Integrand size = 26, antiderivative size = 130 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{3/2}} \, dx=\frac {6 a}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1355, 660, 45} \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{3/2}} \, dx=-\frac {3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {6 a}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
[In]
[Out]
Rule 45
Rule 660
Rule 1355
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^2}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {\left (3 b^3 \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \frac {x^2}{\left (a b+b^2 x\right )^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \\ & = \frac {\left (3 b^3 \left (a+b \sqrt [3]{x}\right )\right ) \text {Subst}\left (\int \left (\frac {a^2}{b^5 (a+b x)^3}-\frac {2 a}{b^5 (a+b x)^2}+\frac {1}{b^5 (a+b x)}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \\ & = \frac {6 a}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac {3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac {3 \left (a+b \sqrt [3]{x}\right ) \log \left (a+b \sqrt [3]{x}\right )}{b^3 \sqrt {a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{3/2}} \, dx=\frac {3 a \left (3 a+4 b \sqrt [3]{x}\right )+6 \left (a+b \sqrt [3]{x}\right )^2 \log \left (a+b \sqrt [3]{x}\right )}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt {\left (a+b \sqrt [3]{x}\right )^2}} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.62
method | result | size |
derivativedivides | \(\frac {3 \left (2 \ln \left (a +b \,x^{\frac {1}{3}}\right ) b^{2} x^{\frac {2}{3}}+4 \ln \left (a +b \,x^{\frac {1}{3}}\right ) a b \,x^{\frac {1}{3}}+2 a^{2} \ln \left (a +b \,x^{\frac {1}{3}}\right )+4 a b \,x^{\frac {1}{3}}+3 a^{2}\right ) \left (a +b \,x^{\frac {1}{3}}\right )}{2 b^{3} {\left (\left (a +b \,x^{\frac {1}{3}}\right )^{2}\right )}^{\frac {3}{2}}}\) | \(81\) |
default | \(\frac {\left (8 x^{3} \ln \left (b^{3} x +a^{3}\right ) a^{3} b^{9}-8 x^{3} \ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right ) a^{3} b^{9}+16 x^{3} \ln \left (a +b \,x^{\frac {1}{3}}\right ) a^{3} b^{9}-27 x^{\frac {4}{3}} a^{8} b^{4}+3 x^{\frac {2}{3}} a^{10} b^{2}-6 x^{\frac {1}{3}} a^{11} b +2 \ln \left (b^{3} x +a^{3}\right ) a^{12}-2 \ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right ) a^{12}+4 \ln \left (a +b \,x^{\frac {1}{3}}\right ) a^{12}+9 a^{12}+36 a^{9} b^{3} x +18 a^{3} b^{9} x^{3}+45 a^{6} b^{6} x^{2}+18 x^{\frac {5}{3}} a^{7} b^{5}+12 x^{\frac {11}{3}} a \,b^{11}+27 x^{\frac {8}{3}} a^{4} b^{8}-15 x^{\frac {10}{3}} a^{2} b^{10}-36 x^{\frac {7}{3}} a^{5} b^{7}+8 x \ln \left (b^{3} x +a^{3}\right ) a^{9} b^{3}-8 x \ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right ) a^{9} b^{3}+16 x \ln \left (a +b \,x^{\frac {1}{3}}\right ) a^{9} b^{3}+2 x^{4} \ln \left (b^{3} x +a^{3}\right ) b^{12}-2 x^{4} \ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right ) b^{12}+4 x^{4} \ln \left (a +b \,x^{\frac {1}{3}}\right ) b^{12}+12 x^{2} \ln \left (b^{3} x +a^{3}\right ) a^{6} b^{6}-12 x^{2} \ln \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right ) a^{6} b^{6}+24 x^{2} \ln \left (a +b \,x^{\frac {1}{3}}\right ) a^{6} b^{6}\right ) \left (a +b \,x^{\frac {1}{3}}\right )}{2 b^{3} \left (b^{2} x^{\frac {2}{3}}-a b \,x^{\frac {1}{3}}+a^{2}\right )^{2} \left (b^{3} x +a^{3}\right )^{2} \left (a^{2}+2 a b \,x^{\frac {1}{3}}+b^{2} x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\) | \(502\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{3/2}} \, dx=\frac {3 \, {\left (6 \, a^{3} b^{3} x + 3 \, a^{6} + 2 \, {\left (b^{6} x^{2} + 2 \, a^{3} b^{3} x + a^{6}\right )} \log \left (b x^{\frac {1}{3}} + a\right ) + {\left (4 \, a b^{5} x + a^{4} b^{2}\right )} x^{\frac {2}{3}} - {\left (5 \, a^{2} b^{4} x + 2 \, a^{5} b\right )} x^{\frac {1}{3}}\right )}}{2 \, {\left (b^{9} x^{2} + 2 \, a^{3} b^{6} x + a^{6} b^{3}\right )}} \]
[In]
[Out]
\[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{3/2}} \, dx=\int \frac {1}{\left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{3/2}} \, dx=\frac {3 \, \log \left (x^{\frac {1}{3}} + \frac {a}{b}\right )}{b^{3}} + \frac {6 \, a x^{\frac {1}{3}}}{b^{4} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{2}} + \frac {9 \, a^{2}}{2 \, b^{5} {\left (x^{\frac {1}{3}} + \frac {a}{b}\right )}^{2}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{3/2}} \, dx=\frac {3 \, \log \left ({\left | b x^{\frac {1}{3}} + a \right |}\right )}{b^{3} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )} + \frac {3 \, {\left (4 \, a x^{\frac {1}{3}} + \frac {3 \, a^{2}}{b}\right )}}{2 \, {\left (b x^{\frac {1}{3}} + a\right )}^{2} b^{2} \mathrm {sgn}\left (b x^{\frac {1}{3}} + a\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\left (a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}\right )^{3/2}} \, dx=\int \frac {1}{{\left (a^2+b^2\,x^{2/3}+2\,a\,b\,x^{1/3}\right )}^{3/2}} \,d x \]
[In]
[Out]