Optimal. Leaf size=605 \[ \frac{\tan ^{-1}\left (\frac{-\sqrt{8 a^2-4 a c+b^2}+4 a x+b}{\sqrt{2} \sqrt{-b \left (b-\sqrt{8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (b-\sqrt{8 a^2-4 a c+b^2}\right )-C \sqrt{8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (b-\sqrt{8 a^2-4 a c+b^2}\right )+4 a^2 B\right )}{\sqrt{2} a \sqrt{8 a^2-4 a c+b^2} \sqrt{-b \left (b-\sqrt{8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}-\frac{\tan ^{-1}\left (\frac{\sqrt{8 a^2-4 a c+b^2}+4 a x+b}{\sqrt{2} \sqrt{-b \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+C \sqrt{8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+4 a^2 B\right )}{\sqrt{2} a \sqrt{8 a^2-4 a c+b^2} \sqrt{-b \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}-\frac{\log \left (x \left (b-\sqrt{8 a^2-4 a c+b^2}\right )+2 a x^2+2 a\right ) \left (D \left (b-\sqrt{8 a^2-4 a c+b^2}\right )+2 a (A-C)\right )}{4 a \sqrt{8 a^2-4 a c+b^2}}+\frac{\log \left (x \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+2 a x^2+2 a\right ) \left (D \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+2 a (A-C)\right )}{4 a \sqrt{8 a^2-4 a c+b^2}} \]
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Rubi [A] time = 4.53531, antiderivative size = 605, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {2086, 634, 618, 204, 628} \[ \frac{\tan ^{-1}\left (\frac{-\sqrt{8 a^2-4 a c+b^2}+4 a x+b}{\sqrt{2} \sqrt{-b \left (b-\sqrt{8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (b-\sqrt{8 a^2-4 a c+b^2}\right )-C \sqrt{8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (b-\sqrt{8 a^2-4 a c+b^2}\right )+4 a^2 B\right )}{\sqrt{2} a \sqrt{8 a^2-4 a c+b^2} \sqrt{-b \left (b-\sqrt{8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}-\frac{\tan ^{-1}\left (\frac{\sqrt{8 a^2-4 a c+b^2}+4 a x+b}{\sqrt{2} \sqrt{-b \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+C \sqrt{8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+4 a^2 B\right )}{\sqrt{2} a \sqrt{8 a^2-4 a c+b^2} \sqrt{-b \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}-\frac{\log \left (x \left (b-\sqrt{8 a^2-4 a c+b^2}\right )+2 a x^2+2 a\right ) \left (D \left (b-\sqrt{8 a^2-4 a c+b^2}\right )+2 a (A-C)\right )}{4 a \sqrt{8 a^2-4 a c+b^2}}+\frac{\log \left (x \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+2 a x^2+2 a\right ) \left (D \left (\sqrt{8 a^2-4 a c+b^2}+b\right )+2 a (A-C)\right )}{4 a \sqrt{8 a^2-4 a c+b^2}} \]
Antiderivative was successfully verified.
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Rule 2086
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx &=-\frac{\int \frac{A b-2 a B-A \sqrt{8 a^2+b^2-4 a c}+2 a D+\left (2 a A-2 a C+b D-\sqrt{8 a^2+b^2-4 a c} D\right ) x}{2 a+\left (b-\sqrt{8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{\sqrt{8 a^2+b^2-4 a c}}+\frac{\int \frac{A b-2 a B+A \sqrt{8 a^2+b^2-4 a c}+2 a D+\left (2 a A-2 a C+b D+\sqrt{8 a^2+b^2-4 a c} D\right ) x}{2 a+\left (b+\sqrt{8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{\sqrt{8 a^2+b^2-4 a c}}\\ &=-\frac{\left (2 a (A-C)+\left (b-\sqrt{8 a^2+b^2-4 a c}\right ) D\right ) \int \frac{b-\sqrt{8 a^2+b^2-4 a c}+4 a x}{2 a+\left (b-\sqrt{8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{4 a \sqrt{8 a^2+b^2-4 a c}}+\frac{\left (2 a (A-C)+\left (b+\sqrt{8 a^2+b^2-4 a c}\right ) D\right ) \int \frac{b+\sqrt{8 a^2+b^2-4 a c}+4 a x}{2 a+\left (b+\sqrt{8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{4 a \sqrt{8 a^2+b^2-4 a c}}+\frac{\left (4 a^2 B+b \left (b-\sqrt{8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b-\sqrt{8 