Optimal. Leaf size=131 \[ -\frac {8 b^2 \sqrt {b x^2+c x^4} (6 b B-7 A c)}{105 c^4 x}+\frac {4 b x \sqrt {b x^2+c x^4} (6 b B-7 A c)}{105 c^3}-\frac {x^3 \sqrt {b x^2+c x^4} (6 b B-7 A c)}{35 c^2}+\frac {B x^5 \sqrt {b x^2+c x^4}}{7 c} \]
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Rubi [A] time = 0.24, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2039, 2016, 1588} \[ -\frac {8 b^2 \sqrt {b x^2+c x^4} (6 b B-7 A c)}{105 c^4 x}-\frac {x^3 \sqrt {b x^2+c x^4} (6 b B-7 A c)}{35 c^2}+\frac {4 b x \sqrt {b x^2+c x^4} (6 b B-7 A c)}{105 c^3}+\frac {B x^5 \sqrt {b x^2+c x^4}}{7 c} \]
Antiderivative was successfully verified.
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Rule 1588
Rule 2016
Rule 2039
Rubi steps
\begin {align*} \int \frac {x^6 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx &=\frac {B x^5 \sqrt {b x^2+c x^4}}{7 c}-\frac {(6 b B-7 A c) \int \frac {x^6}{\sqrt {b x^2+c x^4}} \, dx}{7 c}\\ &=-\frac {(6 b B-7 A c) x^3 \sqrt {b x^2+c x^4}}{35 c^2}+\frac {B x^5 \sqrt {b x^2+c x^4}}{7 c}+\frac {(4 b (6 b B-7 A c)) \int \frac {x^4}{\sqrt {b x^2+c x^4}} \, dx}{35 c^2}\\ &=\frac {4 b (6 b B-7 A c) x \sqrt {b x^2+c x^4}}{105 c^3}-\frac {(6 b B-7 A c) x^3 \sqrt {b x^2+c x^4}}{35 c^2}+\frac {B x^5 \sqrt {b x^2+c x^4}}{7 c}-\frac {\left (8 b^2 (6 b B-7 A c)\right ) \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx}{105 c^3}\\ &=-\frac {8 b^2 (6 b B-7 A c) \sqrt {b x^2+c x^4}}{105 c^4 x}+\frac {4 b (6 b B-7 A c) x \sqrt {b x^2+c x^4}}{105 c^3}-\frac {(6 b B-7 A c) x^3 \sqrt {b x^2+c x^4}}{35 c^2}+\frac {B x^5 \sqrt {b x^2+c x^4}}{7 c}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 85, normalized size = 0.65 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (8 b^2 c \left (7 A+3 B x^2\right )-2 b c^2 x^2 \left (14 A+9 B x^2\right )+3 c^3 x^4 \left (7 A+5 B x^2\right )-48 b^3 B\right )}{105 c^4 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 83, normalized size = 0.63 \[ \frac {{\left (15 \, B c^{3} x^{6} - 3 \, {\left (6 \, B b c^{2} - 7 \, A c^{3}\right )} x^{4} - 48 \, B b^{3} + 56 \, A b^{2} c + 4 \, {\left (6 \, B b^{2} c - 7 \, A b c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{105 \, c^{4} x} \]
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 \[ \int \frac {{\left (B x^{2} + A\right )} x^{6}}{\sqrt {c x^{4} + b x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 89, normalized size = 0.68 \[ \frac {\left (c \,x^{2}+b \right ) \left (15 B \,c^{3} x^{6}+21 A \,c^{3} x^{4}-18 B b \,c^{2} x^{4}-28 A b \,c^{2} x^{2}+24 B \,b^{2} c \,x^{2}+56 A \,b^{2} c -48 B \,b^{3}\right ) x}{105 \sqrt {c \,x^{4}+b \,x^{2}}\, c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.53, size = 106, normalized size = 0.81 \[ \frac {{\left (3 \, c^{3} x^{6} - b c^{2} x^{4} + 4 \, b^{2} c x^{2} + 8 \, b^{3}\right )} A}{15 \, \sqrt {c x^{2} + b} c^{3}} + \frac {{\left (5 \, c^{4} x^{8} - b c^{3} x^{6} + 2 \, b^{2} c^{2} x^{4} - 8 \, b^{3} c x^{2} - 16 \, b^{4}\right )} B}{35 \, \sqrt {c x^{2} + b} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 87, normalized size = 0.66 \[ -\frac {\sqrt {c\,x^4+b\,x^2}\,\left (\frac {48\,B\,b^3-56\,A\,b^2\,c}{105\,c^4}-\frac {B\,x^6}{7\,c}-\frac {x^4\,\left (21\,A\,c^3-18\,B\,b\,c^2\right )}{105\,c^4}+\frac {4\,b\,x^2\,\left (7\,A\,c-6\,B\,b\right )}{105\,c^3}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6} \left (A + B x^{2}\right )}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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