Optimal. Leaf size=37 \[ \frac {2}{5} x^{5/2} (A c+b B)+2 A b \sqrt {x}+\frac {2}{9} B c x^{9/2} \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1584, 448} \[ \frac {2}{5} x^{5/2} (A c+b B)+2 A b \sqrt {x}+\frac {2}{9} B c x^{9/2} \]
Antiderivative was successfully verified.
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Rule 448
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x^{5/2}} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )}{\sqrt {x}} \, dx\\ &=\int \left (\frac {A b}{\sqrt {x}}+(b B+A c) x^{3/2}+B c x^{7/2}\right ) \, dx\\ &=2 A b \sqrt {x}+\frac {2}{5} (b B+A c) x^{5/2}+\frac {2}{9} B c x^{9/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.89 \[ \frac {2}{45} \sqrt {x} \left (9 x^2 (A c+b B)+45 A b+5 B c x^4\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 29, normalized size = 0.78 \[ \frac {2}{45} \, {\left (5 \, B c x^{4} + 9 \, {\left (B b + A c\right )} x^{2} + 45 \, A b\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 29, normalized size = 0.78 \[ \frac {2}{9} \, B c x^{\frac {9}{2}} + \frac {2}{5} \, B b x^{\frac {5}{2}} + \frac {2}{5} \, A c x^{\frac {5}{2}} + 2 \, A b \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 0.86 \[ \frac {2 \left (5 B c \,x^{4}+9 A c \,x^{2}+9 B b \,x^{2}+45 A b \right ) \sqrt {x}}{45} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 27, normalized size = 0.73 \[ \frac {2}{9} \, B c x^{\frac {9}{2}} + \frac {2}{5} \, {\left (B b + A c\right )} x^{\frac {5}{2}} + 2 \, A b \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 31, normalized size = 0.84 \[ \frac {2\,\sqrt {x}\,\left (45\,A\,b+9\,A\,c\,x^2+9\,B\,b\,x^2+5\,B\,c\,x^4\right )}{45} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.62, size = 44, normalized size = 1.19 \[ 2 A b \sqrt {x} + \frac {2 A c x^{\frac {5}{2}}}{5} + \frac {2 B b x^{\frac {5}{2}}}{5} + \frac {2 B c x^{\frac {9}{2}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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