Optimal. Leaf size=126 \[ -\frac {256 b^4}{35 a^5 \sqrt {x} \sqrt {a+\frac {b}{x}}}-\frac {128 b^3 \sqrt {x}}{35 a^4 \sqrt {a+\frac {b}{x}}}+\frac {32 b^2 x^{3/2}}{35 a^3 \sqrt {a+\frac {b}{x}}}-\frac {16 b x^{5/2}}{35 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{7/2}}{7 a \sqrt {a+\frac {b}{x}}} \]
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Rubi [A] time = 0.05, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac {32 b^2 x^{3/2}}{35 a^3 \sqrt {a+\frac {b}{x}}}-\frac {256 b^4}{35 a^5 \sqrt {x} \sqrt {a+\frac {b}{x}}}-\frac {128 b^3 \sqrt {x}}{35 a^4 \sqrt {a+\frac {b}{x}}}-\frac {16 b x^{5/2}}{35 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{7/2}}{7 a \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx &=\frac {2 x^{7/2}}{7 a \sqrt {a+\frac {b}{x}}}-\frac {(8 b) \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx}{7 a}\\ &=-\frac {16 b x^{5/2}}{35 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{7/2}}{7 a \sqrt {a+\frac {b}{x}}}+\frac {\left (48 b^2\right ) \int \frac {\sqrt {x}}{\left (a+\frac {b}{x}\right )^{3/2}} \, dx}{35 a^2}\\ &=\frac {32 b^2 x^{3/2}}{35 a^3 \sqrt {a+\frac {b}{x}}}-\frac {16 b x^{5/2}}{35 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{7/2}}{7 a \sqrt {a+\frac {b}{x}}}-\frac {\left (64 b^3\right ) \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}} \, dx}{35 a^3}\\ &=-\frac {128 b^3 \sqrt {x}}{35 a^4 \sqrt {a+\frac {b}{x}}}+\frac {32 b^2 x^{3/2}}{35 a^3 \sqrt {a+\frac {b}{x}}}-\frac {16 b x^{5/2}}{35 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{7/2}}{7 a \sqrt {a+\frac {b}{x}}}+\frac {\left (128 b^4\right ) \int \frac {1}{\left (a+\frac {b}{x}\right )^{3/2} x^{3/2}} \, dx}{35 a^4}\\ &=-\frac {256 b^4}{35 a^5 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {128 b^3 \sqrt {x}}{35 a^4 \sqrt {a+\frac {b}{x}}}+\frac {32 b^2 x^{3/2}}{35 a^3 \sqrt {a+\frac {b}{x}}}-\frac {16 b x^{5/2}}{35 a^2 \sqrt {a+\frac {b}{x}}}+\frac {2 x^{7/2}}{7 a \sqrt {a+\frac {b}{x}}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 64, normalized size = 0.51 \[ \frac {2 \left (5 a^4 x^4-8 a^3 b x^3+16 a^2 b^2 x^2-64 a b^3 x-128 b^4\right )}{35 a^5 \sqrt {x} \sqrt {a+\frac {b}{x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 70, normalized size = 0.56 \[ \frac {2 \, {\left (5 \, a^{4} x^{4} - 8 \, a^{3} b x^{3} + 16 \, a^{2} b^{2} x^{2} - 64 \, a b^{3} x - 128 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{35 \, {\left (a^{6} x + a^{5} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 85, normalized size = 0.67 \[ \frac {256 \, b^{\frac {7}{2}}}{35 \, a^{5}} - \frac {2 \, b^{4}}{\sqrt {a x + b} a^{5}} + \frac {2 \, {\left (5 \, {\left (a x + b\right )}^{\frac {7}{2}} a^{30} - 28 \, {\left (a x + b\right )}^{\frac {5}{2}} a^{30} b + 70 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{30} b^{2} - 140 \, \sqrt {a x + b} a^{30} b^{3}\right )}}{35 \, a^{35}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 0.52 \[ \frac {2 \left (a x +b \right ) \left (5 a^{4} x^{4}-8 a^{3} x^{3} b +16 a^{2} x^{2} b^{2}-64 a x \,b^{3}-128 b^{4}\right )}{35 \left (\frac {a x +b}{x}\right )^{\frac {3}{2}} a^{5} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 90, normalized size = 0.71 \[ -\frac {2 \, b^{4}}{\sqrt {a + \frac {b}{x}} a^{5} \sqrt {x}} + \frac {2 \, {\left (5 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} x^{\frac {7}{2}} - 28 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b x^{\frac {5}{2}} + 70 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} x^{\frac {3}{2}} - 140 \, \sqrt {a + \frac {b}{x}} b^{3} \sqrt {x}\right )}}{35 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.63, size = 67, normalized size = 0.53 \[ -\frac {2\,\sqrt {x}\,\sqrt {\frac {b+a\,x}{x}}\,\left (-5\,a^4\,x^4+8\,a^3\,b\,x^3-16\,a^2\,b^2\,x^2+64\,a\,b^3\,x+128\,b^4\right )}{35\,a^5\,\left (b+a\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 14.00, size = 614, normalized size = 4.87 \[ \frac {10 a^{7} b^{\frac {33}{2}} x^{7} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} + \frac {14 a^{6} b^{\frac {35}{2}} x^{6} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} + \frac {14 a^{5} b^{\frac {37}{2}} x^{5} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac {70 a^{4} b^{\frac {39}{2}} x^{4} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac {560 a^{3} b^{\frac {41}{2}} x^{3} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac {1120 a^{2} b^{\frac {43}{2}} x^{2} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac {896 a b^{\frac {45}{2}} x \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} - \frac {256 b^{\frac {47}{2}} \sqrt {\frac {a x}{b} + 1}}{35 a^{9} b^{16} x^{4} + 140 a^{8} b^{17} x^{3} + 210 a^{7} b^{18} x^{2} + 140 a^{6} b^{19} x + 35 a^{5} b^{20}} \]
Verification of antiderivative is not currently implemented for this CAS.
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