Optimal. Leaf size=74 \[ \frac {16 b^2 x^{7/2} \left (a+\frac {b}{x}\right )^{7/2}}{693 a^3}-\frac {8 b x^{9/2} \left (a+\frac {b}{x}\right )^{7/2}}{99 a^2}+\frac {2 x^{11/2} \left (a+\frac {b}{x}\right )^{7/2}}{11 a} \]
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Rubi [A] time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac {16 b^2 x^{7/2} \left (a+\frac {b}{x}\right )^{7/2}}{693 a^3}-\frac {8 b x^{9/2} \left (a+\frac {b}{x}\right )^{7/2}}{99 a^2}+\frac {2 x^{11/2} \left (a+\frac {b}{x}\right )^{7/2}}{11 a} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^{5/2} x^{9/2} \, dx &=\frac {2 \left (a+\frac {b}{x}\right )^{7/2} x^{11/2}}{11 a}-\frac {(4 b) \int \left (a+\frac {b}{x}\right )^{5/2} x^{7/2} \, dx}{11 a}\\ &=-\frac {8 b \left (a+\frac {b}{x}\right )^{7/2} x^{9/2}}{99 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{7/2} x^{11/2}}{11 a}+\frac {\left (8 b^2\right ) \int \left (a+\frac {b}{x}\right )^{5/2} x^{5/2} \, dx}{99 a^2}\\ &=\frac {16 b^2 \left (a+\frac {b}{x}\right )^{7/2} x^{7/2}}{693 a^3}-\frac {8 b \left (a+\frac {b}{x}\right )^{7/2} x^{9/2}}{99 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{7/2} x^{11/2}}{11 a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 51, normalized size = 0.69 \[ \frac {2}{693} x^{7/2} \sqrt {a+\frac {b}{x}} \left (\frac {b}{a x}+1\right )^3 \left (63 a^2 x^2-28 a b x+8 b^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 71, normalized size = 0.96 \[ \frac {2 \, {\left (63 \, a^{5} x^{5} + 161 \, a^{4} b x^{4} + 113 \, a^{3} b^{2} x^{3} + 3 \, a^{2} b^{3} x^{2} - 4 \, a b^{4} x + 8 \, b^{5}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{693 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 194, normalized size = 2.62 \[ -\frac {2}{105} \, b^{2} {\left (\frac {8 \, b^{\frac {7}{2}}}{a^{3}} - \frac {15 \, {\left (a x + b\right )}^{\frac {7}{2}} - 42 \, {\left (a x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2}}{a^{3}}\right )} \mathrm {sgn}\relax (x) + \frac {4}{315} \, a b {\left (\frac {16 \, b^{\frac {9}{2}}}{a^{4}} + \frac {35 \, {\left (a x + b\right )}^{\frac {9}{2}} - 135 \, {\left (a x + b\right )}^{\frac {7}{2}} b + 189 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{2} - 105 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{3}}{a^{4}}\right )} \mathrm {sgn}\relax (x) - \frac {2}{3465} \, a^{2} {\left (\frac {128 \, b^{\frac {11}{2}}}{a^{5}} - \frac {315 \, {\left (a x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (a x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (a x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{4}}{a^{5}}\right )} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 44, normalized size = 0.59 \[ \frac {2 \left (a x +b \right ) \left (63 a^{2} x^{2}-28 a b x +8 b^{2}\right ) \left (\frac {a x +b}{x}\right )^{\frac {5}{2}} x^{\frac {5}{2}}}{693 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 52, normalized size = 0.70 \[ \frac {2 \, {\left (63 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}} x^{\frac {11}{2}} - 154 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} b x^{\frac {9}{2}} + 99 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b^{2} x^{\frac {7}{2}}\right )}}{693 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 67, normalized size = 0.91 \[ \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,a^2\,x^{11/2}}{11}+\frac {226\,b^2\,x^{7/2}}{693}+\frac {2\,b^3\,x^{5/2}}{231\,a}-\frac {8\,b^4\,x^{3/2}}{693\,a^2}+\frac {16\,b^5\,\sqrt {x}}{693\,a^3}+\frac {46\,a\,b\,x^{9/2}}{99}\right ) \]
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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