\(\int \frac {x}{\sqrt {d x^2} (a+b x^2)} \, dx\) [673]

Optimal result

Integrand size = 20, antiderivative size = 34 \[ \int \frac {x}{\sqrt {d x^2} \left (a+b x^2\right )} \, dx=\frac {x \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {d x^2}} \]

[Out]

x*arctan(x*b^(1/2)/a^(1/2))/a^(1/2)/b^(1/2)/(d*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 211} \[ \int \frac {x}{\sqrt {d x^2} \left (a+b x^2\right )} \, dx=\frac {x \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {d x^2}} \]

[In]

Int[x/(Sqrt[d*x^2]*(a + b*x^2)),x]

[Out]

(x*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Sqrt[d*x^2])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

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]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {x}{\sqrt {d x^2} \left (a+b x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \sqrt {d x^2}}{\sqrt {a} \sqrt {d}}\right )}{\sqrt {a} \sqrt {b} \sqrt {d}} \]

[In]

Integrate[x/(Sqrt[d*x^2]*(a + b*x^2)),x]

[Out]

ArcTan[(Sqrt[b]*Sqrt[d*x^2])/(Sqrt[a]*Sqrt[d])]/(Sqrt[a]*Sqrt[b]*Sqrt[d])

Maple [A] (verified)

Time = 2.89 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71

method	result	size
default	\(\frac {x \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {d \,x^{2}}\, \sqrt {a b}}\)	\(24\)
pseudoelliptic	\(\frac {\arctan \left (\frac {b \sqrt {d \,x^{2}}}{\sqrt {a b d}}\right )}{\sqrt {a b d}}\)	\(24\)
risch	\(-\frac {x \ln \left (b x +\sqrt {-a b}\right )}{2 \sqrt {d \,x^{2}}\, \sqrt {-a b}}+\frac {x \ln \left (-b x +\sqrt {-a b}\right )}{2 \sqrt {d \,x^{2}}\, \sqrt {-a b}}\)	\(57\)

[In]

int(x/(b*x^2+a)/(d*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/(d*x^2)^(1/2)*x/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.76 \[ \int \frac {x}{\sqrt {d x^2} \left (a+b x^2\right )} \, dx=\left [-\frac {\sqrt {-a b d} \log \left (\frac {b d x^{2} - a d - 2 \, \sqrt {-a b d} \sqrt {d x^{2}}}{b x^{2} + a}\right )}{2 \, a b d}, \frac {\sqrt {a b d} \arctan \left (\frac {\sqrt {a b d} \sqrt {d x^{2}}}{a d}\right )}{a b d}\right ] \]

[In]

integrate(x/(b*x^2+a)/(d*x^2)^(1/2),x, algorithm="fricas")

[Out]

[-1/2*sqrt(-a*b*d)*log((b*d*x^2 - a*d - 2*sqrt(-a*b*d)*sqrt(d*x^2))/(b*x^2 + a))/(a*b*d), sqrt(a*b*d)*arctan(s
qrt(a*b*d)*sqrt(d*x^2)/(a*d))/(a*b*d)]

Sympy [A] (verification not implemented)

Time = 1.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {x}{\sqrt {d x^2} \left (a+b x^2\right )} \, dx=\begin {cases} \frac {\operatorname {atan}{\left (\frac {\sqrt {d x^{2}}}{\sqrt {\frac {a d}{b}}} \right )}}{b \sqrt {\frac {a d}{b}}} & \text {for}\: d \neq 0 \\\tilde {\infty } x^{2} & \text {otherwise} \end {cases} \]

[In]

integrate(x/(b*x**2+a)/(d*x**2)**(1/2),x)

[Out]

Piecewise((atan(sqrt(d*x**2)/sqrt(a*d/b))/(b*sqrt(a*d/b)), Ne(d, 0)), (zoo*x**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int \frac {x}{\sqrt {d x^2} \left (a+b x^2\right )} \, dx=\frac {\arctan \left (\frac {\sqrt {d x^{2}} b}{\sqrt {a b d}}\right )}{\sqrt {a b d}} \]

[In]

integrate(x/(b*x^2+a)/(d*x^2)^(1/2),x, algorithm="maxima")

[Out]

arctan(sqrt(d*x^2)*b/sqrt(a*b*d))/sqrt(a*b*d)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.65 \[ \int \frac {x}{\sqrt {d x^2} \left (a+b x^2\right )} \, dx=\frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} \sqrt {d} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(x/(b*x^2+a)/(d*x^2)^(1/2),x, algorithm="giac")

[Out]

arctan(b*x/sqrt(a*b))/(sqrt(a*b)*sqrt(d)*sgn(x))

Mupad [B] (verification not implemented)

Time = 5.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.68 \[ \int \frac {x}{\sqrt {d x^2} \left (a+b x^2\right )} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x^2}}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}\,\sqrt {d}} \]

[In]

int(x/((a + b*x^2)*(d*x^2)^(1/2)),x)

[Out]

atan((b^(1/2)*(x^2)^(1/2))/a^(1/2))/(a^(1/2)*b^(1/2)*d^(1/2))