3.31.61 \(\int \frac {b^2+a x}{(-b^2+a x) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\) [3061]

3.31.61.1 Optimal result

Integrand size = 38, antiderivative size = 471 \[ \int \frac {b^2+a x}{\left (-b^2+a x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {\sqrt {2} \left (-b+\sqrt {a} x\right ) \sqrt {-\sqrt {a} x+\sqrt {b^2+a x^2}}}{\sqrt {a} b}+\frac {\sqrt {2} \sqrt {b^2+a x^2} \sqrt {-\sqrt {a} x+\sqrt {b^2+a x^2}}}{\sqrt {a} b}-\frac {2 \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {-\sqrt {a} x+\sqrt {b^2+a x^2}}}{\sqrt {b}}\right )}{\sqrt {a}}+\frac {2 \left (\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {b+\sqrt {a+b^2}}-\sqrt {2} b^{3/2} \sqrt {b+\sqrt {a+b^2}}+\sqrt {2} \sqrt {b} \sqrt {a+b^2} \sqrt {b+\sqrt {a+b^2}}\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt {-\sqrt {a} x+\sqrt {b^2+a x^2}}}{\sqrt {b} \sqrt {b+\sqrt {a+b^2}}}\right )}{a^{5/4}}-\frac {2 \left (-\sqrt {2} \sqrt {a} \sqrt {b} \sqrt {-b+\sqrt {a+b^2}}+\sqrt {2} b^{3/2} \sqrt {-b+\sqrt {a+b^2}}+\sqrt {2} \sqrt {b} \sqrt {a+b^2} \sqrt {-b+\sqrt {a+b^2}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {-\sqrt {a} x+\sqrt {b^2+a x^2}}}{\sqrt {b} \sqrt {-b+\sqrt {a+b^2}}}\right )}{a^{5/4}} \]

output

2^(1/2)*(-b+x*a^(1/2))*(-x*a^(1/2)+(a*x^2+b^2)^(1/2))^(1/2)/a^(1/2)/b+2^(1 
/2)*(a*x^2+b^2)^(1/2)*(-x*a^(1/2)+(a*x^2+b^2)^(1/2))^(1/2)/a^(1/2)/b-2*2^( 
1/2)*b^(1/2)*arctan((-x*a^(1/2)+(a*x^2+b^2)^(1/2))^(1/2)/b^(1/2))/a^(1/2)+ 
2*(2^(1/2)*a^(1/2)*b^(1/2)*(b+(b^2+a)^(1/2))^(1/2)-2^(1/2)*b^(3/2)*(b+(b^2 
+a)^(1/2))^(1/2)+2^(1/2)*b^(1/2)*(b^2+a)^(1/2)*(b+(b^2+a)^(1/2))^(1/2))*ar 
ctan(a^(1/4)*(-x*a^(1/2)+(a*x^2+b^2)^(1/2))^(1/2)/b^(1/2)/(b+(b^2+a)^(1/2) 
)^(1/2))/a^(5/4)-2*(-2^(1/2)*a^(1/2)*b^(1/2)*(-b+(b^2+a)^(1/2))^(1/2)+2^(1 
/2)*b^(3/2)*(-b+(b^2+a)^(1/2))^(1/2)+2^(1/2)*b^(1/2)*(b^2+a)^(1/2)*(-b+(b^ 
2+a)^(1/2))^(1/2))*arctanh(a^(1/4)*(-x*a^(1/2)+(a*x^2+b^2)^(1/2))^(1/2)/b^ 
(1/2)/(-b+(b^2+a)^(1/2))^(1/2))/a^(5/4)
 

3.31.61.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.15 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.40 \[ \int \frac {b^2+a x}{\left (-b^2+a x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}}-\frac {i \sqrt {b} \log \left (b+\sqrt {b^2+a x^2}\right )}{\sqrt {2} \sqrt {a}}+\frac {i \sqrt {2} \sqrt {b} \log \left (\sqrt {2} \sqrt {a} \sqrt {b}-\frac {i a x}{\sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}-i \sqrt {2} a^{3/2} \sqrt {b} \text {RootSum}\left [a^2 b-4 i a^2 \text {$\#$1}+2 a b \text {$\#$1}^2+4 i a \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {-\log \left (b+\sqrt {b^2+a x^2}\right )+2 \log \left (\sqrt {2} \sqrt {a} \sqrt {b}-\frac {i a x}{\sqrt {b+\sqrt {b^2+a x^2}}}+\sqrt {b+\sqrt {b^2+a x^2}} \text {$\#$1}\right )}{a^2+i a b \text {$\#$1}-3 a \text {$\#$1}^2+i b \text {$\#$1}^3}\&\right ]+2 i \sqrt {2} \sqrt {a} b^{3/2} \text {RootSum}\left [a^2 b-4 i a^2 \text {$\#$1}+2 a b \text {$\#$1}^2+4 i a \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {\log \left (b+\sqrt {b^2+a x^2}\right ) \text {$\#$1}-2 \log \left (\sqrt {2} \sqrt {a} \sqrt {b}-\frac {i a x}{\sqrt {b+\sqrt {b^2+a x^2}}}+\sqrt {b+\sqrt {b^2+a x^2}} \text {$\#$1}\right ) \text {$\#$1}}{-i a^2+a b \text {$\#$1}+3 i a \text {$\#$1}^2+b \text {$\#$1}^3}\&\right ]-\sqrt {2} \sqrt {a} \sqrt {b} \text {RootSum}\left [a^2 b-4 i a^2 \text {$\#$1}+2 a b \text {$\#$1}^2+4 i a \text {$\#$1}^3+b \text {$\#$1}^4\&,\frac {\log \left (b+\sqrt {b^2+a x^2}\right ) \text {$\#$1}^2-2 \log \left (\sqrt {2} \sqrt {a} \sqrt {b}-\frac {i a x}{\sqrt {b+\sqrt {b^2+a x^2}}}+\sqrt {b+\sqrt {b^2+a x^2}} \text {$\#$1}\right ) \text {$\#$1}^2}{-i a^2+a b \text {$\#$1}+3 i a \text {$\#$1}^2+b \text {$\#$1}^3}\&\right ] \]

