\(\int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx\) [2222]

Optimal result

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

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

(2*c^2*Sqrt[b + a*x])/a - (4*(c + Sqrt[b + a*x])^(3/2))/3 - (2*c*(c + Sqrt[b + a*x])^2)/a + (2*(c + Sqrt[b + a
*x])^3)/(3*a) + a*Log[b - c^2 - a*Sqrt[c + Sqrt[b + a*x]] + 2*c*(c + Sqrt[b + a*x]) - (c + Sqrt[b + a*x])^2] -
a^2*Defer[Subst][Defer[Int][(-b + c^2 + a*x - 2*c*x^2 + x^4)^(-1), x], x, Sqrt[c + Sqrt[b + a*x]]] + 4*a*c*De
fer[Subst][Defer[Int][x/(-b + c^2 + a*x - 2*c*x^2 + x^4), x], x, Sqrt[c + Sqrt[b + a*x]]] - 4*b*Defer[Subst][D
efer[Int][x^2/(-b + c^2 + a*x - 2*c*x^2 + x^4), x], x, Sqrt[c + Sqrt[b + a*x]]]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^2 \left (-b+x^2\right )}{-b+x^2+a \sqrt {c+x}} \, dx,x,\sqrt {b+a x}\right )}{a} \\ & = \frac {4 \text {Subst}\left (\int \frac {x \left (c-x^2\right )^2 \left (-b+\left (c-x^2\right )^2\right )}{-b+a x+\left (c-x^2\right )^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = \frac {4 \text {Subst}\left (\int \frac {x \left (c-x^2\right )^2 \left (b-c^2+2 c x^2-x^4\right )}{b-c^2-a x+2 c x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = \frac {4 \text {Subst}\left (\int \left (c^2 x-a x^2-2 c x^3+x^5+\frac {x^2 \left (a b-a^2 x\right )}{b-c^2-a x+2 c x^2-x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = \frac {2 c^2 \sqrt {b+a x}}{a}-\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}+\frac {4 \text {Subst}\left (\int \frac {x^2 \left (a b-a^2 x\right )}{b-c^2-a x+2 c x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = \frac {2 c^2 \sqrt {b+a x}}{a}-\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}+a \log \left (b-c^2-a \sqrt {c+\sqrt {b+a x}}+2 c \left (c+\sqrt {b+a x}\right )-\left (c+\sqrt {b+a x}\right )^2\right )-\frac {\text {Subst}\left (\int \frac {-a^3+4 a^2 c x-4 a b x^2}{b-c^2-a x+2 c x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = \frac {2 c^2 \sqrt {b+a x}}{a}-\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}+a \log \left (b-c^2-a \sqrt {c+\sqrt {b+a x}}+2 c \left (c+\sqrt {b+a x}\right )-\left (c+\sqrt {b+a x}\right )^2\right )-\frac {\text {Subst}\left (\int \left (\frac {a^3}{-b+c^2+a x-2 c x^2+x^4}-\frac {4 a^2 c x}{-b+c^2+a x-2 c x^2+x^4}+\frac {4 a b x^2}{-b+c^2+a x-2 c x^2+x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )}{a} \\ & = \frac {2 c^2 \sqrt {b+a x}}{a}-\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\frac {2 c \left (c+\sqrt {b+a x}\right )^2}{a}+\frac {2 \left (c+\sqrt {b+a x}\right )^3}{3 a}+a \log \left (b-c^2-a \sqrt {c+\sqrt {b+a x}}+2 c \left (c+\sqrt {b+a x}\right )-\left (c+\sqrt {b+a x}\right )^2\right )-a^2 \text {Subst}\left (\int \frac {1}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-(4 b) \text {Subst}\left (\int \frac {x^2}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )+(4 a c) \text {Subst}\left (\int \frac {x}{-b+c^2+a x-2 c x^2+x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.86 \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=-\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}+\frac {2 \left (c^3+(b+a x)^{3/2}\right )}{3 a}+4 \text {RootSum}\left [b-c^2-a \text {$\#$1}+2 c \text {$\#$1}^2-\text {$\#$1}^4\&,\frac {-b \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^2+a \log \left (\sqrt {c+\sqrt {b+a x}}-\text {$\#$1}\right ) \text {$\#$1}^3}{a-4 c \text {$\#$1}+4 \text {$\#$1}^3}\&\right ] \]

[In]

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

[Out]

(-4*(c + Sqrt[b + a*x])^(3/2))/3 + (2*(c^3 + (b + a*x)^(3/2)))/(3*a) + 4*RootSum[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) & ]

Maple [N/A] (verified)

Time = 0.10 (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	\(-\frac {2 \left (-\frac {\left (c +\sqrt {a x +b}\right )^{3}}{3}+c \left (c +\sqrt {a x +b}\right )^{2}+\frac {2 a \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}-c^{2} \left (c +\sqrt {a x +b}\right )-2 a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 c \,\textit {\_Z}^{2}+a \textit {\_Z} +c^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{3} a -\textit {\_R}^{2} b \right ) \ln \left (\sqrt {c +\sqrt {a x +b}}-\textit {\_R} \right )}{4 \textit {\_R}^{3}-4 \textit {\_R} c +a}\right )\right )}{a}\)	\(129\)

[In]

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

[Out]

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(-2)]

Exception generated. \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\text {Exception raised: AttributeError} \]

[In]

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

[Out]

Exception raised: AttributeError

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {\sqrt {a x + b} x}{x + \sqrt {c + \sqrt {a x + b}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int { \frac {\sqrt {a x + b} x}{x + \sqrt {c + \sqrt {a x + b}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {x \sqrt {b+a x}}{x+\sqrt {c+\sqrt {b+a x}}} \, dx=\int \frac {x\,\sqrt {b+a\,x}}{x+\sqrt {c+\sqrt {b+a\,x}}} \,d x \]

[In]

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

[Out]

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