Optimal. Leaf size=69 \[ \frac {5 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}+\frac {5 a}{b^3 \sqrt {x}}+\frac {1}{b x^{3/2} (a x+b)}-\frac {5}{3 b^2 x^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {263, 51, 63, 205} \[ \frac {5 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}+\frac {5 a}{b^3 \sqrt {x}}+\frac {1}{b x^{3/2} (a x+b)}-\frac {5}{3 b^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rule 263
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^2 x^{9/2}} \, dx &=\int \frac {1}{x^{5/2} (b+a x)^2} \, dx\\ &=\frac {1}{b x^{3/2} (b+a x)}+\frac {5 \int \frac {1}{x^{5/2} (b+a x)} \, dx}{2 b}\\ &=-\frac {5}{3 b^2 x^{3/2}}+\frac {1}{b x^{3/2} (b+a x)}-\frac {(5 a) \int \frac {1}{x^{3/2} (b+a x)} \, dx}{2 b^2}\\ &=-\frac {5}{3 b^2 x^{3/2}}+\frac {5 a}{b^3 \sqrt {x}}+\frac {1}{b x^{3/2} (b+a x)}+\frac {\left (5 a^2\right ) \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{2 b^3}\\ &=-\frac {5}{3 b^2 x^{3/2}}+\frac {5 a}{b^3 \sqrt {x}}+\frac {1}{b x^{3/2} (b+a x)}+\frac {\left (5 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {5}{3 b^2 x^{3/2}}+\frac {5 a}{b^3 \sqrt {x}}+\frac {1}{b x^{3/2} (b+a x)}+\frac {5 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 27, normalized size = 0.39 \[ -\frac {2 \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};-\frac {a x}{b}\right )}{3 b^2 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 184, normalized size = 2.67 \[ \left [\frac {15 \, {\left (a^{2} x^{3} + a b x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {a x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - b}{a x + b}\right ) + 2 \, {\left (15 \, a^{2} x^{2} + 10 \, a b x - 2 \, b^{2}\right )} \sqrt {x}}{6 \, {\left (a b^{3} x^{3} + b^{4} x^{2}\right )}}, -\frac {15 \, {\left (a^{2} x^{3} + a b x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {\frac {a}{b}}}{a \sqrt {x}}\right ) - {\left (15 \, a^{2} x^{2} + 10 \, a b x - 2 \, b^{2}\right )} \sqrt {x}}{3 \, {\left (a b^{3} x^{3} + b^{4} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 58, normalized size = 0.84 \[ \frac {5 \, a^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {a^{2} \sqrt {x}}{{\left (a x + b\right )} b^{3}} + \frac {2 \, {\left (6 \, a x - b\right )}}{3 \, b^{3} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 60, normalized size = 0.87 \[ \frac {5 a^{2} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{3}}+\frac {a^{2} \sqrt {x}}{\left (a x +b \right ) b^{3}}+\frac {4 a}{b^{3} \sqrt {x}}-\frac {2}{3 b^{2} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.24, size = 65, normalized size = 0.94 \[ \frac {a^{2}}{{\left (a b^{3} + \frac {b^{4}}{x}\right )} \sqrt {x}} - \frac {5 \, a^{2} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} b^{3}} + \frac {2 \, {\left (\frac {6 \, a}{\sqrt {x}} - \frac {b}{x^{\frac {3}{2}}}\right )}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 58, normalized size = 0.84 \[ \frac {\frac {5\,a^2\,x^2}{b^3}-\frac {2}{3\,b}+\frac {10\,a\,x}{3\,b^2}}{a\,x^{5/2}+b\,x^{3/2}}+\frac {5\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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