\(\int \frac {x^4}{b x^2+c x^4} \, dx\) [180]

Optimal result

Integrand size = 17, antiderivative size = 31 \[ \int \frac {x^4}{b x^2+c x^4} \, dx=\frac {x}{c}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{3/2}} \]

[Out]

x/c-arctan(x*c^(1/2)/b^(1/2))*b^(1/2)/c^(3/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 327, 211} \[ \int \frac {x^4}{b x^2+c x^4} \, dx=\frac {x}{c}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{3/2}} \]

[In]

Int[x^4/(b*x^2 + c*x^4),x]

[Out]

x/c - (Sqrt[b]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(3/2)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{b x^2+c x^4} \, dx=\frac {x}{c}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{3/2}} \]

[In]

Integrate[x^4/(b*x^2 + c*x^4),x]

[Out]

x/c - (Sqrt[b]*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/c^(3/2)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87

method	result	size
default	\(\frac {x}{c}-\frac {b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{c \sqrt {b c}}\)	\(27\)
risch	\(\frac {x}{c}+\frac {\sqrt {-b c}\, \ln \left (-\sqrt {-b c}\, x -b \right )}{2 c^{2}}-\frac {\sqrt {-b c}\, \ln \left (\sqrt {-b c}\, x -b \right )}{2 c^{2}}\)	\(56\)

[In]

int(x^4/(c*x^4+b*x^2),x,method=_RETURNVERBOSE)

[Out]

x/c-b/c/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.65 \[ \int \frac {x^4}{b x^2+c x^4} \, dx=\left [\frac {\sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) + 2 \, x}{2 \, c}, -\frac {\sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) - x}{c}\right ] \]

[In]

integrate(x^4/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

[1/2*(sqrt(-b/c)*log((c*x^2 - 2*c*x*sqrt(-b/c) - b)/(c*x^2 + b)) + 2*x)/c, -(sqrt(b/c)*arctan(c*x*sqrt(b/c)/b)
- x)/c]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).

Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {x^4}{b x^2+c x^4} \, dx=\frac {\sqrt {- \frac {b}{c^{3}}} \log {\left (- c \sqrt {- \frac {b}{c^{3}}} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{c^{3}}} \log {\left (c \sqrt {- \frac {b}{c^{3}}} + x \right )}}{2} + \frac {x}{c} \]

[In]

integrate(x**4/(c*x**4+b*x**2),x)

[Out]

sqrt(-b/c**3)*log(-c*sqrt(-b/c**3) + x)/2 - sqrt(-b/c**3)*log(c*sqrt(-b/c**3) + x)/2 + x/c

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {x^4}{b x^2+c x^4} \, dx=-\frac {b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c} + \frac {x}{c} \]

[In]

integrate(x^4/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

-b*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c) + x/c

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {x^4}{b x^2+c x^4} \, dx=-\frac {b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c} + \frac {x}{c} \]

[In]

integrate(x^4/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

-b*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c) + x/c

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int \frac {x^4}{b x^2+c x^4} \, dx=\frac {x}{c}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{c^{3/2}} \]

[In]

int(x^4/(b*x^2 + c*x^4),x)

[Out]

x/c - (b^(1/2)*atan((c^(1/2)*x)/b^(1/2)))/c^(3/2)