Optimal. Leaf size=37 \[ -\frac {2 a A}{3 x^{3/2}}+2 (A b+a B) \sqrt {x}+\frac {2}{5} b B x^{5/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459}
\begin {gather*} 2 \sqrt {x} (a B+A b)-\frac {2 a A}{3 x^{3/2}}+\frac {2}{5} b B x^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right ) \left (A+B x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac {a A}{x^{5/2}}+\frac {A b+a B}{\sqrt {x}}+b B x^{3/2}\right ) \, dx\\ &=-\frac {2 a A}{3 x^{3/2}}+2 (A b+a B) \sqrt {x}+\frac {2}{5} b B x^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 35, normalized size = 0.95 \begin {gather*} -\frac {2 \left (5 a A-15 A b x^2-15 a B x^2-3 b B x^4\right )}{15 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 30, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {2 b B \,x^{\frac {5}{2}}}{5}+2 A b \sqrt {x}+2 B a \sqrt {x}-\frac {2 a A}{3 x^{\frac {3}{2}}}\) | \(30\) |
default | \(\frac {2 b B \,x^{\frac {5}{2}}}{5}+2 A b \sqrt {x}+2 B a \sqrt {x}-\frac {2 a A}{3 x^{\frac {3}{2}}}\) | \(30\) |
gosper | \(-\frac {2 \left (-3 b B \,x^{4}-15 A b \,x^{2}-15 B a \,x^{2}+5 A a \right )}{15 x^{\frac {3}{2}}}\) | \(32\) |
trager | \(-\frac {2 \left (-3 b B \,x^{4}-15 A b \,x^{2}-15 B a \,x^{2}+5 A a \right )}{15 x^{\frac {3}{2}}}\) | \(32\) |
risch | \(-\frac {2 \left (-3 b B \,x^{4}-15 A b \,x^{2}-15 B a \,x^{2}+5 A a \right )}{15 x^{\frac {3}{2}}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 27, normalized size = 0.73 \begin {gather*} \frac {2}{5} \, B b x^{\frac {5}{2}} + 2 \, {\left (B a + A b\right )} \sqrt {x} - \frac {2 \, A a}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.98, size = 29, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (3 \, B b x^{4} + 15 \, {\left (B a + A b\right )} x^{2} - 5 \, A a\right )}}{15 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 42, normalized size = 1.14 \begin {gather*} - \frac {2 A a}{3 x^{\frac {3}{2}}} + 2 A b \sqrt {x} + 2 B a \sqrt {x} + \frac {2 B b x^{\frac {5}{2}}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.89, size = 29, normalized size = 0.78 \begin {gather*} \frac {2}{5} \, B b x^{\frac {5}{2}} + 2 \, B a \sqrt {x} + 2 \, A b \sqrt {x} - \frac {2 \, A a}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 31, normalized size = 0.84 \begin {gather*} \frac {30\,A\,b\,x^2-10\,A\,a+30\,B\,a\,x^2+6\,B\,b\,x^4}{15\,x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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