3.3.77 \(\int \frac {x}{(b x^2+c x^4)^{3/2}} \, dx\) [277]

Optimal. Leaf size=28 \[ -\frac {b+2 c x^2}{b^2 \sqrt {b x^2+c x^4}} \]

[Out]

(-2*c*x^2-b)/b^2/(c*x^4+b*x^2)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2038, 627} \begin {gather*} -\frac {b+2 c x^2}{b^2 \sqrt {b x^2+c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(b*x^2 + c*x^4)^(3/2),x]

[Out]

-((b + 2*c*x^2)/(b^2*Sqrt[b*x^2 + c*x^4]))

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2038

Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[(a*x^Simplify[j/n]
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && IntegerQ[Simplify[j
/n]] && EqQ[Simplify[m - n + 1], 0]

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 29, normalized size = 1.04 \begin {gather*} \frac {-b-2 c x^2}{b^2 \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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Maple [A]
time = 0.10, size = 37, normalized size = 1.32

method	result	size
gosper	\(-\frac {x^{2} \left (c \,x^{2}+b \right ) \left (2 c \,x^{2}+b \right )}{b^{2} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}\)	\(37\)
default	\(-\frac {x^{2} \left (c \,x^{2}+b \right ) \left (2 c \,x^{2}+b \right )}{b^{2} \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}\)	\(37\)
trager	\(-\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}}}{\left (c \,x^{2}+b \right ) b^{2} x^{2}}\)	\(39\)
risch	\(-\frac {c \,x^{2}+b}{b^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}-\frac {c \,x^{2}}{b^{2} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(c*x^4+b*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-x^2*(c*x^2+b)*(2*c*x^2+b)/b^2/(c*x^4+b*x^2)^(3/2)

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Maxima [A]
time = 0.29, size = 41, normalized size = 1.46 \begin {gather*} -\frac {2 \, c x^{2}}{\sqrt {c x^{4} + b x^{2}} b^{2}} - \frac {1}{\sqrt {c x^{4} + b x^{2}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(c*x^4+b*x^2)^(3/2),x, algorithm="maxima")

[Out]

-2*c*x^2/(sqrt(c*x^4 + b*x^2)*b^2) - 1/(sqrt(c*x^4 + b*x^2)*b)

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Fricas [A]
time = 0.42, size = 41, normalized size = 1.46 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}} {\left (2 \, c x^{2} + b\right )}}{b^{2} c x^{4} + b^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")

[Out]

-sqrt(c*x^4 + b*x^2)*(2*c*x^2 + b)/(b^2*c*x^4 + b^3*x^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(c*x**4+b*x**2)**(3/2),x)

[Out]

Integral(x/(x**2*(b + c*x**2))**(3/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
time = 6.11, size = 58, normalized size = 2.07 \begin {gather*} -\frac {c x}{\sqrt {c x^{2} + b} b^{2} \mathrm {sgn}\left (x\right )} + \frac {2 \, \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )} b \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")

[Out]

-c*x/(sqrt(c*x^2 + b)*b^2*sgn(x)) + 2*sqrt(c)/(((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)*b*sgn(x))

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Mupad [B]
time = 4.13, size = 26, normalized size = 0.93 \begin {gather*} -\frac {2\,c\,x^2+b}{b^2\,\sqrt {c\,x^4+b\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^2 + c*x^4)^(3/2),x)

[Out]

-(b + 2*c*x^2)/(b^2*(b*x^2 + c*x^4)^(1/2))

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