input

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

output

(2*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] - (I*Sqrt[b]*Log[b + Sqrt[b^2 + a*x^2]]) 
/(Sqrt[2]*Sqrt[a]) + (I*Sqrt[2]*Sqrt[b]*Log[Sqrt[2]*Sqrt[a]*Sqrt[b] - (I*a 
*x)/Sqrt[b + Sqrt[b^2 + a*x^2]]])/Sqrt[a] - I*Sqrt[2]*a^(3/2)*Sqrt[b]*Root 
Sum[a^2*b - (4*I)*a^2*#1 + 2*a*b*#1^2 + (4*I)*a*#1^3 + b*#1^4 & , (-Log[b 
+ Sqrt[b^2 + a*x^2]] + 2*Log[Sqrt[2]*Sqrt[a]*Sqrt[b] - (I*a*x)/Sqrt[b + Sq 
rt[b^2 + a*x^2]] + Sqrt[b + Sqrt[b^2 + a*x^2]]*#1])/(a^2 + I*a*b*#1 - 3*a* 
#1^2 + I*b*#1^3) & ] + (2*I)*Sqrt[2]*Sqrt[a]*b^(3/2)*RootSum[a^2*b - (4*I) 
*a^2*#1 + 2*a*b*#1^2 + (4*I)*a*#1^3 + b*#1^4 & , (Log[b + Sqrt[b^2 + a*x^2 
]]*#1 - 2*Log[Sqrt[2]*Sqrt[a]*Sqrt[b] - (I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2] 
] + Sqrt[b + Sqrt[b^2 + a*x^2]]*#1]*#1)/((-I)*a^2 + a*b*#1 + (3*I)*a*#1^2 
+ b*#1^3) & ] - Sqrt[2]*Sqrt[a]*Sqrt[b]*RootSum[a^2*b - (4*I)*a^2*#1 + 2*a 
*b*#1^2 + (4*I)*a*#1^3 + b*#1^4 & , (Log[b + Sqrt[b^2 + a*x^2]]*#1^2 - 2*L 
og[Sqrt[2]*Sqrt[a]*Sqrt[b] - (I*a*x)/Sqrt[b + Sqrt[b^2 + a*x^2]] + Sqrt[b 
+ Sqrt[b^2 + a*x^2]]*#1]*#1^2)/((-I)*a^2 + a*b*#1 + (3*I)*a*#1^2 + b*#1^3) 
 & ]
 

3.31.61.3 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {1}{\sqrt {\sqrt {a x^2+b^2}+b}}-\frac {2 b^2}{\left (b^2-a x\right ) \sqrt {\sqrt {a x^2+b^2}+b}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}}dx-2 b^2 \int \frac {1}{\left (b^2-a x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}dx\)

input

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

output

$Aborted
 

3.31.61.3.1 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] 
]
 

3.31.61.4 Maple [F]

\[\int \frac {a x +b^{2}}{\left (a x -b^{2}\right ) \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}d x\]

input

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

output

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

3.31.61.5 Fricas [F(-1)]

Timed out. \[ \int \frac {b^2+a x}{\left (-b^2+a x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.31.61.6 Sympy [F]

\[ \int \frac {b^2+a x}{\left (-b^2+a x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int \frac {a x + b^{2}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}} \left (a x - b^{2}\right )}\, dx \]

input

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

output

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

3.31.61.7 Maxima [F]

\[ \int \frac {b^2+a x}{\left (-b^2+a x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int { -\frac {b^{2} + a x}{{\left (b^{2} - a x\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}} \,d x } \]

input

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

output

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

3.31.61.8 Giac [F]

\[ \int \frac {b^2+a x}{\left (-b^2+a x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\int { -\frac {b^{2} + a x}{{\left (b^{2} - a x\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}} \,d x } \]

input

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

output

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

3.31.61.9 Mupad [F(-1)]

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

input

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

output

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