Optimal. Leaf size=64 \[ \frac {a^3}{b^4 \left (a+b \sqrt {x}\right )^2}-\frac {6 a^2}{b^4 \left (a+b \sqrt {x}\right )}-\frac {6 a \log \left (a+b \sqrt {x}\right )}{b^4}+\frac {2 \sqrt {x}}{b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 64, 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} \[ \frac {a^3}{b^4 \left (a+b \sqrt {x}\right )^2}-\frac {6 a^2}{b^4 \left (a+b \sqrt {x}\right )}-\frac {6 a \log \left (a+b \sqrt {x}\right )}{b^4}+\frac {2 \sqrt {x}}{b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sqrt {x}\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{(a+b x)^3} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{b^3}-\frac {a^3}{b^3 (a+b x)^3}+\frac {3 a^2}{b^3 (a+b x)^2}-\frac {3 a}{b^3 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^3}{b^4 \left (a+b \sqrt {x}\right )^2}-\frac {6 a^2}{b^4 \left (a+b \sqrt {x}\right )}+\frac {2 \sqrt {x}}{b^3}-\frac {6 a \log \left (a+b \sqrt {x}\right )}{b^4}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 57, normalized size = 0.89 \[ \frac {\frac {a^3}{\left (a+b \sqrt {x}\right )^2}-\frac {6 a^2}{a+b \sqrt {x}}-6 a \log \left (a+b \sqrt {x}\right )+2 b \sqrt {x}}{b^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 101, normalized size = 1.58 \[ \frac {7 \, a^{3} b^{2} x - 5 \, a^{5} - 6 \, {\left (a b^{4} x^{2} - 2 \, a^{3} b^{2} x + a^{5}\right )} \log \left (b \sqrt {x} + a\right ) + 2 \, {\left (b^{5} x^{2} - 5 \, a^{2} b^{3} x + 3 \, a^{4} b\right )} \sqrt {x}}{b^{8} x^{2} - 2 \, a^{2} b^{6} x + a^{4} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 0.83 \[ -\frac {6 \, a \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{4}} + \frac {2 \, \sqrt {x}}{b^{3}} - \frac {6 \, a^{2} b \sqrt {x} + 5 \, a^{3}}{{\left (b \sqrt {x} + a\right )}^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.89 \[ \frac {a^{3}}{\left (b \sqrt {x}+a \right )^{2} b^{4}}-\frac {6 a^{2}}{\left (b \sqrt {x}+a \right ) b^{4}}-\frac {6 a \ln \left (b \sqrt {x}+a \right )}{b^{4}}+\frac {2 \sqrt {x}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 60, normalized size = 0.94 \[ -\frac {6 \, a \log \left (b \sqrt {x} + a\right )}{b^{4}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}}{b^{4}} - \frac {6 \, a^{2}}{{\left (b \sqrt {x} + a\right )} b^{4}} + \frac {a^{3}}{{\left (b \sqrt {x} + a\right )}^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 66, normalized size = 1.03 \[ \frac {2\,\sqrt {x}}{b^3}-\frac {\frac {5\,a^3}{b}+6\,a^2\,\sqrt {x}}{b^5\,x+a^2\,b^3+2\,a\,b^4\,\sqrt {x}}-\frac {6\,a\,\ln \left (a+b\,\sqrt {x}\right )}{b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.95, size = 233, normalized size = 3.64 \[ \begin {cases} - \frac {6 a^{3} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} - \frac {9 a^{3}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} - \frac {12 a^{2} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} - \frac {12 a^{2} b \sqrt {x}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} - \frac {6 a b^{2} x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} + \frac {2 b^{3} x^{\frac {3}{2}}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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