Optimal. Leaf size=339 \[ \frac {x \left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} F_1\left (\frac {1}{2};1,-q;\frac {3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{c^2 \left (b-\sqrt {b^2-4 a c}\right )}+\frac {x \left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} F_1\left (\frac {1}{2};1,-q;\frac {3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{c^2 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {b x \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac {1}{2},-q;\frac {3}{2};-\frac {e x^2}{d}\right )}{c^2}+\frac {x^3 \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac {3}{2},-q;\frac {5}{2};-\frac {e x^2}{d}\right )}{3 c} \]
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Rubi [A] time = 0.63, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1303, 246, 245, 365, 364, 1692, 430, 429} \[ \frac {x \left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} F_1\left (\frac {1}{2};1,-q;\frac {3}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{c^2 \left (b-\sqrt {b^2-4 a c}\right )}+\frac {x \left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} F_1\left (\frac {1}{2};1,-q;\frac {3}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},-\frac {e x^2}{d}\right )}{c^2 \left (\sqrt {b^2-4 a c}+b\right )}-\frac {b x \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac {1}{2},-q;\frac {3}{2};-\frac {e x^2}{d}\right )}{c^2}+\frac {x^3 \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \, _2F_1\left (\frac {3}{2},-q;\frac {5}{2};-\frac {e x^2}{d}\right )}{3 c} \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 364
Rule 365
Rule 429
Rule 430
Rule 1303
Rule 1692
Rubi steps
\begin {align*} \int \frac {x^6 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx &=\int \left (-\frac {b \left (d+e x^2\right )^q}{c^2}+\frac {x^2 \left (d+e x^2\right )^q}{c}+\frac {\left (a b+\left (b^2-a c\right ) x^2\right ) \left (d+e x^2\right )^q}{c^2 \left (a+b x^2+c x^4\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (a b+\left (b^2-a c\right ) x^2\right ) \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx}{c^2}-\frac {b \int \left (d+e x^2\right )^q \, dx}{c^2}+\frac {\int x^2 \left (d+e x^2\right )^q \, dx}{c}\\ &=\frac {\int \left (\frac {\left (b^2-a c+\frac {b \left (-b^2+3 a c\right )}{\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^2-a c-\frac {b \left (-b^2+3 a c\right )}{\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^2}-\frac {\left (b \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^2}+\frac {\left (\left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q}\right ) \int x^2 \left (1+\frac {e x^2}{d}\right )^q \, dx}{c}\\ &=-\frac {b 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^2}+\frac {x^3 \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \, _2F_1\left (\frac {3}{2},-q;\frac {5}{2};-\frac {e x^2}{d}\right )}{3 c}+\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\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^2}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\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^2}\\ &=-\frac {b 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^2}+\frac {x^3 \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \, _2F_1\left (\frac {3}{2},-q;\frac {5}{2};-\frac {e x^2}{d}\right )}{3 c}+\frac {\left (\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\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^2}+\frac {\left (\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\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^2}\\ &=\frac {\left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\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^2 \left (b-\sqrt {b^2-4 a c}\right )}+\frac {\left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\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^2 \left (b+\sqrt {b^2-4 a c}\right )}-\frac {b 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^2}+\frac {x^3 \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \, _2F_1\left (\frac {3}{2},-q;\frac {5}{2};-\frac {e x^2}{d}\right )}{3 c}\\ \end {align*}
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Mathematica [F] time = 0.49, size = 0, normalized size = 0.00 \[ \int \frac {x^6 \left (d+e x^2\right )^q}{a+b x^2+c x^4} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x^{2} + d\right )}^{q} x^{6}}{c x^{4} + b x^{2} + a}, x\right ) \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{q} x^{6}}{c x^{4} + b x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {x^{6} \left (e \,x^{2}+d \right )^{q}}{c \,x^{4}+b \,x^{2}+a}\, dx \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{q} x^{6}}{c x^{4} + b x^{2} + a}\,{d x} \]
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 \[ \int \frac {x^6\,{\left (e\,x^2+d\right )}^q}{c\,x^4+b\,x^2+a} \,d x \]
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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