Optimal. Leaf size=107 \[ -\frac {(b B-2 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c^2}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{6 c}+\frac {b^2 (b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{5/2}} \]
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Rubi [A]
time = 0.10, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2059, 654, 626,
634, 212} \begin {gather*} \frac {b^2 (b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{5/2}}-\frac {\left (b+2 c x^2\right ) \sqrt {b x^2+c x^4} (b B-2 A c)}{16 c^2}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 634
Rule 654
Rule 2059
Rubi steps
\begin {align*} \int x \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int (A+B x) \sqrt {b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {B \left (b x^2+c x^4\right )^{3/2}}{6 c}+\frac {(-b B+2 A c) \text {Subst}\left (\int \sqrt {b x+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {(b B-2 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c^2}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{6 c}+\frac {\left (b^2 (b B-2 A c)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{32 c^2}\\ &=-\frac {(b B-2 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c^2}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{6 c}+\frac {\left (b^2 (b B-2 A c)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^2}\\ &=-\frac {(b B-2 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{16 c^2}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{6 c}+\frac {b^2 (b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{16 c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 123, normalized size = 1.15 \begin {gather*} \frac {x \left (\sqrt {c} x \left (b+c x^2\right ) \left (-3 b^2 B+2 b c \left (3 A+B x^2\right )+4 c^2 x^2 \left (3 A+2 B x^2\right )\right )-3 b^2 (b B-2 A c) \sqrt {b+c x^2} \log \left (-\sqrt {c} x+\sqrt {b+c x^2}\right )\right )}{48 c^{5/2} \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 164, normalized size = 1.53
method | result | size |
risch | \(\frac {\left (8 B \,c^{2} x^{4}+12 A \,c^{2} x^{2}+2 b B \,x^{2} c +6 A b c -3 b^{2} B \right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{48 c^{2}}+\frac {\left (-\frac {b^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) A}{8 c^{\frac {3}{2}}}+\frac {b^{3} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) B}{16 c^{\frac {5}{2}}}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{x \sqrt {c \,x^{2}+b}}\) | \(135\) |
default | \(\frac {\sqrt {x^{4} c +b \,x^{2}}\, \left (8 B \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {3}{2}} x^{3}+12 A \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {3}{2}} x -6 B \left (c \,x^{2}+b \right )^{\frac {3}{2}} \sqrt {c}\, b x -6 A \sqrt {c \,x^{2}+b}\, c^{\frac {3}{2}} b x +3 B \sqrt {c \,x^{2}+b}\, \sqrt {c}\, b^{2} x -6 A \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{2} c +3 B \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{3}\right )}{48 x \sqrt {c \,x^{2}+b}\, c^{\frac {5}{2}}}\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 177, normalized size = 1.65 \begin {gather*} \frac {1}{16} \, {\left (4 \, \sqrt {c x^{4} + b x^{2}} x^{2} - \frac {b^{2} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {c x^{4} + b x^{2}} b}{c}\right )} A - \frac {1}{96} \, {\left (\frac {12 \, \sqrt {c x^{4} + b x^{2}} b x^{2}}{c} - \frac {3 \, b^{3} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {5}{2}}} + \frac {6 \, \sqrt {c x^{4} + b x^{2}} b^{2}}{c^{2}} - \frac {16 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{c}\right )} B \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.86, size = 223, normalized size = 2.08 \begin {gather*} \left [-\frac {3 \, {\left (B b^{3} - 2 \, A b^{2} c\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, {\left (8 \, B c^{3} x^{4} - 3 \, B b^{2} c + 6 \, A b c^{2} + 2 \, {\left (B b c^{2} + 6 \, A c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{96 \, c^{3}}, -\frac {3 \, {\left (B b^{3} - 2 \, A b^{2} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) - {\left (8 \, B c^{3} x^{4} - 3 \, B b^{2} c + 6 \, A b c^{2} + 2 \, {\left (B b c^{2} + 6 \, A c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{48 \, c^{3}}\right ] \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 \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 = 0.90, size = 140, normalized size = 1.31 \begin {gather*} \frac {1}{48} \, {\left (2 \, {\left (4 \, B x^{2} \mathrm {sgn}\left (x\right ) + \frac {B b c^{3} \mathrm {sgn}\left (x\right ) + 6 \, A c^{4} \mathrm {sgn}\left (x\right )}{c^{4}}\right )} x^{2} - \frac {3 \, {\left (B b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 2 \, A b c^{3} \mathrm {sgn}\left (x\right )\right )}}{c^{4}}\right )} \sqrt {c x^{2} + b} x - \frac {{\left (B b^{3} \mathrm {sgn}\left (x\right ) - 2 \, A b^{2} c \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{16 \, c^{\frac {5}{2}}} + \frac {{\left (B b^{3} \log \left ({\left | b \right |}\right ) - 2 \, A b^{2} c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{32 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.75, size = 140, normalized size = 1.31 \begin {gather*} \frac {A\,\left (\frac {b}{4\,c}+\frac {x^2}{2}\right )\,\sqrt {c\,x^4+b\,x^2}}{2}+\frac {B\,b^3\,\ln \left (b+2\,c\,x^2+2\,\sqrt {c}\,\left |x\right |\,\sqrt {c\,x^2+b}\right )}{32\,c^{5/2}}-\frac {A\,b^2\,\ln \left (\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}+\sqrt {c\,x^4+b\,x^2}\right )}{16\,c^{3/2}}+\frac {B\,\sqrt {c\,x^4+b\,x^2}\,\left (-3\,b^2+2\,b\,c\,x^2+8\,c^2\,x^4\right )}{48\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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