Optimal. Leaf size=243 \[ \frac {2 c (f x)^{1+m} \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} F_1\left (\frac {1+m}{2};1,-q;\frac {3+m}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) f (1+m)}-\frac {2 c (f x)^{1+m} \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} F_1\left (\frac {1+m}{2};1,-q;\frac {3+m}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) f (1+m)} \]
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Rubi [A]
time = 0.44, 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 = {1319, 525, 524}
\begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 1319
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 = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f x)^m \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.02, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (f\,x\right )}^m\,{\left (e\,x^2+d\right )}^q}{c\,x^4+b\,x^2+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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