Optimal. Leaf size=273 \[ -\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} F_1\left (\frac {1}{2};1,-q;\frac {3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{c \left (b-\sqrt {b^2-4 a c}\right )}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} F_1\left (\frac {1}{2};1,-q;\frac {3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{c \left (b+\sqrt {b^2-4 a c}\right )}+\frac {x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \, _2F_1\left (\frac {1}{2},-q;\frac {3}{2};-\frac {e x^2}{d}\right )}{c} \]
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Rubi [A]
time = 0.27, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1317, 252,
251, 1706, 441, 440} \begin {gather*} -\frac {x \left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} F_1\left (\frac {1}{2};1,-q;\frac {3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{c \left (b-\sqrt {b^2-4 a c}\right )}-\frac {x \left (\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}+b\right ) \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} F_1\left (\frac {1}{2};1,-q;\frac {3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{c \left (\sqrt {b^2-4 a c}+b\right )}+\frac {x \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac {1}{2},-q;\frac {3}{2};-\frac {e x^2}{d}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 440
Rule 441
Rule 1317
Rule 1706
Rubi steps
\begin {align*} \int \frac {x^4 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx &=\int \left (\frac {\left (d+e x^2\right )^q}{c}-\frac {\left (a+b x^2\right ) \left (d+e x^2\right )^q}{c \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=\frac {\int \left (d+e x^2\right )^q \, dx}{c}-\frac {\int \frac {\left (a+b x^2\right ) \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx}{c}\\ &=-\frac {\int \left (\frac {\left (b+\frac {-b^2+2 a c}{\sqrt {b^2-4 a c}}\right ) \left (d+e x^2\right )^q}{b-\sqrt {b^2-4 a c}+2 c x^2}+\frac {\left (b-\frac {-b^2+2 a c}{\sqrt {b^2-4 a c}}\right ) \left (d+e x^2\right )^q}{b+\sqrt {b^2-4 a c}+2 c x^2}\right ) \, dx}{c}+\frac {\left (\left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q}\right ) \int \left (1+\frac {e x^2}{d}\right )^q \, dx}{c}\\ &=\frac {x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \, _2F_1\left (\frac {1}{2},-q;\frac {3}{2};-\frac {e x^2}{d}\right )}{c}-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \int \frac {\left (d+e x^2\right )^q}{b-\sqrt {b^2-4 a c}+2 c x^2} \, dx}{c}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \int \frac {\left (d+e x^2\right )^q}{b+\sqrt {b^2-4 a c}+2 c x^2} \, dx}{c}\\ &=\frac {x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \, _2F_1\left (\frac {1}{2},-q;\frac {3}{2};-\frac {e x^2}{d}\right )}{c}-\frac {\left (\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q}\right ) \int \frac {\left (1+\frac {e x^2}{d}\right )^q}{b-\sqrt {b^2-4 a c}+2 c x^2} \, dx}{c}-\frac {\left (\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q}\right ) \int \frac {\left (1+\frac {e x^2}{d}\right )^q}{b+\sqrt {b^2-4 a c}+2 c x^2} \, dx}{c}\\ &=-\frac {\left (b-\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} F_1\left (\frac {1}{2};1,-q;\frac {3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{c \left (b-\sqrt {b^2-4 a c}\right )}-\frac {\left (b+\frac {b^2-2 a c}{\sqrt {b^2-4 a c}}\right ) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} F_1\left (\frac {1}{2};1,-q;\frac {3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{c \left (b+\sqrt {b^2-4 a c}\right )}+\frac {x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \, _2F_1\left (\frac {1}{2},-q;\frac {3}{2};-\frac {e x^2}{d}\right )}{c}\\ \end {align*}
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Mathematica [F]
time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{4} \left (e \,x^{2}+d \right )^{q}}{c \,x^{4}+b \,x^{2}+a}\, dx\]
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 {x^4\,{\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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