Integrand size = 56, antiderivative size = 266 \[ \int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {3 b \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{2 a}-\frac {\sqrt {3} b \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{2} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} a}+\frac {b \log \left (-1+\sqrt [3]{2} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{2 \sqrt [3]{2} a}-\frac {b \log \left (1+\sqrt [3]{2} \sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}+2^{2/3} \left (a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}\right )^{2/3}\right )}{4 \sqrt [3]{2} a} \] Output:
3/2*b*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3)/a-1/4*3^(1/2)*b*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)/a+1/4*b*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-1/8*b*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)/a
Time = 19.03 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {b \sqrt [3]{x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \left (6 \sqrt [3]{2} \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}-2 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{a} \log \left (-\sqrt [3]{a}+\sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}\right )-\sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2}+\left (a+\left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )^2\right )^{2/3}\right )\right )}{4 \sqrt [3]{2} a \sqrt [3]{a x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}} \] Input:
Integrate[(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3)/Sqrt[-(a/b^2) + (a^2*x^2)/b^2],x]
Output:
(b*(x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3)*(6*2^(1/3)*(a*x*(a*x + b *Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3) - 2*Sqrt[3]*a^(1/3)*ArcTan[(1 + (2*(a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(1/3))/a^(1/3))/Sqrt[3]] + 2*a^( 1/3)*Log[-a^(1/3) + (a + (a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])^2)^(1/3)] - 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)]))/(4*2^(1/3)* a*(a*x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2]))^(1/3))
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.
\(\displaystyle \int \frac {\sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}} \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \frac {\sqrt [3]{b x \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x^2}}{\sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}}dx\) |
Input:
Int[(a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])^(1/3)/Sqrt[-(a/b^2) + (a^ 2*x^2)/b^2],x]
Output:
$Aborted
\[\int \frac {\left (a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}\right )^{\frac {1}{3}}}{\sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}d x\]
Input:
int((a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3)/(-a/b^2+a^2*x^2/b^2)^(1/2 ),x)
Output:
int((a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3)/(-a/b^2+a^2*x^2/b^2)^(1/2 ),x)
Timed out. \[ \int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\text {Timed out} \] Input:
integrate((a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3)/(-a/b^2+a^2*x^2/b^2 )^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int \frac {\sqrt [3]{x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}{\sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}\, dx \] Input:
integrate((a*x**2+b*x*(-a/b**2+a**2*x**2/b**2)**(1/2))**(1/3)/(-a/b**2+a** 2*x**2/b**2)**(1/2),x)
Output:
Integral((x*(a*x + b*sqrt(a**2*x**2/b**2 - a/b**2)))**(1/3)/sqrt(a*(a*x**2 - 1)/b**2), x)
\[ \int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int { \frac {{\left (a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x\right )}^{\frac {1}{3}}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}} \,d x } \] Input:
integrate((a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3)/(-a/b^2+a^2*x^2/b^2 )^(1/2),x, algorithm="maxima")
Output:
integrate((a*x^2 + sqrt(a^2*x^2/b^2 - a/b^2)*b*x)^(1/3)/sqrt(a^2*x^2/b^2 - a/b^2), x)
\[ \int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int { \frac {{\left (a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x\right )}^{\frac {1}{3}}}{\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}} \,d x } \] Input:
integrate((a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3)/(-a/b^2+a^2*x^2/b^2 )^(1/2),x, algorithm="giac")
Output:
integrate((a*x^2 + sqrt(a^2*x^2/b^2 - a/b^2)*b*x)^(1/3)/sqrt(a^2*x^2/b^2 - a/b^2), x)
Timed out. \[ \int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int \frac {{\left (a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}\right )}^{1/3}}{\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}} \,d x \] Input:
int((a*x^2 + b*x*((a^2*x^2)/b^2 - a/b^2)^(1/2))^(1/3)/((a^2*x^2)/b^2 - a/b ^2)^(1/2),x)
Output:
int((a*x^2 + b*x*((a^2*x^2)/b^2 - a/b^2)^(1/2))^(1/3)/((a^2*x^2)/b^2 - a/b ^2)^(1/2), x)
\[ \int \frac {\sqrt [3]{a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {x^{\frac {1}{3}} \sqrt {a \,x^{2}-1}\, \left (\sqrt {a}\, \sqrt {a \,x^{2}-1}+a x \right )^{\frac {1}{3}}}{a \,x^{2}-1}d x \right ) b}{a} \] Input:
int((a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/3)/(-a/b^2+a^2*x^2/b^2)^(1/2 ),x)
Output:
(sqrt(a)*int((x**(1/3)*sqrt(a*x**2 - 1)*(sqrt(a)*sqrt(a*x**2 - 1) + a*x)** (1/3))/(a*x**2 - 1),x)*b)/a