Optimal. Leaf size=368 \[ \frac{(d x)^{m+1} \left (\frac{2 A c-b C}{\sqrt{b^2-4 a c}}+C\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{d (m+1) \left (b-\sqrt{b^2-4 a c}\right )}+\frac{(d x)^{m+1} \left (C-\frac{2 A c-b C}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \left (\sqrt{b^2-4 a c}+b\right )}+\frac{2 B c (d x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{d^2 (m+2) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 B c (d x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d^2 (m+2) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]
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Rubi [A] time = 0.621653, antiderivative size = 368, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1662, 1285, 364, 12, 1131} \[ \frac{(d x)^{m+1} \left (\frac{2 A c-b C}{\sqrt{b^2-4 a c}}+C\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{d (m+1) \left (b-\sqrt{b^2-4 a c}\right )}+\frac{(d x)^{m+1} \left (C-\frac{2 A c-b C}{\sqrt{b^2-4 a c}}\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1) \left (\sqrt{b^2-4 a c}+b\right )}+\frac{2 B c (d x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{d^2 (m+2) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 B c (d x)^{m+2} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{d^2 (m+2) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]
Antiderivative was successfully verified.
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Rule 1662
Rule 1285
Rule 364
Rule 12
Rule 1131
Rubi steps
\begin{align*} \int \frac{(d x)^m \left (A+B x+C x^2\right )}{a+b x^2+c x^4} \, dx &=\frac{\int \frac{B (d x)^{1+m}}{a+b x^2+c x^4} \, dx}{d}+\int \frac{(d x)^m \left (A+C x^2\right )}{a+b x^2+c x^4} \, dx\\ &=\frac{1}{2} \left (C-\frac{2 A c-b C}{\sqrt{b^2-4 a c}}\right ) \int \frac{(d x)^m}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx+\frac{1}{2} \left (C+\frac{2 A c-b C}{\sqrt{b^2-4 a c}}\right ) \int \frac{(d x)^m}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx+\frac{B \int \frac{(d x)^{1+m}}{a+b x^2+c x^4} \, dx}{d}\\ &=\frac{\left (C+\frac{2 A c-b C}{\sqrt{b^2-4 a c}}\right ) (d x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{\left (b-\sqrt{b^2-4 a c}\right ) d (1+m)}+\frac{\left (C-\frac{2 A c-b C}{\sqrt{b^2-4 a c}}\right ) (d x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{\left (b+\sqrt{b^2-4 a c}\right ) d (1+m)}+\frac{(B c) \int \frac{(d x)^{1+m}}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{\sqrt{b^2-4 a c} d}-\frac{(B c) \int \frac{(d x)^{1+m}}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^2} \, dx}{\sqrt{b^2-4 a c} d}\\ &=\frac{\left (C+\frac{2 A c-b C}{\sqrt{b^2-4 a c}}\right ) (d x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{\left (b-\sqrt{b^2-4 a c}\right ) d (1+m)}+\frac{\left (C-\frac{2 A c-b C}{\sqrt{b^2-4 a c}}\right ) (d x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{\left (b+\sqrt{b^2-4 a c}\right ) d (1+m)}+\frac{2 B c (d x)^{2+m} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right ) d^2 (2+m)}-\frac{2 B c (d x)^{2+m} \, _2F_1\left (1,\frac{2+m}{2};\frac{4+m}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right ) d^2 (2+m)}\\ \end{align*}
Mathematica [C] time = 0.22039, size = 168, normalized size = 0.46 \[ \frac{1}{2} x (d x)^m \left (\frac{A \text{RootSum}\left [\text{$\#$1}^2 b+\text{$\#$1}^4 c+a\& ,\frac{\, _2F_1\left (1,m+1;m+2;\frac{x}{\text{$\#$1}}\right )}{\text{$\#$1}^2 b+2 a}\& \right ]}{m+1}+x \left (\frac{B \text{RootSum}\left [\text{$\#$1}^2 b+\text{$\#$1}^4 c+a\& ,\frac{\, _2F_1\left (1,m+2;m+3;\frac{x}{\text{$\#$1}}\right )}{\text{$\#$1}^2 b+2 a}\& \right ]}{m+2}+\frac{C x \text{RootSum}\left [\text{$\#$1}^2 b+\text{$\#$1}^4 c+a\& ,\frac{\, _2F_1\left (1,m+3;m+4;\frac{x}{\text{$\#$1}}\right )}{\text{$\#$1}^2 b+2 a}\& \right ]}{m+3}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m} \left ( C{x}^{2}+Bx+A \right ) }{c{x}^{4}+b{x}^{2}+a}}\, dx \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{{\left (C x^{2} + B x + A\right )} \left (d x\right )^{m}}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C x^{2} + B x + A\right )} \left (d x\right )^{m}}{c x^{4} + b x^{2} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d x\right )^{m} \left (A + B x + C x^{2}\right )}{a + b x^{2} + c x^{4}}\, dx \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{{\left (C x^{2} + B x + A\right )} \left (d x\right )^{m}}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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