\(\int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx\) [2638]

Optimal result

Integrand size = 59, antiderivative size = 233 \[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \left (-255-136 a x^2+384 a^2 x^4\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{1920 a^2}+\frac {\left (85 x+568 a x^3-384 a^2 x^5\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{1920 a b}+\frac {17 \log \left (-a x-b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{128 \sqrt {2} a^{3/2} b} \]

[Out]

1/1920*(-a/b^2+a^2*x^2/b^2)^(1/2)*(384*a^2*x^4-136*a*x^2-255)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2)/a^2
+1/1920*(-384*a^2*x^5+568*a*x^3+85*x)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2)/a/b+17/256*ln(-a*x-b*(-a/b^
2+a^2*x^2/b^2)^(1/2)+2^(1/2)*a^(1/2)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2))*2^(1/2)/a^(3/2)/b

Rubi [F]

\[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \]

[In]

Int[(x^3*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])/Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]],x]

[Out]

Defer[Int][(x^3*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])/Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 9.29 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.86 \[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {\sqrt {x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \left (-2 x \left (384 a^3 x^5+255 b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}+17 a x \left (-5+8 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )-8 a^2 x^3 \left (71+48 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )\right )+255 \sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \arctan \left (\sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}\right )\right )}{3840 a^2 b x} \]

[In]

Integrate[(x^3*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])/Sqrt[a*x^2 + b*x*Sqrt[-(a/b^2) + (a^2*x^2)/b^2]],x]

[Out]

(Sqrt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(-2*x*(384*a^3*x^5 + 255*b*Sqrt[(a*(-1 + a*x^2))/b^2] + 17*a*x*(
-5 + 8*b*x*Sqrt[(a*(-1 + a*x^2))/b^2]) - 8*a^2*x^3*(71 + 48*b*x*Sqrt[(a*(-1 + a*x^2))/b^2])) + 255*Sqrt[2]*Sqr
t[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*ArcTan[Sqrt[2]*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^2])]]))
/(3840*a^2*b*x)

Maple [F]

\[\int \frac {x^{3} \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}{\sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}d x\]

[In]

int(x^3*(-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))^(1/2),x)

[Out]

int(x^3*(-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))^(1/2),x)

Fricas [A] (verification not implemented)

none

Time = 21.79 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.54 \[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\left [\frac {255 \, \sqrt {2} \sqrt {a} \log \left (-4 \, a^{2} x^{2} - 4 \, a b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} + a\right ) - 4 \, {\left (384 \, a^{3} x^{5} - 568 \, a^{2} x^{3} - 85 \, a x - {\left (384 \, a^{2} b x^{4} - 136 \, a b x^{2} - 255 \, b\right )} \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{7680 \, a^{2} b}, \frac {255 \, \sqrt {2} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (384 \, a^{3} x^{5} - 568 \, a^{2} x^{3} - 85 \, a x - {\left (384 \, a^{2} b x^{4} - 136 \, a b x^{2} - 255 \, b\right )} \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{3840 \, a^{2} b}\right ] \]

[In]

integrate(x^3*(-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))^(1/2),x, algorithm="fricas")

[Out]

[1/7680*(255*sqrt(2)*sqrt(a)*log(-4*a^2*x^2 - 4*a*b*x*sqrt((a^2*x^2 - a)/b^2) + 2*(sqrt(2)*a^(3/2)*x + sqrt(2)
*sqrt(a)*b*sqrt((a^2*x^2 - a)/b^2))*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2)) + a) - 4*(384*a^3*x^5 - 568*a^2*
x^3 - 85*a*x - (384*a^2*b*x^4 - 136*a*b*x^2 - 255*b)*sqrt((a^2*x^2 - a)/b^2))*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 -
a)/b^2)))/(a^2*b), 1/3840*(255*sqrt(2)*sqrt(-a)*arctan(1/2*sqrt(2)*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*
sqrt(-a)/(a*x)) - 2*(384*a^3*x^5 - 568*a^2*x^3 - 85*a*x - (384*a^2*b*x^4 - 136*a*b*x^2 - 255*b)*sqrt((a^2*x^2
- a)/b^2))*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2)))/(a^2*b)]

Sympy [F]

\[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {x^{3} \sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}{\sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \]

[In]

integrate(x**3*(-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))**(1/2),x)

[Out]

Integral(x**3*sqrt(a*(a*x**2 - 1)/b**2)/sqrt(x*(a*x + b*sqrt(a**2*x**2/b**2 - a/b**2))), x)

Maxima [F]

\[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} x^{3}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x}} \,d x } \]

[In]

integrate(x^3*(-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))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a^2*x^2/b^2 - a/b^2)*x^3/sqrt(a*x^2 + sqrt(a^2*x^2/b^2 - a/b^2)*b*x), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^3*(-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))^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {x^3\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}{\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}} \,d x \]

[In]

int((x^3*((a^2*x^2)/b^2 - a/b^2)^(1/2))/(a*x^2 + b*x*((a^2*x^2)/b^2 - a/b^2)^(1/2))^(1/2),x)

[Out]

int((x^3*((a^2*x^2)/b^2 - a/b^2)^(1/2))/(a*x^2 + b*x*((a^2*x^2)/b^2 - a/b^2)^(1/2))^(1/2), x)