4+20736\,a\,e^4,z,k\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.48, size = 192, normalized size = 0.72 \[ \operatorname {RootSum} {\left (268435456 t^{4} a^{11} c^{2} + 4718592 t^{2} a^{6} c e^{2} - 2709504 t a^{3} c d^{2} e + 20736 a e^{4} + 194481 c d^{4}, \left (t \mapsto t \log {\left (x + \frac {- 67108864 t^{3} a^{9} c e^{2} - 9633792 t^{2} a^{6} c d^{2} e - 589824 t a^{4} e^{4} - 2765952 t a^{3} c d^{4} + 423360 a d^{2} e^{3}}{193536 a d e^{4} - 453789 c d^{5}} \right )} \right )\right )} + \frac {11 a d x + 10 a e x^{2} + 7 c d x^{5} + 6 c e x^{6}}{32 a^{4} + 64 a^{3} c x^{4} + 32 a^{2} c^{2} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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