Optimal. Leaf size=47 \[ -\frac {2 a^3}{5 x^{5/2}}-\frac {6 a^2 b}{\sqrt {x}}+2 a b^2 x^{3/2}+\frac {2}{7} b^3 x^{7/2} \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ -\frac {6 a^2 b}{\sqrt {x}}-\frac {2 a^3}{5 x^{5/2}}+2 a b^2 x^{3/2}+\frac {2}{7} b^3 x^{7/2} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^3}{x^{7/2}} \, dx &=\int \left (\frac {a^3}{x^{7/2}}+\frac {3 a^2 b}{x^{3/2}}+3 a b^2 \sqrt {x}+b^3 x^{5/2}\right ) \, dx\\ &=-\frac {2 a^3}{5 x^{5/2}}-\frac {6 a^2 b}{\sqrt {x}}+2 a b^2 x^{3/2}+\frac {2}{7} b^3 x^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 0.87 \[ \frac {2 \left (-7 a^3-105 a^2 b x^2+35 a b^2 x^4+5 b^3 x^6\right )}{35 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 37, normalized size = 0.79 \[ \frac {2 \, {\left (5 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} - 105 \, a^{2} b x^{2} - 7 \, a^{3}\right )}}{35 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 36, normalized size = 0.77 \[ \frac {2}{7} \, b^{3} x^{\frac {7}{2}} + 2 \, a b^{2} x^{\frac {3}{2}} - \frac {2 \, {\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.81 \[ -\frac {2 \left (-5 b^{3} x^{6}-35 a \,b^{2} x^{4}+105 a^{2} b \,x^{2}+7 a^{3}\right )}{35 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 36, normalized size = 0.77 \[ \frac {2}{7} \, b^{3} x^{\frac {7}{2}} + 2 \, a b^{2} x^{\frac {3}{2}} - \frac {2 \, {\left (15 \, a^{2} b x^{2} + a^{3}\right )}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 37, normalized size = 0.79 \[ -\frac {14\,a^3+210\,a^2\,b\,x^2-70\,a\,b^2\,x^4-10\,b^3\,x^6}{35\,x^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.86, size = 46, normalized size = 0.98 \[ - \frac {2 a^{3}}{5 x^{\frac {5}{2}}} - \frac {6 a^{2} b}{\sqrt {x}} + 2 a b^{2} x^{\frac {3}{2}} + \frac {2 b^{3} x^{\frac {7}{2}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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