Optimal. Leaf size=55 \[ \frac {2 a^2}{3 b^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 a}{b^3 \sqrt {a+\frac {b}{x}}}-\frac {2 \sqrt {a+\frac {b}{x}}}{b^3} \]
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Rubi [A] time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {2 a^2}{3 b^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 a}{b^3 \sqrt {a+\frac {b}{x}}}-\frac {2 \sqrt {a+\frac {b}{x}}}{b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^4} \, dx &=-\operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^{5/2}} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a^2}{b^2 (a+b x)^{5/2}}-\frac {2 a}{b^2 (a+b x)^{3/2}}+\frac {1}{b^2 \sqrt {a+b x}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {2 a^2}{3 b^3 \left (a+\frac {b}{x}\right )^{3/2}}-\frac {4 a}{b^3 \sqrt {a+\frac {b}{x}}}-\frac {2 \sqrt {a+\frac {b}{x}}}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.80 \[ -\frac {2 \sqrt {a+\frac {b}{x}} \left (8 a^2 x^2+12 a b x+3 b^2\right )}{3 b^3 (a x+b)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 55, normalized size = 1.00 \[ -\frac {2 \, {\left (8 \, a^{2} x^{2} + 12 \, a b x + 3 \, b^{2}\right )} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 54, normalized size = 0.98 \[ \frac {2 \, {\left (\frac {{\left (a^{2} - \frac {6 \, {\left (a x + b\right )} a}{x}\right )} x}{{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}} - 3 \, \sqrt {\frac {a x + b}{x}}\right )}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 0.80 \[ -\frac {2 \left (a x +b \right ) \left (8 a^{2} x^{2}+12 a b x +3 b^{2}\right )}{3 \left (\frac {a x +b}{x}\right )^{\frac {5}{2}} b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 47, normalized size = 0.85 \[ -\frac {2 \, \sqrt {a + \frac {b}{x}}}{b^{3}} - \frac {4 \, a}{\sqrt {a + \frac {b}{x}} b^{3}} + \frac {2 \, a^{2}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 40, normalized size = 0.73 \[ -\frac {2\,\sqrt {a+\frac {b}{x}}\,\left (8\,a^2\,x^2+12\,a\,b\,x+3\,b^2\right )}{3\,b^3\,{\left (b+a\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.27, size = 136, normalized size = 2.47 \[ \begin {cases} - \frac {16 a^{2} x^{2}}{3 a b^{3} x^{2} \sqrt {a + \frac {b}{x}} + 3 b^{4} x \sqrt {a + \frac {b}{x}}} - \frac {24 a b x}{3 a b^{3} x^{2} \sqrt {a + \frac {b}{x}} + 3 b^{4} x \sqrt {a + \frac {b}{x}}} - \frac {6 b^{2}}{3 a b^{3} x^{2} \sqrt {a + \frac {b}{x}} + 3 b^{4} x \sqrt {a + \frac {b}{x}}} & \text {for}\: b \neq 0 \\- \frac {1}{3 a^{\frac {5}{2}} x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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