\(\int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx\) [873]

Optimal result

Integrand size = 21, antiderivative size = 66 \[ \int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx=-\frac {2}{15} \sqrt {-1+x^2} \sqrt {x^2 \left (x+\sqrt {-1+x^2}\right )}+\frac {4 \left (-1+2 x^2\right ) \sqrt {x^2 \left (x+\sqrt {-1+x^2}\right )}}{15 x} \]

[Out]

-2/15*(x^2-1)^(1/2)*(x^2*(x+(x^2-1)^(1/2)))^(1/2)+4/15*(2*x^2-1)*(x^2*(x+(x^2-1)^(1/2)))^(1/2)/x

Rubi [F]

\[ \int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx=\int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx \]

[In]

Int[Sqrt[x^3 + x^2*Sqrt[-1 + x^2]],x]

[Out]

Defer[Int][Sqrt[x^3 + x^2*Sqrt[-1 + x^2]], x]

Rubi steps \begin{align*} \text {integral}& = \int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx=\frac {2 x^3 \left (2-6 x^2+6 x^4-3 x \sqrt {-1+x^2}+6 x^3 \sqrt {-1+x^2}\right )}{15 \left (x^2 \left (x+\sqrt {-1+x^2}\right )\right )^{3/2}} \]

[In]

Integrate[Sqrt[x^3 + x^2*Sqrt[-1 + x^2]],x]

[Out]

(2*x^3*(2 - 6*x^2 + 6*x^4 - 3*x*Sqrt[-1 + x^2] + 6*x^3*Sqrt[-1 + x^2]))/(15*(x^2*(x + Sqrt[-1 + x^2]))^(3/2))

Maple [F]

\[\int \sqrt {x^{3}+x^{2} \sqrt {x^{2}-1}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.59 \[ \int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx=\frac {2 \, \sqrt {x^{3} + \sqrt {x^{2} - 1} x^{2}} {\left (4 \, x^{2} - \sqrt {x^{2} - 1} x - 2\right )}}{15 \, x} \]

[In]

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

[Out]

2/15*sqrt(x^3 + sqrt(x^2 - 1)*x^2)*(4*x^2 - sqrt(x^2 - 1)*x - 2)/x

Sympy [F]

\[ \int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx=\int \sqrt {x^{3} + x^{2} \sqrt {x^{2} - 1}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx=\int { \sqrt {x^{3} + \sqrt {x^{2} - 1} x^{2}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(x^3 + sqrt(x^2 - 1)*x^2), x)

Giac [F]

\[ \int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx=\int { \sqrt {x^{3} + \sqrt {x^{2} - 1} x^{2}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(x^3 + sqrt(x^2 - 1)*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {x^3+x^2 \sqrt {-1+x^2}} \, dx=\int \sqrt {x^2\,\sqrt {x^2-1}+x^3} \,d x \]

[In]

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

[Out]

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