\(\int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx\) [2223]

Optimal result

Integrand size = 30, antiderivative size = 165 \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=-\frac {4}{3} c \sqrt {c+\sqrt {b+a x}}+\sqrt {b+a x} \left (\frac {2 (b+a x)}{3 a}-\frac {4}{3} \sqrt {c+\sqrt {b+a x}}\right )+4 \text {RootSum}\left [b-c^2-a \text {$\#$1}+2 c \text {$\#$1}^2-\text {$\#$1}^4\&,\frac {-b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2+a \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a-4 c \text {$\#$1}+4 \text {$\#$1}^3}\&\right ] \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.86 \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=-\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}+\frac {2 \left (c^3+(b+a x)^{3/2}\right )}{3 a}+4 \text {RootSum}\left [b-c^2-a \text {$\#$1}+2 c \text {$\#$1}^2-\text {$\#$1}^4\&,\frac {-b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2+a \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a-4 c \text {$\#$1}+4 \text {$\#$1}^3}\&\right ] \] Input:

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

Output:

(-4*(c + Sqrt[b + a*x])^(3/2))/3 + (2*(c^3 + (b + a*x)^(3/2)))/(3*a) + 4*R 
ootSum[b - c^2 - a*#1 + 2*c*#1^2 - #1^4 & , (-(b*Log[Sqrt[c + Sqrt[b + a*x 
]] - #1]*#1^2) + a*Log[Sqrt[c + Sqrt[b + a*x]] - #1]*#1^3)/(a - 4*c*#1 + 4 
*#1^3) & ]
 

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.

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

\(\Big \downarrow \) 7267

\(\displaystyle \frac {2 \int -\frac {a x (b+a x)}{-x a-\sqrt {c+\sqrt {b+a x}} a}d\sqrt {b+a x}}{a}\)

\(\Big \downarrow \) 7267

\(\displaystyle \frac {4 \int \frac {(-b+c-a x)^2 \left (b-(-b+c-a x)^2\right ) \sqrt {c+\sqrt {b+a x}}}{-(-b+c-a x)^2+b-a \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}}}{a}\)

\(\Big \downarrow \) 7292

\(\displaystyle \frac {4 \int \frac {(-b+c-a x)^2 \sqrt {c+\sqrt {b+a x}} \left (-c^2+2 (b+a x) c-(b+a x)^2+b\right )}{-c^2+2 (b+a x) c-(b+a x)^2+b-a \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}}}{a}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {4 \int \left ((b+a x)^{5/2}-2 c (b+a x)^{3/2}-a (b+a x)+\frac {\left (a b-a^2 \sqrt {c+\sqrt {b+a x}}\right ) (b+a x)}{-c^2+2 (b+a x) c-(b+a x)^2+b-a \sqrt {c+\sqrt {b+a x}}}+c^2 \sqrt {c+\sqrt {b+a x}}\right )d\sqrt {c+\sqrt {b+a x}}}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 \left (-\frac {1}{4} a^3 \int \frac {1}{c^2-2 (b+a x) c+(b+a x)^2-b+a \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}}+a^2 c \int \frac {\sqrt {c+\sqrt {b+a x}}}{c^2-2 (b+a x) c+(b+a x)^2-b+a \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}}-a b \int \frac {b+a x}{c^2-2 (b+a x) c+(b+a x)^2-b+a \sqrt {c+\sqrt {b+a x}}}d\sqrt {c+\sqrt {b+a x}}+\frac {1}{4} a^2 \log \left (2 c (a x+b)-a \sqrt {\sqrt {a x+b}+c}-(a x+b)^2+b-c^2\right )+\frac {1}{2} c^2 (a x+b)-\frac {1}{2} c (a x+b)^2-\frac {1}{3} a (a x+b)^{3/2}+\frac {1}{6} (a x+b)^3\right )}{a}\)

Input:

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

Output:

$Aborted
 

Maple [N/A] (verified)

Time = 0.13 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {\frac {2 \left (c +\sqrt {a x +b}\right )^{3}}{3}-2 c \left (c +\sqrt {a x +b}\right )^{2}-\frac {4 a \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+2 c^{2} \left (c +\sqrt {a x +b}\right )+4 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 c \,\textit {\_Z}^{2}+a \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{3} a -\textit {\_R}^{2} b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-4 \textit {\_R} c +a}\right )}{a}\)	\(129\)
default	\(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{3}}{3}+c \left (c +\sqrt {a x +b}\right )^{2}+\frac {2 a \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}-c^{2} \left (c +\sqrt {a x +b}\right )-2 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 c \,\textit {\_Z}^{2}+a \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{3} a -\textit {\_R}^{2} b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-4 \textit {\_R} c +a}\right )\right )}{a}\)	\(129\)

Input:

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

Output:

2/a*(1/3*(c+(a*x+b)^(1/2))^3-c*(c+(a*x+b)^(1/2))^2-2/3*a*(c+(a*x+b)^(1/2)) 
^(3/2)+c^2*(c+(a*x+b)^(1/2))+2*a*sum((_R^3*a-_R^2*b)/(4*_R^3-4*_R*c+a)*ln( 
(c+(a*x+b)^(1/2))^(1/2)-_R),_R=RootOf(_Z^4-2*_Z^2*c+_Z*a+c^2-b)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {\sqrt {a x + b} x}{x + \sqrt {c + \sqrt {a x + b}}} \,d x } \] Input:

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

Output:

integrate(sqrt(a*x + b)*x/(x + sqrt(c + sqrt(a*x + b))), x)
 

Giac [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {\sqrt {a x + b} x}{x + \sqrt {c + \sqrt {a x + b}}} \,d x } \] Input:

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

Output:

integrate(sqrt(a*x + b)*x/(x + sqrt(c + sqrt(a*x + b))), x)
 

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {x\,\sqrt {b+a\,x}}{x+\sqrt {c+\sqrt {b+a\,x}}} \,d x \] Input:

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

Output:

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

Reduce [N/A]

Not integrable

Time = 200.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {x \sqrt {a x +b}}{x +\sqrt {c +\sqrt {a x +b}}}d x \] Input:

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

Output:

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