Optimal. Leaf size=45 \[ -\frac {a^3}{4 x^4}-\frac {6 a^2 b}{7 x^{7/2}}-\frac {a b^2}{x^3}-\frac {2 b^3}{5 x^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 45, 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^3}{4 x^4}-\frac {6 a^2 b}{7 x^{7/2}}-\frac {a b^2}{x^3}-\frac {2 b^3}{5 x^{5/2}} \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 )^3}{x^5} \, dx &=2 \text {Subst}\left (\int \frac {(a+b x)^3}{x^9} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (\frac {a^3}{x^9}+\frac {3 a^2 b}{x^8}+\frac {3 a b^2}{x^7}+\frac {b^3}{x^6}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^3}{4 x^4}-\frac {6 a^2 b}{7 x^{7/2}}-\frac {a b^2}{x^3}-\frac {2 b^3}{5 x^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.91 \begin {gather*} \frac {-35 a^3-120 a^2 b \sqrt {x}-140 a b^2 x-56 b^3 x^{3/2}}{140 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 36, normalized size = 0.80
method | result | size |
derivativedivides | \(-\frac {a^{3}}{4 x^{4}}-\frac {6 a^{2} b}{7 x^{\frac {7}{2}}}-\frac {a \,b^{2}}{x^{3}}-\frac {2 b^{3}}{5 x^{\frac {5}{2}}}\) | \(36\) |
default | \(-\frac {a^{3}}{4 x^{4}}-\frac {6 a^{2} b}{7 x^{\frac {7}{2}}}-\frac {a \,b^{2}}{x^{3}}-\frac {2 b^{3}}{5 x^{\frac {5}{2}}}\) | \(36\) |
trager | \(\frac {\left (x -1\right ) \left (a^{2} x^{3}+4 b^{2} x^{3}+a^{2} x^{2}+4 b^{2} x^{2}+a^{2} x +4 b^{2} x +a^{2}\right ) a}{4 x^{4}}-\frac {2 \left (7 b^{2} x +15 a^{2}\right ) b}{35 x^{\frac {7}{2}}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 35, normalized size = 0.78 \begin {gather*} -\frac {56 \, b^{3} x^{\frac {3}{2}} + 140 \, a b^{2} x + 120 \, a^{2} b \sqrt {x} + 35 \, a^{3}}{140 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 36, normalized size = 0.80 \begin {gather*} -\frac {140 \, a b^{2} x + 35 \, a^{3} + 8 \, {\left (7 \, b^{3} x + 15 \, a^{2} b\right )} \sqrt {x}}{140 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 42, normalized size = 0.93 \begin {gather*} - \frac {a^{3}}{4 x^{4}} - \frac {6 a^{2} b}{7 x^{\frac {7}{2}}} - \frac {a b^{2}}{x^{3}} - \frac {2 b^{3}}{5 x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 35, normalized size = 0.78 \begin {gather*} -\frac {56 \, b^{3} x^{\frac {3}{2}} + 140 \, a b^{2} x + 120 \, a^{2} b \sqrt {x} + 35 \, a^{3}}{140 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 35, normalized size = 0.78 \begin {gather*} -\frac {35\,a^3+56\,b^3\,x^{3/2}+120\,a^2\,b\,\sqrt {x}+140\,a\,b^2\,x}{140\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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