Optimal. Leaf size=190 \[ \frac{3 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^2 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.118063, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1341, 1355, 263, 43} \[ \frac{3 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^3 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^2 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{3 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^4 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1341
Rule 1355
Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}} \, dx &=3 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a^2+\frac{b^2}{x^2}+\frac{2 a b}{x}}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{\left (3 \left (a b+\frac{b^2}{\sqrt [3]{x}}\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{a b+\frac{b^2}{x}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}\\ &=\frac{\left (3 \left (a b+\frac{b^2}{\sqrt [3]{x}}\right )\right ) \operatorname{Subst}\left (\int \frac{x^3}{b^2+a b x} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}\\ &=\frac{\left (3 \left (a b+\frac{b^2}{\sqrt [3]{x}}\right )\right ) \operatorname{Subst}\left (\int \left (\frac{b}{a^3}-\frac{x}{a^2}+\frac{x^2}{a b}-\frac{b^2}{a^3 (b+a x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}\\ &=\frac{3 \left (a b^2+\frac{b^3}{\sqrt [3]{x}}\right ) \sqrt [3]{x}}{a^3 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}-\frac{3 \left (a b+\frac{b^2}{\sqrt [3]{x}}\right ) x^{2/3}}{2 a^2 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}+\frac{\left (a+\frac{b}{\sqrt [3]{x}}\right ) x}{a \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}-\frac{3 \left (a b^3+\frac{b^4}{\sqrt [3]{x}}\right ) \log \left (b+a \sqrt [3]{x}\right )}{a^4 \sqrt{a^2+\frac{b^2}{x^{2/3}}+\frac{2 a b}{\sqrt [3]{x}}}}\\ \end{align*}
Mathematica [A] time = 0.0410827, size = 86, normalized size = 0.45 \[ \frac{\left (a \sqrt [3]{x}+b\right ) \left (-3 a^2 b x^{2/3}+2 a^3 x+6 a b^2 \sqrt [3]{x}-6 b^3 \log \left (a \sqrt [3]{x}+b\right )\right )}{2 a^4 \sqrt [3]{x} \sqrt{\frac{\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 78, normalized size = 0.4 \begin{align*} -{\frac{1}{2\,{a}^{4}} \left ( b+a\sqrt [3]{x} \right ) \left ( 3\,{x}^{2/3}{a}^{2}b-6\,a{b}^{2}\sqrt [3]{x}+6\,{b}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) -2\,{a}^{3}x \right ){\frac{1}{\sqrt{{ \left ({a}^{2}{x}^{{\frac{2}{3}}}+2\,ab\sqrt [3]{x}+{b}^{2} \right ){x}^{-{\frac{2}{3}}}}}}}{\frac{1}{\sqrt [3]{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01861, size = 59, normalized size = 0.31 \begin{align*} -\frac{3 \, b^{3} \log \left (a x^{\frac{1}{3}} + b\right )}{a^{4}} + \frac{2 \, a^{2} x - 3 \, a b x^{\frac{2}{3}} + 6 \, b^{2} x^{\frac{1}{3}}}{2 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.20538, size = 104, normalized size = 0.55 \begin{align*} -\frac{3 \, b^{3} \log \left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{4} \mathrm{sgn}\left (a x + b x^{\frac{2}{3}}\right ) \mathrm{sgn}\left (x\right )} + \frac{2 \, a^{2} x - 3 \, a b x^{\frac{2}{3}} + 6 \, b^{2} x^{\frac{1}{3}}}{2 \, a^{3} \mathrm{sgn}\left (a x + b x^{\frac{2}{3}}\right ) \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]