Optimal. Leaf size=189 \[ \frac {\sqrt {2} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {e \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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Rubi [A] time = 0.21, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1673, 12, 1093, 205, 1107, 618, 206} \[ \frac {\sqrt {2} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {e \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 206
Rule 618
Rule 1093
Rule 1107
Rule 1673
Rubi steps
\begin {align*} \int \frac {d+e x}{a+b x^2+c x^4} \, dx &=\int \frac {d}{a+b x^2+c x^4} \, dx+\int \frac {e x}{a+b x^2+c x^4} \, dx\\ &=d \int \frac {1}{a+b x^2+c x^4} \, dx+e \int \frac {x}{a+b x^2+c x^4} \, dx\\ &=\frac {(c d) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{\sqrt {b^2-4 a c}}-\frac {(c d) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{\sqrt {b^2-4 a c}}+\frac {1}{2} e \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {\sqrt {2} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}-e \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )\\ &=\frac {\sqrt {2} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {e \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 194, normalized size = 1.03 \[ \frac {\frac {2 \sqrt {2} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}}}-\frac {2 \sqrt {2} \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {\sqrt {b^2-4 a c}+b}}+e \left (\log \left (\sqrt {b^2-4 a c}-b-2 c x^2\right )-\log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right )\right )}{2 \sqrt {b^2-4 a c}} \]
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 [B] time = 4.59, size = 1248, normalized size = 6.60 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 231, normalized size = 1.22 \[ \frac {2 \sqrt {-4 a c +b^{2}}\, \sqrt {2}\, c d \arctanh \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {2 \sqrt {-4 a c +b^{2}}\, \sqrt {2}\, c d \arctan \left (\frac {\sqrt {2}\, c x}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\left (8 a c -2 b^{2}\right ) \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\sqrt {-4 a c +b^{2}}\, e \ln \left (-2 c \,x^{2}-b +\sqrt {-4 a c +b^{2}}\right )}{8 a c -2 b^{2}}+\frac {\sqrt {-4 a c +b^{2}}\, e \ln \left (2 c \,x^{2}+b +\sqrt {-4 a c +b^{2}}\right )}{8 a c -2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e x + d}{c x^{4} + b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 1308, normalized size = 6.92 \[ \sum _{k=1}^4\ln \left (c^2\,\left (d\,e^2+e^3\,x+{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,b^2\,d\,4-{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^3\,b^3\,x\,8-{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,a\,c\,d\,16+\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )\,b\,e^2\,x\,2-\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )\,c\,d^2\,x\,4-{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,b^2\,e\,x\,4+\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )\,b\,d\,e\,4+{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^3\,a\,b\,c\,x\,32+{\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right )}^2\,a\,c\,e\,x\,16\right )\right )\,\mathrm {root}\left (128\,a^2\,b^2\,c\,z^4-256\,a^3\,c^2\,z^4-16\,a\,b^4\,z^4+16\,a\,b\,c\,d^2\,z^2-32\,a^2\,c\,e^2\,z^2+8\,a\,b^2\,e^2\,z^2-4\,b^3\,d^2\,z^2+16\,a\,c\,d^2\,e\,z-4\,b^2\,d^2\,e\,z-b\,d^2\,e^2-c\,d^4-a\,e^4,z,k\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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