Optimal. Leaf size=131 \[ -\frac {8 b^2 (2 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{315 c^4 x^3}+\frac {4 b (2 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{105 c^3 x}-\frac {(2 b B-3 A c) x \left (b x^2+c x^4\right )^{3/2}}{21 c^2}+\frac {B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c} \]
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Rubi [A]
time = 0.14, 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 = {2064, 2041,
2025} \begin {gather*} -\frac {8 b^2 \left (b x^2+c x^4\right )^{3/2} (2 b B-3 A c)}{315 c^4 x^3}+\frac {4 b \left (b x^2+c x^4\right )^{3/2} (2 b B-3 A c)}{105 c^3 x}-\frac {x \left (b x^2+c x^4\right )^{3/2} (2 b B-3 A c)}{21 c^2}+\frac {B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 2025
Rule 2041
Rule 2064
Rubi steps
\begin {align*} \int x^4 \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx &=\frac {B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c}-\frac {(6 b B-9 A c) \int x^4 \sqrt {b x^2+c x^4} \, dx}{9 c}\\ &=-\frac {(2 b B-3 A c) x \left (b x^2+c x^4\right )^{3/2}}{21 c^2}+\frac {B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c}+\frac {(4 b (2 b B-3 A c)) \int x^2 \sqrt {b x^2+c x^4} \, dx}{21 c^2}\\ &=\frac {4 b (2 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{105 c^3 x}-\frac {(2 b B-3 A c) x \left (b x^2+c x^4\right )^{3/2}}{21 c^2}+\frac {B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c}-\frac {\left (8 b^2 (2 b B-3 A c)\right ) \int \sqrt {b x^2+c x^4} \, dx}{105 c^3}\\ &=-\frac {8 b^2 (2 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{315 c^4 x^3}+\frac {4 b (2 b B-3 A c) \left (b x^2+c x^4\right )^{3/2}}{105 c^3 x}-\frac {(2 b B-3 A c) x \left (b x^2+c x^4\right )^{3/2}}{21 c^2}+\frac {B x^3 \left (b x^2+c x^4\right )^{3/2}}{9 c}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 82, normalized size = 0.63 \begin {gather*} \frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (-16 b^3 B+24 b^2 c \left (A+B x^2\right )-6 b c^2 x^2 \left (6 A+5 B x^2\right )+5 c^3 x^4 \left (9 A+7 B x^2\right )\right )}{315 c^4 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 91, normalized size = 0.69
method | result | size |
gosper | \(\frac {\left (c \,x^{2}+b \right ) \left (35 B \,c^{3} x^{6}+45 A \,c^{3} x^{4}-30 B b \,c^{2} x^{4}-36 A b \,c^{2} x^{2}+24 B \,b^{2} c \,x^{2}+24 A \,b^{2} c -16 B \,b^{3}\right ) \sqrt {x^{4} c +b \,x^{2}}}{315 c^{4} x}\) | \(91\) |
default | \(\frac {\left (c \,x^{2}+b \right ) \left (35 B \,c^{3} x^{6}+45 A \,c^{3} x^{4}-30 B b \,c^{2} x^{4}-36 A b \,c^{2} x^{2}+24 B \,b^{2} c \,x^{2}+24 A \,b^{2} c -16 B \,b^{3}\right ) \sqrt {x^{4} c +b \,x^{2}}}{315 c^{4} x}\) | \(91\) |
trager | \(\frac {\left (35 B \,c^{4} x^{8}+45 A \,c^{4} x^{6}+5 B b \,c^{3} x^{6}+9 A b \,c^{3} x^{4}-6 B \,b^{2} c^{2} x^{4}-12 A \,b^{2} c^{2} x^{2}+8 B \,b^{3} c \,x^{2}+24 A \,b^{3} c -16 B \,b^{4}\right ) \sqrt {x^{4} c +b \,x^{2}}}{315 c^{4} x}\) | \(108\) |
risch | \(\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (35 B \,c^{4} x^{8}+45 A \,c^{4} x^{6}+5 B b \,c^{3} x^{6}+9 A b \,c^{3} x^{4}-6 B \,b^{2} c^{2} x^{4}-12 A \,b^{2} c^{2} x^{2}+8 B \,b^{3} c \,x^{2}+24 A \,b^{3} c -16 B \,b^{4}\right )}{315 x \,c^{4}}\) | \(108\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 106, normalized size = 0.81 \begin {gather*} \frac {{\left (15 \, c^{3} x^{6} + 3 \, b c^{2} x^{4} - 4 \, b^{2} c x^{2} + 8 \, b^{3}\right )} \sqrt {c x^{2} + b} A}{105 \, c^{3}} + \frac {{\left (35 \, c^{4} x^{8} + 5 \, b c^{3} x^{6} - 6 \, b^{2} c^{2} x^{4} + 8 \, b^{3} c x^{2} - 16 \, b^{4}\right )} \sqrt {c x^{2} + b} B}{315 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.34, size = 106, normalized size = 0.81 \begin {gather*} \frac {{\left (35 \, B c^{4} x^{8} + 5 \, {\left (B b c^{3} + 9 \, A c^{4}\right )} x^{6} - 16 \, B b^{4} + 24 \, A b^{3} c - 3 \, {\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{4} + 4 \, {\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{315 \, c^{4} 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 x^{4} \sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.63, size = 140, normalized size = 1.07 \begin {gather*} \frac {8 \, {\left (2 \, B b^{\frac {9}{2}} - 3 \, A b^{\frac {7}{2}} c\right )} \mathrm {sgn}\left (x\right )}{315 \, c^{4}} + \frac {35 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} B \mathrm {sgn}\left (x\right ) - 135 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} B b \mathrm {sgn}\left (x\right ) + 189 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B b^{2} \mathrm {sgn}\left (x\right ) - 105 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b^{3} \mathrm {sgn}\left (x\right ) + 45 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} A c \mathrm {sgn}\left (x\right ) - 126 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} A b c \mathrm {sgn}\left (x\right ) + 105 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A b^{2} c \mathrm {sgn}\left (x\right )}{315 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 103, normalized size = 0.79 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2}\,\left (\frac {B\,x^8}{9}-\frac {16\,B\,b^4-24\,A\,b^3\,c}{315\,c^4}+\frac {x^6\,\left (45\,A\,c^4+5\,B\,b\,c^3\right )}{315\,c^4}-\frac {4\,b^2\,x^2\,\left (3\,A\,c-2\,B\,b\right )}{315\,c^3}+\frac {b\,x^4\,\left (3\,A\,c-2\,B\,b\right )}{105\,c^2}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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