\(\int \frac {x^3 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx\) [2084]

Optimal result

Integrand size = 32, antiderivative size = 150 \[ \int \frac {x^3 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {\sqrt {x+x^2} \left (495495-264264 x+201344 x^2-168960 x^3-1146880 x^4\right ) \sqrt {x \left (x+\sqrt {x+x^2}\right )}}{5160960 x}+\sqrt {x \left (x+\sqrt {x+x^2}\right )} \left (\frac {-165165+37752 x-18304 x^2+1387520 x^3+1146880 x^4}{5160960}-\frac {1573 \sqrt {-x+\sqrt {x+x^2}} \text {arctanh}\left (\sqrt {2} \sqrt {-x+\sqrt {x+x^2}}\right )}{16384 \sqrt {2} x}\right ) \]

[Out]

1/5160960*(x^2+x)^(1/2)*(-1146880*x^4-168960*x^3+201344*x^2-264264*x+495495)*(x*(x+(x^2+x)^(1/2)))^(1/2)/x+(x*
(x+(x^2+x)^(1/2)))^(1/2)*(2/9*x^4+271/1008*x^3-143/40320*x^2+1573/215040*x-1573/49152-1573/32768*2^(1/2)*(-x+(
x^2+x)^(1/2))^(1/2)*arctanh(2^(1/2)*(-x+(x^2+x)^(1/2))^(1/2))/x)

Rubi [F]

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

[In]

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

[Out]

(2*Sqrt[x + x^2]*Defer[Subst][Defer[Int][(x^8*Sqrt[1 + x^2])/Sqrt[x^4 + x^2*Sqrt[x^2 + x^4]], x], x, Sqrt[x]])
/(Sqrt[x]*Sqrt[1 + x])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x+x^2} \int \frac {x^{7/2} \sqrt {1+x}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx}{\sqrt {x} \sqrt {1+x}} \\ & = \frac {\left (2 \sqrt {x+x^2}\right ) \text {Subst}\left (\int \frac {x^8 \sqrt {1+x^2}}{\sqrt {x^4+x^2 \sqrt {x^2+x^4}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.34 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.19 \[ \int \frac {x^3 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {(1+x) \sqrt {x \left (x+\sqrt {x (1+x)}\right )} \left (2 x \left (-1146880 x^5-165165 \left (-3+\sqrt {x (1+x)}\right )+4719 x \left (49+8 \sqrt {x (1+x)}\right )-1144 x^2 \left (55+16 \sqrt {x (1+x)}\right )+5120 x^4 \left (-257+224 \sqrt {x (1+x)}\right )+128 x^3 \left (253+10840 \sqrt {x (1+x)}\right )\right )-495495 \sqrt {2} \sqrt {x (1+x)} \sqrt {-x+\sqrt {x (1+x)}} \text {arctanh}\left (\sqrt {-2 x+2 \sqrt {x (1+x)}}\right )\right )}{10321920 (x (1+x))^{3/2}} \]

[In]

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

[Out]

((1 + x)*Sqrt[x*(x + Sqrt[x*(1 + x)])]*(2*x*(-1146880*x^5 - 165165*(-3 + Sqrt[x*(1 + x)]) + 4719*x*(49 + 8*Sqr
t[x*(1 + x)]) - 1144*x^2*(55 + 16*Sqrt[x*(1 + x)]) + 5120*x^4*(-257 + 224*Sqrt[x*(1 + x)]) + 128*x^3*(253 + 10
840*Sqrt[x*(1 + x)])) - 495495*Sqrt[2]*Sqrt[x*(1 + x)]*Sqrt[-x + Sqrt[x*(1 + x)]]*ArcTanh[Sqrt[-2*x + 2*Sqrt[x
*(1 + x)]]]))/(10321920*(x*(1 + x))^(3/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 \sqrt {x+x^2}}{\sqrt {x^2+x \sqrt {x+x^2}}} \, dx=\frac {495495 \, \sqrt {2} x \log \left (\frac {4 \, x^{2} - 2 \, \sqrt {x^{2} + \sqrt {x^{2} + x} x} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + x}\right )} + 4 \, \sqrt {x^{2} + x} x + x}{x}\right ) + 4 \, {\left (1146880 \, x^{5} + 1387520 \, x^{4} - 18304 \, x^{3} + 37752 \, x^{2} - {\left (1146880 \, x^{4} + 168960 \, x^{3} - 201344 \, x^{2} + 264264 \, x - 495495\right )} \sqrt {x^{2} + x} - 165165 \, x\right )} \sqrt {x^{2} + \sqrt {x^{2} + x} x}}{20643840 \, x} \]

[In]

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

[Out]

1/20643840*(495495*sqrt(2)*x*log((4*x^2 - 2*sqrt(x^2 + sqrt(x^2 + x)*x)*(sqrt(2)*x + sqrt(2)*sqrt(x^2 + x)) +
4*sqrt(x^2 + x)*x + x)/x) + 4*(1146880*x^5 + 1387520*x^4 - 18304*x^3 + 37752*x^2 - (1146880*x^4 + 168960*x^3 -
201344*x^2 + 264264*x - 495495)*sqrt(x^2 + x) - 165165*x)*sqrt(x^2 + sqrt(x^2 + x)*x))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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