Optimal. Leaf size=315 \[ \frac {c (d x)^{m+1} \left (b (1-m) \sqrt {b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right ) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {c (d x)^{m+1} \left (-b (1-m) \sqrt {b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right ) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(d x)^{m+1} \left (-2 a c+b^2+b c x^2\right )}{2 a d \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
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Rubi [A] time = 0.70, antiderivative size = 315, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1121, 1285, 364} \[ \frac {c (d x)^{m+1} \left (b (1-m) \sqrt {b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right ) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {c (d x)^{m+1} \left (-b (1-m) \sqrt {b^2-4 a c}-4 a c (3-m)+b^2 (1-m)\right ) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{2 a d (m+1) \left (b^2-4 a c\right )^{3/2} \left (\sqrt {b^2-4 a c}+b\right )}+\frac {(d x)^{m+1} \left (-2 a c+b^2+b c x^2\right )}{2 a d \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
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Rule 364
Rule 1121
Rule 1285
Rubi steps
\begin {align*} \int \frac {(d x)^m}{\left (a+b x^2+c x^4\right )^2} \, dx &=\frac {(d x)^{1+m} \left (b^2-2 a c+b 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 (-b^2 (1-m)+2 a c (3-m)-b c (1-m) x^2\right )}{a+b x^2+c x^4} \, dx}{2 a \left (b^2-4 a c\right )}\\ &=\frac {(d x)^{1+m} \left (b^2-2 a c+b c x^2\right )}{2 a \left (b^2-4 a c\right ) d \left (a+b x^2+c x^4\right )}-\frac {\left (c \left (b^2 (1-m)-b \sqrt {b^2-4 a c} (1-m)-4 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 (b^2 (1-m)+b \sqrt {b^2-4 a c} (1-m)-4 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 {(d x)^{1+m} \left (b^2-2 a c+b 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 (b^2 (1-m)+b \sqrt {b^2-4 a c} (1-m)-4 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 (b^2 (1-m)-b \sqrt {b^2-4 a c} (1-m)-4 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)}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 78, normalized size = 0.25 \[ \frac {x (d x)^m F_1\left (\frac {m+1}{2};2,2;\frac {m+3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{\sqrt {b^2-4 a c}-b}\right )}{a^2 (m+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (d x\right )^{m}}{c^{2} x^{8} + 2 \, b c x^{6} + {\left (b^{2} + 2 \, a c\right )} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x \right )^{m}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x\right )^{m}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d\,x\right )}^m}{{\left (c\,x^4+b\,x^2+a\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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