Optimal. Leaf size=61 \[ \frac {B \left (b x^2+c x^4\right )^{3/2}}{5 c x}-\frac {\left (b x^2+c x^4\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^3} \]
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Rubi [A] time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1145, 2000} \[ \frac {B \left (b x^2+c x^4\right )^{3/2}}{5 c x}-\frac {\left (b x^2+c x^4\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^3} \]
Antiderivative was successfully verified.
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Rule 1145
Rule 2000
Rubi steps
\begin {align*} \int \left (A+B x^2\right ) \sqrt {b x^2+c x^4} \, dx &=\frac {B \left (b x^2+c x^4\right )^{3/2}}{5 c x}-\frac {(2 b B-5 A c) \int \sqrt {b x^2+c x^4} \, dx}{5 c}\\ &=-\frac {(2 b B-5 A c) \left (b x^2+c x^4\right )^{3/2}}{15 c^2 x^3}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{5 c x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.67 \[ \frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (5 A c-2 b B+3 B c x^2\right )}{15 c^2 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 57, normalized size = 0.93 \[ \frac {{\left (3 \, B c^{2} x^{4} - 2 \, B b^{2} + 5 \, A b c + {\left (B b c + 5 \, A c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{15 \, c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 72, normalized size = 1.18 \[ \frac {{\left (2 \, B b^{\frac {5}{2}} - 5 \, A b^{\frac {3}{2}} c\right )} \mathrm {sgn}\relax (x)}{15 \, c^{2}} + \frac {3 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B \mathrm {sgn}\relax (x) - 5 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b \mathrm {sgn}\relax (x) + 5 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A c \mathrm {sgn}\relax (x)}{15 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 45, normalized size = 0.74 \[ \frac {\left (c \,x^{2}+b \right ) \left (3 B c \,x^{2}+5 A c -2 b B \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{15 c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 51, normalized size = 0.84 \[ \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}} A}{3 \, c} + \frac {{\left (3 \, c^{2} x^{4} + b c x^{2} - 2 \, b^{2}\right )} \sqrt {c x^{2} + b} B}{15 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 60, normalized size = 0.98 \[ \frac {\sqrt {c\,x^4+b\,x^2}\,\left (\frac {B\,x^4}{5}-\frac {2\,B\,b^2-5\,A\,b\,c}{15\,c^2}+\frac {x^2\,\left (5\,A\,c^2+B\,b\,c\right )}{15\,c^2}\right )}{x} \]
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 \[ \int \sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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