Optimal. Leaf size=70 \[ -\frac {\left (a+b \sqrt {x}\right )^6}{4 a x^4}+\frac {b \left (a+b \sqrt {x}\right )^6}{14 a^2 x^{7/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^6}{84 a^3 x^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {272, 47, 37}
\begin {gather*} -\frac {b^2 \left (a+b \sqrt {x}\right )^6}{84 a^3 x^3}+\frac {b \left (a+b \sqrt {x}\right )^6}{14 a^2 x^{7/2}}-\frac {\left (a+b \sqrt {x}\right )^6}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^5}{x^5} \, dx &=2 \text {Subst}\left (\int \frac {(a+b x)^5}{x^9} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\left (a+b \sqrt {x}\right )^6}{4 a x^4}-\frac {b \text {Subst}\left (\int \frac {(a+b x)^5}{x^8} \, dx,x,\sqrt {x}\right )}{2 a}\\ &=-\frac {\left (a+b \sqrt {x}\right )^6}{4 a x^4}+\frac {b \left (a+b \sqrt {x}\right )^6}{14 a^2 x^{7/2}}+\frac {b^2 \text {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,\sqrt {x}\right )}{14 a^2}\\ &=-\frac {\left (a+b \sqrt {x}\right )^6}{4 a x^4}+\frac {b \left (a+b \sqrt {x}\right )^6}{14 a^2 x^{7/2}}-\frac {b^2 \left (a+b \sqrt {x}\right )^6}{84 a^3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 65, normalized size = 0.93 \begin {gather*} \frac {-21 a^5-120 a^4 b \sqrt {x}-280 a^3 b^2 x-336 a^2 b^3 x^{3/2}-210 a b^4 x^2-56 b^5 x^{5/2}}{84 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 58, normalized size = 0.83
method | result | size |
derivativedivides | \(-\frac {2 b^{5}}{3 x^{\frac {3}{2}}}-\frac {5 a \,b^{4}}{2 x^{2}}-\frac {10 a^{3} b^{2}}{3 x^{3}}-\frac {4 a^{2} b^{3}}{x^{\frac {5}{2}}}-\frac {10 a^{4} b}{7 x^{\frac {7}{2}}}-\frac {a^{5}}{4 x^{4}}\) | \(58\) |
default | \(-\frac {2 b^{5}}{3 x^{\frac {3}{2}}}-\frac {5 a \,b^{4}}{2 x^{2}}-\frac {10 a^{3} b^{2}}{3 x^{3}}-\frac {4 a^{2} b^{3}}{x^{\frac {5}{2}}}-\frac {10 a^{4} b}{7 x^{\frac {7}{2}}}-\frac {a^{5}}{4 x^{4}}\) | \(58\) |
trager | \(\frac {\left (x -1\right ) \left (3 a^{4} x^{3}+40 a^{2} b^{2} x^{3}+30 x^{3} b^{4}+3 a^{4} x^{2}+40 a^{2} b^{2} x^{2}+30 b^{4} x^{2}+3 a^{4} x +40 a^{2} b^{2} x +3 a^{4}\right ) a}{12 x^{4}}-\frac {2 \left (7 b^{4} x^{2}+42 a^{2} b^{2} x +15 a^{4}\right ) b}{21 x^{\frac {7}{2}}}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 57, normalized size = 0.81 \begin {gather*} -\frac {56 \, b^{5} x^{\frac {5}{2}} + 210 \, a b^{4} x^{2} + 336 \, a^{2} b^{3} x^{\frac {3}{2}} + 280 \, a^{3} b^{2} x + 120 \, a^{4} b \sqrt {x} + 21 \, a^{5}}{84 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 58, normalized size = 0.83 \begin {gather*} -\frac {210 \, a b^{4} x^{2} + 280 \, a^{3} b^{2} x + 21 \, a^{5} + 8 \, {\left (7 \, b^{5} x^{2} + 42 \, a^{2} b^{3} x + 15 \, a^{4} b\right )} \sqrt {x}}{84 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 73, normalized size = 1.04 \begin {gather*} - \frac {a^{5}}{4 x^{4}} - \frac {10 a^{4} b}{7 x^{\frac {7}{2}}} - \frac {10 a^{3} b^{2}}{3 x^{3}} - \frac {4 a^{2} b^{3}}{x^{\frac {5}{2}}} - \frac {5 a b^{4}}{2 x^{2}} - \frac {2 b^{5}}{3 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.01, size = 57, normalized size = 0.81 \begin {gather*} -\frac {56 \, b^{5} x^{\frac {5}{2}} + 210 \, a b^{4} x^{2} + 336 \, a^{2} b^{3} x^{\frac {3}{2}} + 280 \, a^{3} b^{2} x + 120 \, a^{4} b \sqrt {x} + 21 \, a^{5}}{84 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.10, size = 57, normalized size = 0.81 \begin {gather*} -\frac {21\,a^5+56\,b^5\,x^{5/2}+280\,a^3\,b^2\,x+210\,a\,b^4\,x^2+120\,a^4\,b\,\sqrt {x}+336\,a^2\,b^3\,x^{3/2}}{84\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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