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]

Log[1 + Sqrt[a + b*x^2]]/b

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Rubi [A]  time = 0.0717561, antiderivative size = 18, normalized size of antiderivative = 1., 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 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]

Rule 31

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

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*}

Mathematica [A]  time = 0.0413454, size = 18, normalized size = 1. \[ \frac{\log \left (\sqrt{a+b x^2}+1\right )}{b} \]

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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Maple [B]  time = 0.039, size = 1059, normalized size = 58.8 \begin{align*} \text{result too large to display} \end{align*}

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

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Maxima [A]  time = 1.12493, size = 22, normalized size = 1.22 \begin{align*} \frac{\log \left (\sqrt{b x^{2} + a} + 1\right )}{b} \end{align*}

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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Fricas [B]  time = 1.78275, size = 167, normalized size = 9.28 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 1.97963, size = 14, normalized size = 0.78 \begin{align*} \frac{\log{\left (\sqrt{a + b x^{2}} + 1 \right )}}{b} \end{align*}

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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Giac [A]  time = 1.15959, size = 22, normalized size = 1.22 \begin{align*} \frac{\log \left (\sqrt{b x^{2} + a} + 1\right )}{b} \end{align*}

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