Optimal. Leaf size=38 \[ \frac {a}{2 b^2 \left (a+b \sqrt {x}\right )^4}-\frac {2}{3 b^2 \left (a+b \sqrt {x}\right )^3} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac {a}{2 b^2 \left (a+b \sqrt {x}\right )^4}-\frac {2}{3 b^2 \left (a+b \sqrt {x}\right )^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 190
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sqrt {x}\right )^5} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{(a+b x)^5} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^5}+\frac {1}{b (a+b x)^4}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a}{2 b^2 \left (a+b \sqrt {x}\right )^4}-\frac {2}{3 b^2 \left (a+b \sqrt {x}\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 0.74 \[ -\frac {a+4 b \sqrt {x}}{6 b^2 \left (a+b \sqrt {x}\right )^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 96, normalized size = 2.53 \[ \frac {15 \, a b^{4} x^{2} + 10 \, a^{3} b^{2} x - a^{5} - 4 \, {\left (b^{5} x^{2} + 5 \, a^{2} b^{3} x\right )} \sqrt {x}}{6 \, {\left (b^{10} x^{4} - 4 \, a^{2} b^{8} x^{3} + 6 \, a^{4} b^{6} x^{2} - 4 \, a^{6} b^{4} x + a^{8} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 22, normalized size = 0.58 \[ -\frac {4 \, b \sqrt {x} + a}{6 \, {\left (b \sqrt {x} + a\right )}^{4} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 200, normalized size = 5.26 \[ -10 \left (-\frac {a^{2}}{4 \left (b^{2} x -a^{2}\right )^{4} b^{4}}-\frac {1}{3 \left (b^{2} x -a^{2}\right )^{3} b^{4}}\right ) a^{3} b^{2}-5 \left (-\frac {a^{4}}{4 \left (b^{2} x -a^{2}\right )^{4} b^{6}}-\frac {2 a^{2}}{3 \left (b^{2} x -a^{2}\right )^{3} b^{6}}-\frac {1}{2 \left (b^{2} x -a^{2}\right )^{2} b^{6}}\right ) a \,b^{4}+\frac {a^{5}}{4 \left (b^{2} x -a^{2}\right )^{4} b^{2}}+\frac {a}{4 \left (b \sqrt {x}+a \right )^{4} b^{2}}-\frac {a}{4 \left (b \sqrt {x}-a \right )^{4} b^{2}}-\frac {1}{3 \left (b \sqrt {x}+a \right )^{3} b^{2}}-\frac {1}{3 \left (b \sqrt {x}-a \right )^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 30, normalized size = 0.79 \[ -\frac {2}{3 \, {\left (b \sqrt {x} + a\right )}^{3} b^{2}} + \frac {a}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 57, normalized size = 1.50 \[ -\frac {\frac {a}{6\,b^2}+\frac {2\,\sqrt {x}}{3\,b}}{a^4+b^4\,x^2+6\,a^2\,b^2\,x+4\,a^3\,b\,\sqrt {x}+4\,a\,b^3\,x^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.81, size = 121, normalized size = 3.18 \[ \begin {cases} - \frac {a}{6 a^{4} b^{2} + 24 a^{3} b^{3} \sqrt {x} + 36 a^{2} b^{4} x + 24 a b^{5} x^{\frac {3}{2}} + 6 b^{6} x^{2}} - \frac {4 b \sqrt {x}}{6 a^{4} b^{2} + 24 a^{3} b^{3} \sqrt {x} + 36 a^{2} b^{4} x + 24 a b^{5} x^{\frac {3}{2}} + 6 b^{6} x^{2}} & \text {for}\: b \neq 0 \\\frac {x}{a^{5}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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