Optimal. Leaf size=241 \[ -\frac {3 c \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {3 c \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {3 c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {3 c \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}+\frac {x (c+d x)}{4 a \left (a+b x^4\right )} \]
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Rubi [A] time = 0.20, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ -\frac {3 c \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {3 c \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {3 c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {3 c \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}+\frac {x (c+d x)}{4 a \left (a+b x^4\right )} \]
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 {c+d x}{\left (a+b x^4\right )^2} \, dx &=\frac {x (c+d x)}{4 a \left (a+b x^4\right )}-\frac {\int \frac {-3 c-2 d x}{a+b x^4} \, dx}{4 a}\\ &=\frac {x (c+d x)}{4 a \left (a+b x^4\right )}-\frac {\int \left (-\frac {3 c}{a+b x^4}-\frac {2 d x}{a+b x^4}\right ) \, dx}{4 a}\\ &=\frac {x (c+d x)}{4 a \left (a+b x^4\right )}+\frac {(3 c) \int \frac {1}{a+b x^4} \, dx}{4 a}+\frac {d \int \frac {x}{a+b x^4} \, dx}{2 a}\\ &=\frac {x (c+d x)}{4 a \left (a+b x^4\right )}+\frac {(3 c) \int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx}{8 a^{3/2}}+\frac {(3 c) \int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx}{8 a^{3/2}}+\frac {d \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,x^2\right )}{4 a}\\ &=\frac {x (c+d x)}{4 a \left (a+b x^4\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}+\frac {(3 c) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} \sqrt {b}}+\frac {(3 c) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx}{16 a^{3/2} \sqrt {b}}-\frac {(3 c) \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}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(3 c) \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}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}}\\ &=\frac {x (c+d x)}{4 a \left (a+b x^4\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {3 c \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {3 c \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b}}\\ &=\frac {x (c+d x)}{4 a \left (a+b x^4\right )}+\frac {d \tan ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {3 c \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {3 c \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {3 c \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {3 c \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 224, normalized size = 0.93 \[ \frac {\frac {8 a^{3/4} x (c+d x)}{a+b x^4}-\frac {2 \left (4 \sqrt [4]{a} d+3 \sqrt {2} \sqrt [4]{b} c\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {b}}+\frac {2 \left (3 \sqrt {2} \sqrt [4]{b} c-4 \sqrt [4]{a} d\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {b}}-\frac {3 \sqrt {2} c \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt [4]{b}}+\frac {3 \sqrt {2} c \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt [4]{b}}}{32 a^{7/4}} \]
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 = 0.17, size = 238, normalized size = 0.99 \[ \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b} - \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b} + \frac {d x^{2} + c x}{4 \, {\left (b x^{4} + a\right )} a} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{2}} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 188, normalized size = 0.78 \[ \frac {d \,x^{2}}{4 \left (b \,x^{4}+a \right ) a}+\frac {c x}{4 \left (b \,x^{4}+a \right ) a}+\frac {d \arctan \left (\sqrt {\frac {b}{a}}\, x^{2}\right )}{4 \sqrt {a b}\, a}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{16 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{16 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c \ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {\frac {a}{b}}}\right )}{32 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 238, normalized size = 0.99 \[ \frac {d x^{2} + c x}{4 \, {\left (a b x^{4} + a^{2}\right )}} + \frac {\frac {3 \, \sqrt {2} c \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {3 \, \sqrt {2} c \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} c - 4 \, \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {1}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} c + 4 \, \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {1}{4}}}}{32 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.94, size = 282, normalized size = 1.17 \[ \left (\sum _{k=1}^4\ln \left (\frac {b^2\,\left (3\,c\,d^2+2\,d^3\,x-{\mathrm {root}\left (65536\,a^7\,b^2\,z^4+2048\,a^4\,b\,d^2\,z^2-1152\,a^2\,b\,c^2\,d\,z+81\,b\,c^4+16\,a\,d^4,z,k\right )}^2\,a^3\,b\,c\,192+{\mathrm {root}\left (65536\,a^7\,b^2\,z^4+2048\,a^4\,b\,d^2\,z^2-1152\,a^2\,b\,c^2\,d\,z+81\,b\,c^4+16\,a\,d^4,z,k\right )}^2\,a^3\,b\,d\,x\,128-\mathrm {root}\left (65536\,a^7\,b^2\,z^4+2048\,a^4\,b\,d^2\,z^2-1152\,a^2\,b\,c^2\,d\,z+81\,b\,c^4+16\,a\,d^4,z,k\right )\,a\,b\,c^2\,x\,36\right )}{a^3\,16}\right )\,\mathrm {root}\left (65536\,a^7\,b^2\,z^4+2048\,a^4\,b\,d^2\,z^2-1152\,a^2\,b\,c^2\,d\,z+81\,b\,c^4+16\,a\,d^4,z,k\right )\right )+\frac {\frac {d\,x^2}{4\,a}+\frac {c\,x}{4\,a}}{b\,x^4+a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.51, size = 155, normalized size = 0.64 \[ \operatorname {RootSum} {\left (65536 t^{4} a^{7} b^{2} + 2048 t^{2} a^{4} b d^{2} - 1152 t a^{2} b c^{2} d + 16 a d^{4} + 81 b c^{4}, \left (t \mapsto t \log {\left (x + \frac {- 32768 t^{3} a^{6} b d^{2} - 4608 t^{2} a^{4} b c^{2} d - 512 t a^{3} d^{4} - 1296 t a^{2} b c^{4} + 360 a c^{2} d^{3}}{192 a c d^{4} - 243 b c^{5}} \right )} \right )\right )} + \frac {c x + d x^{2}}{4 a^{2} + 4 a b x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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