3.683 \(\int \frac {x}{a+b x^2+\sqrt {a+b x^2}} \, dx\)

Optimal. Leaf size=18 \[ \frac {\log \left (\sqrt {a+b x^2}+1\right )}{b} \]

[Out]

ln(1+(b*x^2+a)^(1/2))/b

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Rubi [A]  time = 0.07, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2155, 31} \[ \frac {\log \left (\sqrt {a+b x^2}+1\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 + Sqrt[a + b*x^2]]/b

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2155

Int[(x_)^(m_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/n, Subst[Int
[x^((m + 1)/n - 1)/(c + d*x + e*Sqrt[a + b*x]), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[b*c
- a*d, 0] && IntegerQ[(m + 1)/n]

Rubi steps

\begin {align*} \int \frac {x}{a+b x^2+\sqrt {a+b x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a+b x+\sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt {a+b x^2}\right )}{b}\\ &=\frac {\log \left (1+\sqrt {a+b x^2}\right )}{b}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 18, normalized size = 1.00 \[ \frac {\log \left (\sqrt {a+b x^2}+1\right )}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 + Sqrt[a + b*x^2]]/b

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fricas [B]  time = 0.47, size = 67, normalized size = 3.72 \[ \frac {2 \, \log \left (b x^{2} + a - 1\right ) + \log \left (\frac {b x^{2} + a + 2 \, \sqrt {b x^{2} + a} + 1}{x^{2}}\right ) - \log \left (\frac {b x^{2} + a - 2 \, \sqrt {b x^{2} + a} + 1}{x^{2}}\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.34, size = 16, normalized size = 0.89 \[ \frac {\log \left (\sqrt {b x^{2} + a} + 1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt(b*x^2 + a) + 1)/b

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maple [B]  time = 0.06, size = 1059, normalized size = 58.83 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b*x^2+(b*x^2+a)^(1/2)),x)

[Out]

-1/2/((-(a-1)*b)^(1/2)+(-a*b)^(1/2))/(-(-(a-1)*b)^(1/2)+(-a*b)^(1/2))*((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(
x+(-a*b)^(1/2)/b))^(1/2)+1/2/((-(a-1)*b)^(1/2)+(-a*b)^(1/2))/(-(-(a-1)*b)^(1/2)+(-a*b)^(1/2))*(-a*b)^(1/2)*ln(
((x+(-a*b)^(1/2)/b)*b-(-a*b)^(1/2))/b^(1/2)+((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b))^(1/2))/
b^(1/2)-1/2/((-(a-1)*b)^(1/2)+(-a*b)^(1/2))/(-(-(a-1)*b)^(1/2)+(-a*b)^(1/2))*((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^
(1/2)*(x-(-a*b)^(1/2)/b))^(1/2)-1/2/((-(a-1)*b)^(1/2)+(-a*b)^(1/2))/(-(-(a-1)*b)^(1/2)+(-a*b)^(1/2))*(-a*b)^(1
/2)*ln(((x-(-a*b)^(1/2)/b)*b+(-a*b)^(1/2))/b^(1/2)+((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a*b)^(1/2)/b))^
(1/2))/b^(1/2)+1/2/((-(a-1)*b)^(1/2)+(-a*b)^(1/2))/(-(-(a-1)*b)^(1/2)+(-a*b)^(1/2))*(b*(x+(-(a-1)*b)^(1/2)/b)^
2-2*(-(a-1)*b)^(1/2)*(x+(-(a-1)*b)^(1/2)/b)+1)^(1/2)-1/2/((-(a-1)*b)^(1/2)+(-a*b)^(1/2))/(-(-(a-1)*b)^(1/2)+(-
a*b)^(1/2))*(-(a-1)*b)^(1/2)*ln((b*(x+(-(a-1)*b)^(1/2)/b)-(-(a-1)*b)^(1/2))/b^(1/2)+(b*(x+(-(a-1)*b)^(1/2)/b)^
2-2*(-(a-1)*b)^(1/2)*(x+(-(a-1)*b)^(1/2)/b)+1)^(1/2))/b^(1/2)-1/2/((-(a-1)*b)^(1/2)+(-a*b)^(1/2))/(-(-(a-1)*b)
^(1/2)+(-a*b)^(1/2))*arctanh(1/2*(2-2*(-(a-1)*b)^(1/2)*(x+(-(a-1)*b)^(1/2)/b))/(b*(x+(-(a-1)*b)^(1/2)/b)^2-2*(
-(a-1)*b)^(1/2)*(x+(-(a-1)*b)^(1/2)/b)+1)^(1/2))+1/2/((-(a-1)*b)^(1/2)+(-a*b)^(1/2))/(-(-(a-1)*b)^(1/2)+(-a*b)
^(1/2))*(b*(x-(-(a-1)*b)^(1/2)/b)^2+2*(-(a-1)*b)^(1/2)*(x-(-(a-1)*b)^(1/2)/b)+1)^(1/2)+1/2/((-(a-1)*b)^(1/2)+(
-a*b)^(1/2))/(-(-(a-1)*b)^(1/2)+(-a*b)^(1/2))*(-(a-1)*b)^(1/2)*ln((b*(x-(-(a-1)*b)^(1/2)/b)+(-(a-1)*b)^(1/2))/
b^(1/2)+(b*(x-(-(a-1)*b)^(1/2)/b)^2+2*(-(a-1)*b)^(1/2)*(x-(-(a-1)*b)^(1/2)/b)+1)^(1/2))/b^(1/2)-1/2/((-(a-1)*b
)^(1/2)+(-a*b)^(1/2))/(-(-(a-1)*b)^(1/2)+(-a*b)^(1/2))*arctanh(1/2*(2+2*(-(a-1)*b)^(1/2)*(x-(-(a-1)*b)^(1/2)/b
))/(b*(x-(-(a-1)*b)^(1/2)/b)^2+2*(-(a-1)*b)^(1/2)*(x-(-(a-1)*b)^(1/2)/b)+1)^(1/2))+1/2/b*ln(b*x^2+a-1)

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maxima [A]  time = 0.57, size = 16, normalized size = 0.89 \[ \frac {\log \left (\sqrt {b x^{2} + a} + 1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt(b*x^2 + a) + 1)/b

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mupad [B]  time = 3.38, size = 26, normalized size = 1.44 \[ \frac {\mathrm {atanh}\left (\sqrt {b\,x^2+a}\right )+\frac {\ln \left (b\,x^2+a-1\right )}{2}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^2 + (a + b*x^2)^(1/2)),x)

[Out]

(atanh((a + b*x^2)^(1/2)) + log(a + b*x^2 - 1)/2)/b

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sympy [A]  time = 3.39, size = 14, normalized size = 0.78 \[ \frac {\log {\left (\sqrt {a + b x^{2}} + 1 \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt(a + b*x**2) + 1)/b

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