Optimal. Leaf size=392 \[ \frac{c (f x)^{m+1} \left (b \left (d (1-m) \sqrt{b^2-4 a c}+4 a e\right )-2 a \left (e (1-m) \sqrt{b^2-4 a c}+2 c d (3-m)\right )+b^2 (d-d m)\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a f (m+1) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (f x)^{m+1} \left (b \left (4 a e-d (1-m) \sqrt{b^2-4 a c}\right )+2 a \left (e (1-m) \sqrt{b^2-4 a c}-2 c d (3-m)\right )+b^2 d (1-m)\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a f (m+1) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f x)^{m+1} \left (c x^2 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a f \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 2.646, antiderivative size = 358, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1277, 1285, 364} \[ \frac{c (f x)^{m+1} \left ((1-m) \sqrt{b^2-4 a c} (b d-2 a e)+4 a b e-4 a c d (3-m)+b^2 (d-d m)\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}\right )}{2 a f (m+1) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (f x)^{m+1} \left (-(1-m) \sqrt{b^2-4 a c} (b d-2 a e)+4 a b e-4 a c d (3-m)+b^2 (d-d m)\right ) \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}}\right )}{2 a f (m+1) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(f x)^{m+1} \left (c x^2 (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a f \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 1277
Rule 1285
Rule 364
Rubi steps
\begin{align*} \int \frac{(f x)^m \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac{(f x)^{1+m} \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^2\right )}{2 a \left (b^2-4 a c\right ) f \left (a+b x^2+c x^4\right )}-\frac{\int \frac{(f x)^m \left (-b^2 d (1-m)+2 a c d (3-m)-a b e (1+m)-c (b d-2 a e) (1-m) x^2\right )}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac{(f x)^{1+m} \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^2\right )}{2 a \left (b^2-4 a c\right ) f \left (a+b x^2+c x^4\right )}+\frac{\left (c \left (4 a b e+b^2 d (1-m)+\sqrt{b^2-4 a c} (b d-2 a e) (1-m)-4 a c d (3-m)\right )\right ) \int \frac{(f 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 e-\sqrt{b^2-4 a c} (b d-2 a e) (1-m)-4 a c d (3-m)+b^2 (d-d m)\right )\right ) \int \frac{(f 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{(f x)^{1+m} \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^2\right )}{2 a \left (b^2-4 a c\right ) f \left (a+b x^2+c x^4\right )}+\frac{c \left (4 a b e+b^2 d (1-m)+\sqrt{b^2-4 a c} (b d-2 a e) (1-m)-4 a c d (3-m)\right ) (f 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 ) f (1+m)}-\frac{c \left (4 a b e-\sqrt{b^2-4 a c} (b d-2 a e) (1-m)-4 a c d (3-m)+b^2 (d-d m)\right ) (f 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 ) f (1+m)}\\ \end{align*}
Mathematica [C] time = 0.217344, size = 160, normalized size = 0.41 \[ \frac{x (f x)^m \left (d (m+3) F_1\left (\frac{m+1}{2};2,2;\frac{m+3}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )+e (m+1) x^2 F_1\left (\frac{m+3}{2};2,2;\frac{m+5}{2};-\frac{2 c x^2}{b+\sqrt{b^2-4 a c}},\frac{2 c x^2}{\sqrt{b^2-4 a c}-b}\right )\right )}{a^2 (m+1) (m+3)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m} \left ( e{x}^{2}+d \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 (e x^{2} + d\right )} \left (f 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 (e x^{2} + d\right )} \left (f 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 (e x^{2} + d\right )} \left (f 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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