Optimal. Leaf size=37 \[ -\frac {2 (a B+A b)}{\sqrt {x}}-\frac {2 a A}{5 x^{5/2}}+\frac {2}{3} b B x^{3/2} \]
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Rubi [A] time = 0.02, 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 = {448} \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 448
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 = 0.01, size = 36, normalized size = 0.97 \begin {gather*} \frac {10 b x^2 \left (B x^2-3 A\right )-6 a \left (A+5 B x^2\right )}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [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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fricas [A] time = 0.85, 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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giac [A] time = 0.37, 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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maple [A] time = 0.00, size = 32, normalized size = 0.86 \begin {gather*} -\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, 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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mupad [B] time = 0.04, 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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sympy [A] time = 1.76, 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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