Optimal. Leaf size=59 \[ -\frac {(2 b B-3 A c) \sqrt {b x^2+c x^4}}{3 c^2 x}+\frac {B x \sqrt {b x^2+c x^4}}{3 c} \]
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Rubi [A]
time = 0.08, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2064, 1602}
\begin {gather*} \frac {B x \sqrt {b x^2+c x^4}}{3 c}-\frac {\sqrt {b x^2+c x^4} (2 b B-3 A c)}{3 c^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 1602
Rule 2064
Rubi steps
\begin {align*} \int \frac {x^2 \left (A+B x^2\right )}{\sqrt {b x^2+c x^4}} \, dx &=\frac {B x \sqrt {b x^2+c x^4}}{3 c}-\frac {(2 b B-3 A c) \int \frac {x^2}{\sqrt {b x^2+c x^4}} \, dx}{3 c}\\ &=-\frac {(2 b B-3 A c) \sqrt {b x^2+c x^4}}{3 c^2 x}+\frac {B x \sqrt {b x^2+c x^4}}{3 c}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 40, normalized size = 0.68 \begin {gather*} \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (-2 b B+3 A c+B c x^2\right )}{3 c^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 42, normalized size = 0.71
method | result | size |
trager | \(\frac {\left (B c \,x^{2}+3 A c -2 B b \right ) \sqrt {x^{4} c +b \,x^{2}}}{3 c^{2} x}\) | \(37\) |
gosper | \(\frac {\left (c \,x^{2}+b \right ) \left (B c \,x^{2}+3 A c -2 B b \right ) x}{3 c^{2} \sqrt {x^{4} c +b \,x^{2}}}\) | \(42\) |
default | \(\frac {\left (c \,x^{2}+b \right ) \left (B c \,x^{2}+3 A c -2 B b \right ) x}{3 c^{2} \sqrt {x^{4} c +b \,x^{2}}}\) | \(42\) |
risch | \(\frac {x \left (c \,x^{2}+b \right ) \left (B c \,x^{2}+3 A c -2 B b \right )}{3 \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, c^{2}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 50, normalized size = 0.85 \begin {gather*} \frac {\sqrt {c x^{2} + b} A}{c} + \frac {{\left (c^{2} x^{4} - b c x^{2} - 2 \, b^{2}\right )} B}{3 \, \sqrt {c x^{2} + b} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.88, size = 36, normalized size = 0.61 \begin {gather*} \frac {\sqrt {c x^{4} + b x^{2}} {\left (B c x^{2} - 2 \, B b + 3 \, A c\right )}}{3 \, c^{2} 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^{2} \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 = 0.48, size = 67, normalized size = 1.14 \begin {gather*} \frac {{\left (2 \, B b^{\frac {3}{2}} - 3 \, A \sqrt {b} c\right )} \mathrm {sgn}\left (x\right )}{3 \, c^{2}} + \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}} B}{3 \, c^{2} \mathrm {sgn}\left (x\right )} - \frac {\sqrt {c x^{2} + b} {\left (B b - A c\right )}}{c^{2} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 41, normalized size = 0.69 \begin {gather*} \frac {\left (\frac {3\,A\,c-2\,B\,b}{3\,c^2}+\frac {B\,x^2}{3\,c}\right )\,\sqrt {c\,x^4+b\,x^2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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