3.7.83 \(\int \frac {x}{a+b x^2+\sqrt {a+b x^2}} \, dx\) [683]

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

[Out]

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

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Rubi [A]
time = 0.05, 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 = {2186, 31} \begin {gather*} \frac {\log \left (\sqrt {a+b x^2}+1\right )}{b} \end {gather*}

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 2186

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} \text {Subst}\left (\int \frac {1}{a+b x+\sqrt {a+b x}} \, dx,x,x^2\right )\\ &=\frac {\text {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 = 0.02, size = 20, normalized size = 1.11 \begin {gather*} \frac {\log \left (b+b \sqrt {a+b x^2}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[b + b*Sqrt[a + b*x^2]]/b

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1058\) vs. \(2(16)=32\).
time = 0.05, size = 1059, normalized size = 58.83

method	result	size
default	\(\frac {\sqrt {b \left (x +\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )^{2}-2 \sqrt {-\left (a -1\right ) b}\, \left (x +\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )+1}}{2 \left (\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \left (-\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right )}-\frac {\sqrt {-\left (a -1\right ) b}\, \ln \left (\frac {\left (x +\frac {\sqrt {-\left (a -1\right ) b}}{b}\right ) b -\sqrt {-\left (a -1\right ) b}}{\sqrt {b}}+\sqrt {b \left (x +\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )^{2}-2 \sqrt {-\left (a -1\right ) b}\, \left (x +\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )+1}\right )}{2 \left (\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \left (-\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \sqrt {b}}-\frac {\arctanh \left (\frac {2-2 \sqrt {-\left (a -1\right ) b}\, \left (x +\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )}{2 \sqrt {b \left (x +\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )^{2}-2 \sqrt {-\left (a -1\right ) b}\, \left (x +\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )+1}}\right )}{2 \left (\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \left (-\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right )}+\frac {\sqrt {b \left (x -\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )^{2}+2 \sqrt {-\left (a -1\right ) b}\, \left (x -\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )+1}}{2 \left (\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \left (-\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right )}+\frac {\sqrt {-\left (a -1\right ) b}\, \ln \left (\frac {\left (x -\frac {\sqrt {-\left (a -1\right ) b}}{b}\right ) b +\sqrt {-\left (a -1\right ) b}}{\sqrt {b}}+\sqrt {b \left (x -\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )^{2}+2 \sqrt {-\left (a -1\right ) b}\, \left (x -\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )+1}\right )}{2 \left (\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \left (-\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \sqrt {b}}-\frac {\arctanh \left (\frac {2+2 \sqrt {-\left (a -1\right ) b}\, \left (x -\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )}{2 \sqrt {b \left (x -\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )^{2}+2 \sqrt {-\left (a -1\right ) b}\, \left (x -\frac {\sqrt {-\left (a -1\right ) b}}{b}\right )+1}}\right )}{2 \left (\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \left (-\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right )}-\frac {\sqrt {b \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{2 \left (\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \left (-\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right )}-\frac {\sqrt {-a b}\, \ln \left (\frac {\left (x -\frac {\sqrt {-a b}}{b}\right ) b +\sqrt {-a b}}{\sqrt {b}}+\sqrt {b \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}\right )}{2 \left (\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \left (-\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \sqrt {b}}-\frac {\sqrt {b \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{2 \left (\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \left (-\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right )}+\frac {\sqrt {-a b}\, \ln \left (\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b -\sqrt {-a b}}{\sqrt {b}}+\sqrt {b \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}\right )}{2 \left (\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \left (-\sqrt {-\left (a -1\right ) b}+\sqrt {-a b}\right ) \sqrt {b}}+\frac {\ln \left (b \,x^{2}+a -1\right )}{2 b}\)	\(1059\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 16, normalized size = 0.89 \begin {gather*} \frac {\log \left (\sqrt {b x^{2} + a} + 1\right )}{b} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (16) = 32\).
time = 0.37, size = 67, normalized size = 3.72 \begin {gather*} \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 {gather*}

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 = 1.66, size = 14, normalized size = 0.78 \begin {gather*} \frac {\log {\left (\sqrt {a + b x^{2}} + 1 \right )}}{b} \end {gather*}

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 = 2.16, size = 16, normalized size = 0.89 \begin {gather*} \frac {\log \left (\sqrt {b x^{2} + a} + 1\right )}{b} \end {gather*}

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

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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