Optimal. Leaf size=368 \[ \frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (-\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{2 c e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
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Rubi [A] time = 0.870422, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {1673, 1178, 1166, 205, 12, 1107, 614, 618, 206} \[ \frac{x \left (c x^2 (b d-2 a f)-a b f-2 a c d+b^2 d\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (-\frac{4 a b f-12 a c d+b^2 d}{\sqrt{b^2-4 a c}}-2 a f+b d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{2 c e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1673
Rule 1178
Rule 1166
Rule 205
Rule 12
Rule 1107
Rule 614
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{d+e x+f x^2}{\left (a+b x^2+c x^4\right )^2} \, dx &=\int \frac{e x}{\left (a+b x^2+c x^4\right )^2} \, dx+\int \frac{d+f x^2}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{\int \frac{-b^2 d+6 a c d-a b f-c (b d-2 a f) x^2}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}+e \int \frac{x}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{\left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )+\frac{\left (c \left (b d-2 a f-\frac{b^2 d-12 a c d+4 a b f}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a \left (b^2-4 a c\right )}+\frac{\left (c \left (b d-2 a f+\frac{b^2 d-12 a c d+4 a b f}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{4 a \left (b^2-4 a c\right )}\\ &=-\frac{e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (b d-2 a f+\frac{b^2 d-12 a c d+4 a b f}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (b d-2 a f-\frac{b^2 d-12 a c d+4 a b f}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}-\frac{(c e) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{b^2-4 a c}\\ &=-\frac{e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (b d-2 a f+\frac{b^2 d-12 a c d+4 a b f}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (b d-2 a f-\frac{b^2 d-12 a c d+4 a b f}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{(2 c e) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{b^2-4 a c}\\ &=-\frac{e \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{x \left (b^2 d-2 a c d-a b f+c (b d-2 a f) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{c} \left (b d-2 a f+\frac{b^2 d-12 a c d+4 a b f}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{c} \left (b d-2 a f-\frac{b^2 d-12 a c d+4 a b f}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b+\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} a \left (b^2-4 a c\right ) \sqrt{b+\sqrt{b^2-4 a c}}}+\frac{2 c e \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 1.35201, size = 398, normalized size = 1.08 \[ \frac{1}{4} \left (\frac{2 a b (e+f x)+4 a c x (d+x (e+f x))-2 b d x \left (b+c x^2\right )}{a \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \sqrt{c} \left (b \left (d \sqrt{b^2-4 a c}+4 a f\right )-2 a \left (f \sqrt{b^2-4 a c}+6 c d\right )+b^2 d\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \sqrt{c} \left (b d \sqrt{b^2-4 a c}-2 a f \sqrt{b^2-4 a c}-4 a b f+12 a c d+b^2 (-d)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}-\frac{4 c e \log \left (\sqrt{b^2-4 a c}-b-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac{4 c e \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.118, size = 1813, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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