Optimal. Leaf size=243 \[ \frac {2 c (f x)^{m+1} \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} F_1\left (\frac {m+1}{2};1,-q;\frac {m+3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{f (m+1) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c (f x)^{m+1} \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} F_1\left (\frac {m+1}{2};1,-q;\frac {m+3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{f (m+1) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )} \]
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Rubi [A] time = 0.65, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1305, 511, 510} \[ \frac {2 c (f x)^{m+1} \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} F_1\left (\frac {m+1}{2};1,-q;\frac {m+3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{f (m+1) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c (f x)^{m+1} \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} F_1\left (\frac {m+1}{2};1,-q;\frac {m+3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{f (m+1) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rule 1305
Rubi steps
\begin {align*} \int \frac {(f x)^m \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx &=\int \left (\frac {2 c (f x)^m \left (d+e x^2\right )^q}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )}-\frac {2 c (f x)^m \left (d+e x^2\right )^q}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}\right ) \, dx\\ &=\frac {(2 c) \int \frac {(f x)^m \left (d+e x^2\right )^q}{b-\sqrt {b^2-4 a c}+2 c x^2} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {(f x)^m \left (d+e x^2\right )^q}{b+\sqrt {b^2-4 a c}+2 c x^2} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\left (2 c \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q}\right ) \int \frac {(f x)^m \left (1+\frac {e x^2}{d}\right )^q}{b-\sqrt {b^2-4 a c}+2 c x^2} \, dx}{\sqrt {b^2-4 a c}}-\frac {\left (2 c \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q}\right ) \int \frac {(f x)^m \left (1+\frac {e x^2}{d}\right )^q}{b+\sqrt {b^2-4 a c}+2 c x^2} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {2 c (f x)^{1+m} \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} F_1\left (\frac {1+m}{2};1,-q;\frac {3+m}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) f (1+m)}-\frac {2 c (f x)^{1+m} \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} F_1\left (\frac {1+m}{2};1,-q;\frac {3+m}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) f (1+m)}\\ \end {align*}
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Mathematica [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {(f x)^m \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x^{2} + d\right )}^{q} \left (f x\right )^{m}}{c x^{4} + b x^{2} + a}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{q} \left (f x\right )^{m}}{c x^{4} + b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x \right )^{m} \left (e \,x^{2}+d \right )^{q}}{c \,x^{4}+b \,x^{2}+a}\, dx \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{q} \left (f x\right )^{m}}{c x^{4} + b x^{2} + a}\,{d x} \]
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 \[ \int \frac {{\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^q}{c\,x^4+b\,x^2+a} \,d x \]
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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