3.23.51 \(\int \frac {a x+\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [2251]

3.23.51.1 Optimal result

Integrand size = 47, antiderivative size = 168 \[ \int \frac {a x+\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {b x}{2}+\frac {\left (3 b^2+a x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{3 a}+\sqrt {b^2+a^2 x^2} \left (-\frac {b}{2 a}+\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{3 a}\right )+\frac {b \log \left (a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}+\frac {\left (-b-b^3\right ) \log \left (b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{a} \]

output

-1/2*b*x+1/3*(a*x+3*b^2)*(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)/a+(a^2*x^2+b^2)^( 
1/2)*(-1/2*b/a+1/3*(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)/a)+1/2*b*ln(a*x+(a^2*x^ 
2+b^2)^(1/2))/a+(-b^3-b)*ln(b+(a*x+(a^2*x^2+b^2)^(1/2))^(1/2))/a
 

3.23.51.2 Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.88 \[ \int \frac {a x+\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {-3 a b x+2 \left (3 b^2+a x\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}+\sqrt {b^2+a^2 x^2} \left (-3 b+2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )+3 b \log \left (a x+\sqrt {b^2+a^2 x^2}\right )-6 \left (b+b^3\right ) \log \left (b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{6 a} \]

input

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

output

(-3*a*b*x + 2*(3*b^2 + a*x)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] + Sqrt[b^2 + a 
^2*x^2]*(-3*b + 2*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]) + 3*b*Log[a*x + Sqrt[b^ 
2 + a^2*x^2]] - 6*(b + b^3)*Log[b + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]])/(6*a 
)
 

3.23.51.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {a x}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}+b}+\frac {\sqrt {a^2 x^2+b^2}}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}+b}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^2}{4 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {x b}{4}-\frac {\text {arctanh}\left (\frac {a x}{\sqrt {b^2+a^2 x^2}}\right ) b}{4 a}+\frac {\left (1+\frac {1}{b^2}\right ) \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{12 a}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{12 a}+\left (1-\frac {1}{b^2}\right ) \int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-b^2+2 a x+1}dx-\frac {b^2+1}{4 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {1}{4 a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {\left (b^2+1\right ) x}{4 b}+\frac {x}{4 b}-\frac {\left (1-b^2\right )^2 \text {arctanh}\left (\frac {a x}{\sqrt {b^2+a^2 x^2}}\right )}{8 a b}+\frac {\left (2 b^2+1\right ) \text {arctanh}\left (\frac {a x}{\sqrt {b^2+a^2 x^2}}\right )}{8 a b}+\frac {\left (1-b^4\right ) \text {arctanh}\left (\frac {a x}{\sqrt {b^2+a^2 x^2}}\right )}{8 a b}-\frac {\text {arctanh}\left (\frac {a x}{\sqrt {b^2+a^2 x^2}}\right )}{8 a b}+\frac {\left (b^2+1\right )^2 \text {arctanh}\left (\frac {2 b^2-a \left (1-b^2\right ) x}{\left (b^2+1\right ) \sqrt {b^2+a^2 x^2}}\right )}{8 a b}-\frac {\left (1-b^4\right ) \text {arctanh}\left (\frac {2 b^2-a \left (1-b^2\right ) x}{\left (b^2+1\right ) \sqrt {b^2+a^2 x^2}}\right )}{8 a b}-\frac {\left (b^2+1\right )^2 \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b}\right )}{4 a b}+\frac {\left (1-b^4\right ) \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b}\right )}{4 a b}-\frac {\left (b^2+1\right )^2 \log \left (-b^2+2 a x+1\right )}{8 a b}+\frac {\left (1-b^4\right ) \log \left (-b^2+2 a x+1\right )}{8 a b}+\frac {\left (1-b^2\right ) \sqrt {b^2+a^2 x^2}}{4 a b}-\frac {\left (b^2+1\right ) \sqrt {b^2+a^2 x^2}}{4 a b}-\frac {x \sqrt {b^2+a^2 x^2}}{4 b}-\frac {(1-a x) \sqrt {b^2+a^2 x^2}}{4 a b}+\frac {\sqrt {b^2+a^2 x^2}}{4 a b}-\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{12 a b^2}-\frac {\left (b^2+1\right )^2 \arctan \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{4 a b^2}+\frac {\left (1-b^4\right ) \arctan \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{4 a b^2}-\frac {\int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-2 a x-1}dx}{b^2}+\frac {\left (b^2+1\right )^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a b^2}-\frac {\left (1-b^4\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{4 a b^2}\)

input

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

output

$Aborted
 

3.23.51.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.23.51.4 Maple [F]

\[\int \frac {a x +\sqrt {a^{2} x^{2}+b^{2}}}{b +\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]

input

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

output

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

3.23.51.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.73 \[ \int \frac {a x+\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {3 \, a b x + 6 \, {\left (b^{3} + b\right )} \log \left (b + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) - 6 \, b \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right ) - 2 \, {\left (3 \, b^{2} + a x + \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} + 3 \, \sqrt {a^{2} x^{2} + b^{2}} b}{6 \, a} \]

input

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

output

-1/6*(3*a*b*x + 6*(b^3 + b)*log(b + sqrt(a*x + sqrt(a^2*x^2 + b^2))) - 6*b 
*log(sqrt(a*x + sqrt(a^2*x^2 + b^2))) - 2*(3*b^2 + a*x + sqrt(a^2*x^2 + b^ 
2))*sqrt(a*x + sqrt(a^2*x^2 + b^2)) + 3*sqrt(a^2*x^2 + b^2)*b)/a
 

3.23.51.6 Sympy [F]

\[ \int \frac {a x+\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {a x + \sqrt {a^{2} x^{2} + b^{2}}}{b + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]

input

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

output

Integral((a*x + sqrt(a**2*x**2 + b**2))/(b + sqrt(a*x + sqrt(a**2*x**2 + b 
**2))), x)
 

3.23.51.7 Maxima [F]

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

input

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

output

1/4*a*x^2/b + 1/2*integrate(sqrt(a^2*x^2 + b^2), x)/b - integrate(-1/2*(a* 
b^2*x - 2*a^2*x^2 - b^2 + sqrt(a^2*x^2 + b^2)*(b^2 - 2*a*x))/(b^3 + a*b*x 
+ 2*sqrt(a*x + sqrt(a^2*x^2 + b^2))*b^2 + sqrt(a^2*x^2 + b^2)*b), x)
 

3.23.51.8 Giac [F]

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

input

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

output

integrate((a*x + sqrt(a^2*x^2 + b^2))/(b + sqrt(a*x + sqrt(a^2*x^2 + b^2)) 
), x)
 

3.23.51.9 Mupad [F(-1)]

Timed out. \[ \int \frac {a x+\sqrt {b^2+a^2 x^2}}{b+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {a\,x+\sqrt {a^2\,x^2+b^2}}{b+\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \]

input

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

output

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