a^2+b^2-4 a c}\right )+b C-\sqrt{8 a^2+b^2-4 a c} C+2 c D\right )\right ) \int \frac{1}{2 a+\left (b-\sqrt{8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{2 a \sqrt{8 a^2+b^2-4 a c}}-\frac{\left (4 a^2 B+b \left (b+\sqrt{8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b+\sqrt{8 a^2+b^2-4 a c}\right )+b C+\sqrt{8 a^2+b^2-4 a c} C+2 c D\right )\right ) \int \frac{1}{2 a+\left (b+\sqrt{8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{2 a \sqrt{8 a^2+b^2-4 a c}}\\ &=-\frac{\left (2 a (A-C)+\left (b-\sqrt{8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b-\sqrt{8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt{8 a^2+b^2-4 a c}}+\frac{\left (2 a (A-C)+\left (b+\sqrt{8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b+\sqrt{8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt{8 a^2+b^2-4 a c}}-\frac{\left (4 a^2 B+b \left (b-\sqrt{8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b-\sqrt{8 a^2+b^2-4 a c}\right )+b C-\sqrt{8 a^2+b^2-4 a c} C+2 c D\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-16 a^2+\left (b-\sqrt{8 a^2+b^2-4 a c}\right )^2-x^2} \, dx,x,b-\sqrt{8 a^2+b^2-4 a c}+4 a x\right )}{a \sqrt{8 a^2+b^2-4 a c}}+\frac{\left (4 a^2 B+b \left (b+\sqrt{8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b+\sqrt{8 a^2+b^2-4 a c}\right )+b C+\sqrt{8 a^2+b^2-4 a c} C+2 c D\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-16 a^2+\left (b+\sqrt{8 a^2+b^2-4 a c}\right )^2-x^2} \, dx,x,b+\sqrt{8 a^2+b^2-4 a c}+4 a x\right )}{a \sqrt{8 a^2+b^2-4 a c}}\\ &=\frac{\left (4 a^2 B+b \left (b-\sqrt{8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b-\sqrt{8 a^2+b^2-4 a c}\right )+b C-\sqrt{8 a^2+b^2-4 a c} C+2 c D\right )\right ) \tan ^{-1}\left (\frac{b-\sqrt{8 a^2+b^2-4 a c}+4 a x}{\sqrt{2} \sqrt{4 a^2+2 a c-b \left (b-\sqrt{8 a^2+b^2-4 a c}\right )}}\right )}{\sqrt{2} a \sqrt{8 a^2+b^2-4 a c} \sqrt{4 a^2+2 a c-b \left (b-\sqrt{8 a^2+b^2-4 a c}\right )}}-\frac{\left (4 a^2 B+b \left (b+\sqrt{8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b+\sqrt{8 a^2+b^2-4 a c}\right )+b C+\sqrt{8 a^2+b^2-4 a c} C+2 c D\right )\right ) \tan ^{-1}\left (\frac{b+\sqrt{8 a^2+b^2-4 a c}+4 a x}{\sqrt{2} \sqrt{4 a^2+2 a c-b \left (b+\sqrt{8 a^2+b^2-4 a c}\right )}}\right )}{\sqrt{2} a \sqrt{8 a^2+b^2-4 a c} \sqrt{4 a^2+2 a c-b \left (b+\sqrt{8 a^2+b^2-4 a c}\right )}}-\frac{\left (2 a (A-C)+\left (b-\sqrt{8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b-\sqrt{8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt{8 a^2+b^2-4 a c}}+\frac{\left (2 a (A-C)+\left (b+\sqrt{8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b+\sqrt{8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt{8 a^2+b^2-4 a c}}\\ \end{align*}
Mathematica [C] time = 0.0750757, size = 98, normalized size = 0.16 \[ \text{RootSum}\left [\text{$\#$1}^4 a+\text{$\#$1}^3 b+\text{$\#$1}^2 c+\text{$\#$1} b+a\& ,\frac{\text{$\#$1}^2 C \log (x-\text{$\#$1})+\text{$\#$1}^3 D \log (x-\text{$\#$1})+A \log (x-\text{$\#$1})+\text{$\#$1} B \log (x-\text{$\#$1})}{4 \text{$\#$1}^3 a+3 \text{$\#$1}^2 b+2 \text{$\#$1} c+b}\& \right ] \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 2105, normalized size = 3.5 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{D x^{3} + C x^{2} + B x + A}{a x^{4} + b x^{3} + c x^{2} + b x + a}\,{d x} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{D x^{3} + C x^{2} + B x + A}{a x^{4} + b x^{3} + c x^{2} + b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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