Optimal. Leaf size=32 \[ -\frac {a^2}{4 x^4}-\frac {4 a b}{7 x^{7/2}}-\frac {b^2}{3 x^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, 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 = {272, 45}
\begin {gather*} -\frac {a^2}{4 x^4}-\frac {4 a b}{7 x^{7/2}}-\frac {b^2}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^2}{x^5} \, dx &=2 \text {Subst}\left (\int \frac {(a+b x)^2}{x^9} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {a^2}{x^9}+\frac {2 a b}{x^8}+\frac {b^2}{x^7}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^2}{4 x^4}-\frac {4 a b}{7 x^{7/2}}-\frac {b^2}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 28, normalized size = 0.88 \begin {gather*} \frac {-21 a^2-48 a b \sqrt {x}-28 b^2 x}{84 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 25, normalized size = 0.78
method | result | size |
derivativedivides | \(-\frac {a^{2}}{4 x^{4}}-\frac {4 a b}{7 x^{\frac {7}{2}}}-\frac {b^{2}}{3 x^{3}}\) | \(25\) |
default | \(-\frac {a^{2}}{4 x^{4}}-\frac {4 a b}{7 x^{\frac {7}{2}}}-\frac {b^{2}}{3 x^{3}}\) | \(25\) |
trager | \(\frac {\left (x -1\right ) \left (3 a^{2} x^{3}+4 b^{2} x^{3}+3 a^{2} x^{2}+4 b^{2} x^{2}+3 a^{2} x +4 b^{2} x +3 a^{2}\right )}{12 x^{4}}-\frac {4 a b}{7 x^{\frac {7}{2}}}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 24, normalized size = 0.75 \begin {gather*} -\frac {28 \, b^{2} x + 48 \, a b \sqrt {x} + 21 \, a^{2}}{84 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 24, normalized size = 0.75 \begin {gather*} -\frac {28 \, b^{2} x + 48 \, a b \sqrt {x} + 21 \, a^{2}}{84 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 29, normalized size = 0.91 \begin {gather*} - \frac {a^{2}}{4 x^{4}} - \frac {4 a b}{7 x^{\frac {7}{2}}} - \frac {b^{2}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.91, size = 24, normalized size = 0.75 \begin {gather*} -\frac {28 \, b^{2} x + 48 \, a b \sqrt {x} + 21 \, a^{2}}{84 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 24, normalized size = 0.75 \begin {gather*} -\frac {28\,b^2\,x+21\,a^2+48\,a\,b\,\sqrt {x}}{84\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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