Integrand size = 56, antiderivative size = 330 \[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\left (-10-a x^2\right ) \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{16 b}+\frac {7}{16} x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}+\frac {5 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{2} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {3}}\right )}{8 \sqrt [3]{2} \sqrt {3} b}-\frac {5 \log \left (-1+\sqrt [3]{2} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{24 \sqrt [3]{2} b}+\frac {5 \log \left (1+\sqrt [3]{2} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}+2^{2/3} \left (a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )^{2/3}\right )}{48 \sqrt [3]{2} b} \]
1/16*(-a*x^2-10)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3)/b+7/16*x*(-a /b^2+a^2*x^2/b^2)^(1/2)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3)+5/48* arctan(1/3*3^(1/2)+2/3*2^(1/3)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3 )*3^(1/2))*2^(2/3)*3^(1/2)/b-5/48*ln(-1+2^(1/3)*(a*x^2+b*x*(-a/b^2+a^2*x^2 /b^2)^(1/2))^(1/3))*2^(2/3)/b+5/96*ln(1+2^(1/3)*(a*x^2+b*x*(-a/b^2+a^2*x^2 /b^2)^(1/2))^(1/3)+2^(2/3)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(2/3))*2 ^(2/3)/b
Leaf count is larger than twice the leaf count of optimal. \(755\) vs. \(2(330)=660\).
Time = 20.23 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.29 \[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\left (-1+a x^2\right ) \sqrt [3]{x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \left (30 \sqrt [3]{2} \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}-99 \sqrt [3]{2} a x^2 \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}+36 \sqrt [3]{2} a^2 x^4 \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}-81 \sqrt [3]{2} b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}} \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}+36 \sqrt [3]{2} a b x^3 \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}} \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}+10 \sqrt {3} \sqrt [3]{a} \left (-1+2 a x^2+2 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-10 \sqrt [3]{a} \left (-1+2 a x^2+2 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right ) \log \left (-\sqrt [3]{a}+\sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}\right )-5 \sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}+\left (a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2\right )^{2/3}\right )+10 a^{4/3} x^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}+\left (a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2\right )^{2/3}\right )+10 \sqrt [3]{a} b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}+\left (a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2\right )^{2/3}\right )\right )}{48 \sqrt [3]{2} b \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \left (-1+a x^2+b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2} \]
Integrate[Sqrt[-(a/b^2) + (a^2*x^2)/b^2]*(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2 *x^2)/b^2])^(1/3),x]
((-1 + a*x^2)*(x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3)*(30*2^(1/3)*( a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3) - 99*2^(1/3)*a*x^2*(a*x*(a *x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3) + 36*2^(1/3)*a^2*x^4*(a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3) - 81*2^(1/3)*b*x*Sqrt[(a*(-1 + a*x^2 ))/b^2]*(a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3) + 36*2^(1/3)*a*b* x^3*Sqrt[(a*(-1 + a*x^2))/b^2]*(a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^ (1/3) + 10*Sqrt[3]*a^(1/3)*(-1 + 2*a*x^2 + 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2 ])*ArcTan[(1 + (2*(a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(1/3))/a^(1 /3))/Sqrt[3]] - 10*a^(1/3)*(-1 + 2*a*x^2 + 2*b*x*Sqrt[(a*(-1 + a*x^2))/b^2 ])*Log[-a^(1/3) + (a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(1/3)] - 5* a^(1/3)*Log[a^(2/3) + a^(1/3)*(a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2) ^(1/3) + (a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(2/3)] + 10*a^(4/3)* x^2*Log[a^(2/3) + a^(1/3)*(a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(1/ 3) + (a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(2/3)] + 10*a^(1/3)*b*x* Sqrt[(a*(-1 + a*x^2))/b^2]*Log[a^(2/3) + a^(1/3)*(a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(1/3) + (a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^( 2/3)]))/(48*2^(1/3)*b*(a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3)*(-1 + a*x^2 + b*x*Sqrt[(a*(-1 + a*x^2))/b^2])^2)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}} \sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}dx\) |
3.30.17.3.1 Defintions of rubi rules used
\[\int \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\, \left (a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\right )^{\frac {1}{3}}d x\]
Timed out. \[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\text {Timed out} \]
integrate((-a/b^2+a^2*x^2/b^2)^(1/2)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2) )^(1/3),x, algorithm="fricas")
\[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int \sqrt [3]{x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )} \sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}\, dx \]
integrate((-a/b**2+a**2*x**2/b**2)**(1/2)*(a*x**2+b*x*(-a/b**2+a**2*x**2/b **2)**(1/2))**(1/3),x)
\[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int { {\left (a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x\right )}^{\frac {1}{3}} \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} \,d x } \]
integrate((-a/b^2+a^2*x^2/b^2)^(1/2)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2) )^(1/3),x, algorithm="maxima")
Exception generated. \[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\text {Exception raised: TypeError} \]
integrate((-a/b^2+a^2*x^2/b^2)^(1/2)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2) )^(1/3),x, algorithm="giac")
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int {\left (a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}\right )}^{1/3}\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}} \,d x \]