Optimal. Leaf size=37 \[ -\frac {2 a A}{5 x^{5/2}}-\frac {2 (A b+a B)}{\sqrt {x}}+\frac {2}{3} b B x^{3/2} \]
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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*} -\frac {2 (a B+A b)}{\sqrt {x}}-\frac {2 a A}{5 x^{5/2}}+\frac {2}{3} b B x^{3/2} \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^{7/2}} \, dx &=\int \left (\frac {a A}{x^{7/2}}+\frac {A b+a B}{x^{3/2}}+b B \sqrt {x}\right ) \, dx\\ &=-\frac {2 a A}{5 x^{5/2}}-\frac {2 (A b+a B)}{\sqrt {x}}+\frac {2}{3} b B x^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 35, normalized size = 0.95 \begin {gather*} -\frac {2 \left (3 a A+15 A b x^2+15 a B x^2-5 b B x^4\right )}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 28, normalized size = 0.76
method | result | size |
derivativedivides | \(-\frac {2 a A}{5 x^{\frac {5}{2}}}+\frac {2 b B \,x^{\frac {3}{2}}}{3}-\frac {2 \left (A b +B a \right )}{\sqrt {x}}\) | \(28\) |
default | \(-\frac {2 a A}{5 x^{\frac {5}{2}}}+\frac {2 b B \,x^{\frac {3}{2}}}{3}-\frac {2 \left (A b +B a \right )}{\sqrt {x}}\) | \(28\) |
gosper | \(-\frac {2 \left (-5 b B \,x^{4}+15 A b \,x^{2}+15 B a \,x^{2}+3 A a \right )}{15 x^{\frac {5}{2}}}\) | \(32\) |
trager | \(-\frac {2 \left (-5 b B \,x^{4}+15 A b \,x^{2}+15 B a \,x^{2}+3 A a \right )}{15 x^{\frac {5}{2}}}\) | \(32\) |
risch | \(-\frac {2 \left (-5 b B \,x^{4}+15 A b \,x^{2}+15 B a \,x^{2}+3 A a \right )}{15 x^{\frac {5}{2}}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 29, normalized size = 0.78 \begin {gather*} \frac {2}{3} \, B b x^{\frac {3}{2}} - \frac {2 \, {\left (5 \, {\left (B a + A b\right )} x^{2} + A a\right )}}{5 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.26, size = 29, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (5 \, B b x^{4} - 15 \, {\left (B a + A b\right )} x^{2} - 3 \, A a\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.35, size = 42, normalized size = 1.14 \begin {gather*} - \frac {2 A a}{5 x^{\frac {5}{2}}} - \frac {2 A b}{\sqrt {x}} - \frac {2 B a}{\sqrt {x}} + \frac {2 B b x^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.74, size = 31, normalized size = 0.84 \begin {gather*} \frac {2}{3} \, B b x^{\frac {3}{2}} - \frac {2 \, {\left (5 \, B a x^{2} + 5 \, A b x^{2} + A a\right )}}{5 \, x^{\frac {5}{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 {6\,A\,a+30\,A\,b\,x^2+30\,B\,a\,x^2-10\,B\,b\,x^4}{15\,x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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