Optimal. Leaf size=51 \[ -\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4}+\frac {2 a^2 \sqrt {x}}{b^3}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b} \]
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Rubi [A] time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {2 a^2 \sqrt {x}}{b^3}-\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x}{a+b \sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{a+b x} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 a^2 \sqrt {x}}{b^3}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b}-\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 1.00 \[ -\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4}+\frac {2 a^2 \sqrt {x}}{b^3}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 43, normalized size = 0.84 \[ -\frac {3 \, a b^{2} x + 6 \, a^{3} \log \left (b \sqrt {x} + a\right ) - 2 \, {\left (b^{3} x + 3 \, a^{2} b\right )} \sqrt {x}}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 45, normalized size = 0.88 \[ -\frac {2 \, a^{3} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{4}} + \frac {2 \, b^{2} x^{\frac {3}{2}} - 3 \, a b x + 6 \, a^{2} \sqrt {x}}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 44, normalized size = 0.86 \[ \frac {2 x^{\frac {3}{2}}}{3 b}-\frac {2 a^{3} \ln \left (b \sqrt {x}+a \right )}{b^{4}}-\frac {a x}{b^{2}}+\frac {2 a^{2} \sqrt {x}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 61, normalized size = 1.20 \[ -\frac {2 \, a^{3} \log \left (b \sqrt {x} + a\right )}{b^{4}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{3}}{3 \, b^{4}} - \frac {3 \, {\left (b \sqrt {x} + a\right )}^{2} a}{b^{4}} + \frac {6 \, {\left (b \sqrt {x} + a\right )} a^{2}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 43, normalized size = 0.84 \[ \frac {2\,x^{3/2}}{3\,b}-\frac {2\,a^3\,\ln \left (a+b\,\sqrt {x}\right )}{b^4}+\frac {2\,a^2\,\sqrt {x}}{b^3}-\frac {a\,x}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 54, normalized size = 1.06 \[ \begin {cases} - \frac {2 a^{3} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{b^{4}} + \frac {2 a^{2} \sqrt {x}}{b^{3}} - \frac {a x}{b^{2}} + \frac {2 x^{\frac {3}{2}}}{3 b} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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