Optimal. Leaf size=94 \[ \frac {2 b (4 b B-5 A c) \sqrt {b x^2+c x^4}}{15 c^3 x}-\frac {(4 b B-5 A c) x \sqrt {b x^2+c x^4}}{15 c^2}+\frac {B x^3 \sqrt {b x^2+c x^4}}{5 c} \]
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Rubi [A]
time = 0.12, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2064, 2041,
1602} \begin {gather*} \frac {2 b \sqrt {b x^2+c x^4} (4 b B-5 A c)}{15 c^3 x}-\frac {x \sqrt {b x^2+c x^4} (4 b B-5 A c)}{15 c^2}+\frac {B x^3 \sqrt {b x^2+c x^4}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 1602
Rule 2041
Rule 2064
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx &=\frac {B x^3 \sqrt {b x^2+c x^4}}{5 c}-\frac {(4 b B-5 A c) \int \frac {x^4}{\sqrt {b x^2+c x^4}} \, dx}{5 c}\\ &=-\frac {(4 b B-5 A c) x \sqrt {b x^2+c x^4}}{15 c^2}+\frac {B x^3 \sqrt {b x^2+c x^4}}{5 c}+\frac {(2 b (4 b B-5 A c)) \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx}{15 c^2}\\ &=\frac {2 b (4 b B-5 A c) \sqrt {b x^2+c x^4}}{15 c^3 x}-\frac {(4 b B-5 A c) x \sqrt {b x^2+c x^4}}{15 c^2}+\frac {B x^3 \sqrt {b x^2+c x^4}}{5 c}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 63, normalized size = 0.67 \begin {gather*} \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (8 b^2 B-2 b c \left (5 A+2 B x^2\right )+c^2 x^2 \left (5 A+3 B x^2\right )\right )}{15 c^3 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 65, normalized size = 0.69
method | result | size |
trager | \(-\frac {\left (-3 B \,c^{2} x^{4}-5 A \,c^{2} x^{2}+4 b B \,x^{2} c +10 A b c -8 b^{2} B \right ) \sqrt {x^{4} c +b \,x^{2}}}{15 c^{3} x}\) | \(60\) |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (-3 B \,c^{2} x^{4}-5 A \,c^{2} x^{2}+4 b B \,x^{2} c +10 A b c -8 b^{2} B \right ) x}{15 c^{3} \sqrt {x^{4} c +b \,x^{2}}}\) | \(65\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) \left (-3 B \,c^{2} x^{4}-5 A \,c^{2} x^{2}+4 b B \,x^{2} c +10 A b c -8 b^{2} B \right ) x}{15 c^{3} \sqrt {x^{4} c +b \,x^{2}}}\) | \(65\) |
risch | \(-\frac {x \left (c \,x^{2}+b \right ) \left (-3 B \,c^{2} x^{4}-5 A \,c^{2} x^{2}+4 b B \,x^{2} c +10 A b c -8 b^{2} B \right )}{15 \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, c^{3}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 83, normalized size = 0.88 \begin {gather*} \frac {{\left (c^{2} x^{4} - b c x^{2} - 2 \, b^{2}\right )} A}{3 \, \sqrt {c x^{2} + b} c^{2}} + \frac {{\left (3 \, c^{3} x^{6} - b c^{2} x^{4} + 4 \, b^{2} c x^{2} + 8 \, b^{3}\right )} B}{15 \, \sqrt {c x^{2} + b} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.38, size = 59, normalized size = 0.63 \begin {gather*} \frac {{\left (3 \, B c^{2} x^{4} + 8 \, B b^{2} - 10 \, A b c - {\left (4 \, B b c - 5 \, A c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{15 \, c^{3} x} \end {gather*}
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 \begin {gather*} \int \frac {x^{4} \left (A + B x^{2}\right )}{\sqrt {x^{2} \left (b + c x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 98, normalized size = 1.04 \begin {gather*} -\frac {2 \, {\left (4 \, B b^{\frac {5}{2}} - 5 \, A b^{\frac {3}{2}} c\right )} \mathrm {sgn}\left (x\right )}{15 \, c^{3}} + \frac {{\left (B b^{2} - A b c\right )} \sqrt {c x^{2} + b}}{c^{3} \mathrm {sgn}\left (x\right )} + \frac {3 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B - 10 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b + 5 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A c}{15 \, c^{3} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 64, normalized size = 0.68 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2}\,\left (\frac {8\,B\,b^2-10\,A\,b\,c}{15\,c^3}+\frac {x^2\,\left (5\,A\,c^2-4\,B\,b\,c\right )}{15\,c^3}+\frac {B\,x^4}{5\,c}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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