Optimal. Leaf size=685 \[ \frac{c (d x)^{m+1} \left (A \left (b (1-m) \sqrt{b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right )+2 a C \left (2 b-(1-m) \sqrt{b^2-4 a c}\right )\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (A \left (-b (1-m) \sqrt{b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right )+2 a C \left ((1-m) \sqrt{b^2-4 a c}+2 b\right )\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(d x)^{m+1} \left (A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C\right )}{2 a d \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{B c (d x)^{m+2} \left (b m \left (\sqrt{b^2-4 a c}+b\right )+4 a c (2-m)\right ) \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a d^2 (m+2) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}+\frac{B c (d x)^{m+2} \left (b m \left (b-\sqrt{b^2-4 a c}\right )+4 a c (2-m)\right ) \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a d^2 (m+2) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{B (d x)^{m+2} \left (-2 a c+b^2+b c x^2\right )}{2 a d^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 2.37779, antiderivative size = 670, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1662, 1277, 1285, 364, 12, 1121} \[ \frac{c (d x)^{m+1} \left (A \left (b (1-m) \sqrt{b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right )+2 a C \left (2 b-(1-m) \sqrt{b^2-4 a c}\right )\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (-(1-m) \sqrt{b^2-4 a c} (A b-2 a C)-4 a A c (3-m)+4 a b C+A b^2 (1-m)\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(d x)^{m+1} \left (A \left (b^2-2 a c\right )+c x^2 (A b-2 a C)-a b C\right )}{2 a d \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{B c (d x)^{m+2} \left (b m \left (\sqrt{b^2-4 a c}+b\right )+4 a c (2-m)\right ) \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a d^2 (m+2) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}+\frac{B c (d x)^{m+2} \left (b m \left (b-\sqrt{b^2-4 a c}\right )+4 a c (2-m)\right ) \, _2F_1\left (1,\frac{m+2}{2};\frac{m+4}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a d^2 (m+2) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{B (d x)^{m+2} \left (-2 a c+b^2+b c x^2\right )}{2 a d^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1662
Rule 1277
Rule 1285
Rule 364
Rule 12
Rule 1121
Rubi steps
\begin{align*} \int \frac{(d x)^m \left (A+B x+C x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac{\int \frac{B (d x)^{1+m}}{\left (a+b x^2+c x^4\right )^2} \, dx}{d}+\int \frac{(d x)^m \left (A+C x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\\ &=\frac{(d x)^{1+m} \left (A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2\right )}{2 a \left (b^2-4 a c\right ) d \left (a+b x^2+c x^4\right )}-\frac{\int \frac{(d x)^m \left (-A b^2 (1-m)+2 a A c (3-m)-a b C (1+m)-c (A b-2 a C) (1-m) x^2\right )}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}+\frac{B \int \frac{(d x)^{1+m}}{\left (a+b x^2+c x^4\right )^2} \, dx}{d}\\ &=\frac{B (d x)^{2+m} \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) d^2 \left (a+b x^2+c x^4\right )}+\frac{(d x)^{1+m} \left (A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2\right )}{2 a \left (b^2-4 a c\right ) d \left (a+b x^2+c x^4\right )}-\frac{B \int \frac{(d x)^{1+m} \left (2 a c (2-m)+b^2 m+b c m x^2\right )}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right ) d}-\frac{\left (c \left (4 a b C+A b^2 (1-m)-\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\right )\right ) \int \frac{(d x)^m}{\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 )^{3/2}}+\frac{\left (c \left (4 a b C+A b^2 (1-m)+\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\right )\right ) \int \frac{(d x)^m}{\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 )^{3/2}}\\ &=\frac{B (d x)^{2+m} \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) d^2 \left (a+b x^2+c x^4\right )}+\frac{(d x)^{1+m} \left (A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2\right )}{2 a \left (b^2-4 a c\right ) d \left (a+b x^2+c x^4\right )}+\frac{c \left (4 a b C+A b^2 (1-m)+\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\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 )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) d (1+m)}-\frac{c \left (4 a b C+A b^2 (1-m)-\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\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 )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) d (1+m)}+\frac{\left (B c \left (4 a c (2-m)+b \left (b-\sqrt{b^2-4 a c}\right ) m\right )\right ) \int \frac{(d x)^{1+m}}{\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 )^{3/2} d}-\frac{\left (B c \left (4 a c (2-m)+b \left (b+\sqrt{b^2-4 a c}\right ) m\right )\right ) \int \frac{(d x)^{1+m}}{\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 )^{3/2} d}\\ &=\frac{B (d x)^{2+m} \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) d^2 \left (a+b x^2+c x^4\right )}+\frac{(d x)^{1+m} \left (A \left (b^2-2 a c\right )-a b C+c (A b-2 a C) x^2\right )}{2 a \left (b^2-4 a c\right ) d \left (a+b x^2+c x^4\right )}+\frac{c \left (4 a b C+A b^2 (1-m)+\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\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 )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) d (1+m)}-\frac{c \left (4 a b C+A b^2 (1-m)-\sqrt{b^2-4 a c} (A b-2 a C) (1-m)-4 a A c (3-m)\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 )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) d (1+m)}-\frac{B c \left (4 a c (2-m)+b \left (b+\sqrt{b^2-4 a c}\right ) m\right ) (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 )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) d^2 (2+m)}+\frac{B c \left (4 a c (2-m)+b \left (b-\sqrt{b^2-4 a c}\right ) m\right ) (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 )}{2 a \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) d^2 (2+m)}\\ \end{align*}
Mathematica [C] time = 0.344501, size = 242, normalized size = 0.35 \[ \frac{x (d x)^m \left (A \left (m^2+5 m+6\right ) F_1\left (\frac{m+1}{2};2,2;\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+(m+1) x \left (B (m+3) F_1\left (\frac{m+2}{2};2,2;\frac{m+4}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+C (m+2) x F_1\left (\frac{m+3}{2};2,2;\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )\right )}{a^2 (m+1) (m+2) (m+3)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m} \left ( C{x}^{2}+Bx+A \right ) }{ \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}}}\, 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}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}}\,{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^{2} x^{8} + 2 \, b c x^{6} +{\left (b^{2} + 2 \, a c\right )} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \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{{\left (C x^{2} + B x + A\right )} \left (d x\right )^{m}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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