Optimal. Leaf size=47 \[ -\frac {a^3}{3 x^3}-\frac {6 a^2 b}{5 x^{5/2}}-\frac {3 a b^2}{2 x^2}-\frac {2 b^3}{3 x^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 47, 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} \[ -\frac {6 a^2 b}{5 x^{5/2}}-\frac {a^3}{3 x^3}-\frac {3 a b^2}{2 x^2}-\frac {2 b^3}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt {x}\right )^3}{x^4} \, dx &=2 \operatorname {Subst}\left (\int \frac {(a+b x)^3}{x^7} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {a^3}{x^7}+\frac {3 a^2 b}{x^6}+\frac {3 a b^2}{x^5}+\frac {b^3}{x^4}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {a^3}{3 x^3}-\frac {6 a^2 b}{5 x^{5/2}}-\frac {3 a b^2}{2 x^2}-\frac {2 b^3}{3 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.87 \[ -\frac {10 a^3+36 a^2 b \sqrt {x}+45 a b^2 x+20 b^3 x^{3/2}}{30 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 36, normalized size = 0.77 \[ -\frac {45 \, a b^{2} x + 10 \, a^{3} + 4 \, {\left (5 \, b^{3} x + 9 \, a^{2} b\right )} \sqrt {x}}{30 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 35, normalized size = 0.74 \[ -\frac {20 \, b^{3} x^{\frac {3}{2}} + 45 \, a b^{2} x + 36 \, a^{2} b \sqrt {x} + 10 \, a^{3}}{30 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.77 \[ -\frac {2 b^{3}}{3 x^{\frac {3}{2}}}-\frac {3 a \,b^{2}}{2 x^{2}}-\frac {6 a^{2} b}{5 x^{\frac {5}{2}}}-\frac {a^{3}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 35, normalized size = 0.74 \[ -\frac {20 \, b^{3} x^{\frac {3}{2}} + 45 \, a b^{2} x + 36 \, a^{2} b \sqrt {x} + 10 \, a^{3}}{30 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 35, normalized size = 0.74 \[ -\frac {10\,a^3+20\,b^3\,x^{3/2}+36\,a^2\,b\,\sqrt {x}+45\,a\,b^2\,x}{30\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.10, size = 46, normalized size = 0.98 \[ - \frac {a^{3}}{3 x^{3}} - \frac {6 a^{2} b}{5 x^{\frac {5}{2}}} - \frac {3 a b^{2}}{2 x^{2}} - \frac {2 b^{3}}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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