Optimal. Leaf size=45 \[ 2 a^3 \sqrt {x}-\frac {6 a^2 b}{\sqrt {x}}-\frac {2 a b^2}{x^{3/2}}-\frac {2 b^3}{5 x^{5/2}} \]
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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 = {263, 43} \[ -\frac {6 a^2 b}{\sqrt {x}}+2 a^3 \sqrt {x}-\frac {2 a b^2}{x^{3/2}}-\frac {2 b^3}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 263
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x}\right )^3}{\sqrt {x}} \, dx &=\int \frac {(b+a x)^3}{x^{7/2}} \, dx\\ &=\int \left (\frac {b^3}{x^{7/2}}+\frac {3 a b^2}{x^{5/2}}+\frac {3 a^2 b}{x^{3/2}}+\frac {a^3}{\sqrt {x}}\right ) \, dx\\ &=-\frac {2 b^3}{5 x^{5/2}}-\frac {2 a b^2}{x^{3/2}}-\frac {6 a^2 b}{\sqrt {x}}+2 a^3 \sqrt {x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 0.87 \[ \frac {2 \left (5 a^3 x^3-15 a^2 b x^2-5 a b^2 x-b^3\right )}{5 x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 35, normalized size = 0.78 \[ \frac {2 \, {\left (5 \, a^{3} x^{3} - 15 \, a^{2} b x^{2} - 5 \, a b^{2} x - b^{3}\right )}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 34, normalized size = 0.76 \[ 2 \, a^{3} \sqrt {x} - \frac {2 \, {\left (15 \, a^{2} b x^{2} + 5 \, a b^{2} x + b^{3}\right )}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.80 \[ \frac {2 a^{3} x^{3}-6 a^{2} b \,x^{2}-2 a \,b^{2} x -\frac {2}{5} b^{3}}{x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 35, normalized size = 0.78 \[ 2 \, a^{3} \sqrt {x} - \frac {6 \, a^{2} b}{\sqrt {x}} - \frac {2 \, a b^{2}}{x^{\frac {3}{2}}} - \frac {2 \, b^{3}}{5 \, x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 35, normalized size = 0.78 \[ -\frac {-10\,a^3\,x^3+30\,a^2\,b\,x^2+10\,a\,b^2\,x+2\,b^3}{5\,x^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 44, normalized size = 0.98 \[ 2 a^{3} \sqrt {x} - \frac {6 a^{2} b}{\sqrt {x}} - \frac {2 a b^{2}}{x^{\frac {3}{2}}} - \frac {2 b^{3}}{5 x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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