Optimal. Leaf size=54 \[ \frac {2 a^3}{b^4 \left (a+b \sqrt {x}\right )}+\frac {6 a^2 \log \left (a+b \sqrt {x}\right )}{b^4}-\frac {4 a \sqrt {x}}{b^3}+\frac {x}{b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 54, 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} \[ \frac {2 a^3}{b^4 \left (a+b \sqrt {x}\right )}+\frac {6 a^2 \log \left (a+b \sqrt {x}\right )}{b^4}-\frac {4 a \sqrt {x}}{b^3}+\frac {x}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sqrt {x}\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{(a+b x)^2} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {2 a}{b^3}+\frac {x}{b^2}-\frac {a^3}{b^3 (a+b x)^2}+\frac {3 a^2}{b^3 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 a^3}{b^4 \left (a+b \sqrt {x}\right )}-\frac {4 a \sqrt {x}}{b^3}+\frac {x}{b^2}+\frac {6 a^2 \log \left (a+b \sqrt {x}\right )}{b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 50, normalized size = 0.93 \[ \frac {\frac {2 a^3}{a+b \sqrt {x}}+6 a^2 \log \left (a+b \sqrt {x}\right )-4 a b \sqrt {x}+b^2 x}{b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.73, size = 82, normalized size = 1.52 \[ \frac {b^{4} x^{2} - a^{2} b^{2} x - 2 \, a^{4} + 6 \, {\left (a^{2} b^{2} x - a^{4}\right )} \log \left (b \sqrt {x} + a\right ) - 2 \, {\left (2 \, a b^{3} x - 3 \, a^{3} b\right )} \sqrt {x}}{b^{6} x - a^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 52, normalized size = 0.96 \[ \frac {6 \, a^{2} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{4}} + \frac {2 \, a^{3}}{{\left (b \sqrt {x} + a\right )} b^{4}} + \frac {b^{2} x - 4 \, a b \sqrt {x}}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 49, normalized size = 0.91 \[ \frac {2 a^{3}}{\left (b \sqrt {x}+a \right ) b^{4}}+\frac {6 a^{2} \ln \left (b \sqrt {x}+a \right )}{b^{4}}+\frac {x}{b^{2}}-\frac {4 a \sqrt {x}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.85, size = 60, normalized size = 1.11 \[ \frac {6 \, a^{2} \log \left (b \sqrt {x} + a\right )}{b^{4}} + \frac {{\left (b \sqrt {x} + a\right )}^{2}}{b^{4}} - \frac {6 \, {\left (b \sqrt {x} + a\right )} a}{b^{4}} + \frac {2 \, a^{3}}{{\left (b \sqrt {x} + a\right )} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 54, normalized size = 1.00 \[ \frac {x}{b^2}+\frac {2\,a^3}{b\,\left (a\,b^3+b^4\,\sqrt {x}\right )}-\frac {4\,a\,\sqrt {x}}{b^3}+\frac {6\,a^2\,\ln \left (a+b\,\sqrt {x}\right )}{b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.48, size = 134, normalized size = 2.48 \[ \begin {cases} \frac {6 a^{3} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a b^{4} + b^{5} \sqrt {x}} + \frac {6 a^{3}}{a b^{4} + b^{5} \sqrt {x}} + \frac {6 a^{2} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a b^{4} + b^{5} \sqrt {x}} - \frac {3 a b^{2} x}{a b^{4} + b^{5} \sqrt {x}} + \frac {b^{3} x^{\frac {3}{2}}}{a b^{4} + b^{5} \sqrt {x}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________