Optimal. Leaf size=61 \[ \frac {B \left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac {\left (b x^2+c x^4\right )^{5/2} (2 b B-7 A c)}{35 c^2 x^5} \]
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Rubi [A] time = 0.16, antiderivative size = 61, 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 = {2039, 2014} \[ \frac {B \left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac {\left (b x^2+c x^4\right )^{5/2} (2 b B-7 A c)}{35 c^2 x^5} \]
Antiderivative was successfully verified.
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Rule 2014
Rule 2039
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx &=\frac {B \left (b x^2+c x^4\right )^{5/2}}{7 c x^3}-\frac {(2 b B-7 A c) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^2} \, dx}{7 c}\\ &=-\frac {(2 b B-7 A c) \left (b x^2+c x^4\right )^{5/2}}{35 c^2 x^5}+\frac {B \left (b x^2+c x^4\right )^{5/2}}{7 c x^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 48, normalized size = 0.79 \[ \frac {x \left (b+c x^2\right )^3 \left (7 A c-2 b B+5 B c x^2\right )}{35 c^2 \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 80, normalized size = 1.31 \[ \frac {{\left (5 \, B c^{3} x^{6} + {\left (8 \, B b c^{2} + 7 \, A c^{3}\right )} x^{4} - 2 \, B b^{3} + 7 \, A b^{2} c + {\left (B b^{2} c + 14 \, A b c^{2}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{35 \, c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 72, normalized size = 1.18 \[ \frac {{\left (2 \, B b^{\frac {7}{2}} - 7 \, A b^{\frac {5}{2}} c\right )} \mathrm {sgn}\relax (x)}{35 \, c^{2}} + \frac {5 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} B \mathrm {sgn}\relax (x) - 7 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B b \mathrm {sgn}\relax (x) + 7 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} A c \mathrm {sgn}\relax (x)}{35 \, c^{2}} \]
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 \[ \frac {\left (c \,x^{2}+b \right ) \left (5 B c \,x^{2}+7 A c -2 b B \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{35 c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 80, normalized size = 1.31 \[ \frac {{\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {c x^{2} + b} A}{5 \, c} + \frac {{\left (5 \, c^{3} x^{6} + 8 \, b c^{2} x^{4} + b^{2} c x^{2} - 2 \, b^{3}\right )} \sqrt {c x^{2} + b} B}{35 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 83, normalized size = 1.36 \[ \frac {\sqrt {c\,x^4+b\,x^2}\,\left (\frac {x^4\,\left (7\,A\,c^3+8\,B\,b\,c^2\right )}{35\,c^2}-\frac {2\,B\,b^3-7\,A\,b^2\,c}{35\,c^2}+\frac {B\,c\,x^6}{7}+\frac {b\,x^2\,\left (14\,A\,c+B\,b\right )}{35\,c}\right )}{x} \]
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 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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