\(\int \frac {\sqrt {x+x^4}}{x^3} \, dx\) [434]

Optimal result

Integrand size = 13, antiderivative size = 35 \[ \int \frac {\sqrt {x+x^4}}{x^3} \, dx=-\frac {2 \sqrt {x+x^4}}{3 x^2}+\frac {2}{3} \text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2045, 2054, 212} \[ \int \frac {\sqrt {x+x^4}}{x^3} \, dx=\frac {2}{3} \text {arctanh}\left (\frac {x^2}{\sqrt {x^4+x}}\right )-\frac {2 \sqrt {x^4+x}}{3 x^2} \]

[In]

Int[Sqrt[x + x^4]/x^3,x]

[Out]

(-2*Sqrt[x + x^4])/(3*x^2) + (2*ArcTanh[x^2/Sqrt[x + x^4]])/3

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2045

Int[((c_.)*(x_))^(m_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b*
x^n)^p/(c*(m + j*p + 1))), x] - Dist[b*p*((n - j)/(c^n*(m + j*p + 1))), Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p -
1), x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && GtQ[p
, 0] && LtQ[m + j*p + 1, 0]

Rule 2054

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {x+x^4}}{3 x^2}+\int \frac {x}{\sqrt {x+x^4}} \, dx \\ & = -\frac {2 \sqrt {x+x^4}}{3 x^2}+\frac {2}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right ) \\ & = -\frac {2 \sqrt {x+x^4}}{3 x^2}+\frac {2}{3} \text {arctanh}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {x+x^4}}{x^3} \, dx=-\frac {2 \sqrt {x+x^4}}{3 x^2}+\frac {2 \sqrt {x+x^4} \log \left (x^{3/2}+\sqrt {1+x^3}\right )}{3 \sqrt {x} \sqrt {1+x^3}} \]

[In]

Integrate[Sqrt[x + x^4]/x^3,x]

[Out]

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

Maple [A] (verified)

Time = 3.81 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89

method	result	size
meijerg	\(-\frac {\frac {4 \sqrt {\pi }\, \sqrt {x^{3}+1}}{x^{\frac {3}{2}}}-4 \sqrt {\pi }\, \operatorname {arcsinh}\left (x^{\frac {3}{2}}\right )}{6 \sqrt {\pi }}\)	\(31\)
default	\(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}-\frac {\ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{3}\)	\(34\)
trager	\(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}-\frac {\ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{3}\)	\(34\)
risch	\(-\frac {2 \left (x^{3}+1\right )}{3 x \sqrt {x \left (x^{3}+1\right )}}-\frac {\ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{3}\)	\(41\)
pseudoelliptic	\(\frac {-2 \sqrt {x^{4}+x}-x^{2} \left (\ln \left (\frac {-x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )-\ln \left (\frac {x^{2}+\sqrt {x^{4}+x}}{x^{2}}\right )\right )}{3 x^{2}}\)	\(58\)
elliptic	\(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}-\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\operatorname {EllipticF}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\operatorname {EllipticPi}\left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\)	\(303\)

[In]

int((x^4+x)^(1/2)/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/6/Pi^(1/2)*(4*Pi^(1/2)/x^(3/2)*(x^3+1)^(1/2)-4*Pi^(1/2)*arcsinh(x^(3/2)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {x+x^4}}{x^3} \, dx=\frac {x^{2} \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x - 1\right ) - 2 \, \sqrt {x^{4} + x}}{3 \, x^{2}} \]

[In]

integrate((x^4+x)^(1/2)/x^3,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {x+x^4}}{x^3} \, dx=\int \frac {\sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}{x^{3}}\, dx \]

[In]

integrate((x**4+x)**(1/2)/x**3,x)

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {x+x^4}}{x^3} \, dx=\int { \frac {\sqrt {x^{4} + x}}{x^{3}} \,d x } \]

[In]

integrate((x^4+x)^(1/2)/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 + x)/x^3, x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x+x^4}}{x^3} \, dx=-\frac {2}{3} \, \sqrt {\frac {1}{x^{3}} + 1} + \frac {1}{3} \, \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{3} \, \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \]

[In]

integrate((x^4+x)^(1/2)/x^3,x, algorithm="giac")

[Out]

-2/3*sqrt(1/x^3 + 1) + 1/3*log(sqrt(1/x^3 + 1) + 1) - 1/3*log(abs(sqrt(1/x^3 + 1) - 1))

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {x+x^4}}{x^3} \, dx=\int \frac {\sqrt {x^4+x}}{x^3} \,d x \]

[In]

int((x + x^4)^(1/2)/x^3,x)

[Out]

int((x + x^4)^(1/2)/x^3, x)