\(\int \frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{(b+a^2 x^2)^{3/2}} \, dx\) [2647]

Optimal result

Integrand size = 62, antiderivative size = 235 \[ \int \frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{a^2 x \sqrt {b+a^2 x^2}+a \left (b+a^2 x^2\right )}-\frac {\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-c \log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right )+3 \log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{c^3 \text {$\#$1}-3 c^2 \text {$\#$1}^3+3 c \text {$\#$1}^5-\text {$\#$1}^7}\&\right ]}{4 a} \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2}} \, dx \]

[In]

Int[(Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(b + a^2*x^2)^(3/2),x]

[Out]

Defer[Int][(Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(b + a^2*x^2)^(3/2), x]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2}} \, dx=-\frac {\frac {4 \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a x \left (a x+\sqrt {b+a^2 x^2}\right )}-8 \text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{-c^3+3 c^2 \text {$\#$1}^2-3 c \text {$\#$1}^4+\text {$\#$1}^6}\&\right ]+\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {c \log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right )+5 \log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-c^3 \text {$\#$1}+3 c^2 \text {$\#$1}^3-3 c \text {$\#$1}^5+\text {$\#$1}^7}\&\right ]}{4 a} \]

[In]

Integrate[(Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(b + a^2*x^2)^(3/2),x]

[Out]

-1/4*((4*Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]])/(b + a*x*(a*x + Sqrt[b + a^2*x
^2])) - 8*RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c*#1^6 + #1^8 & , (Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^
2*x^2]]] - #1]*#1)/(-c^3 + 3*c^2*#1^2 - 3*c*#1^4 + #1^6) & ] + RootSum[b + c^4 - 4*c^3*#1^2 + 6*c^2*#1^4 - 4*c
*#1^6 + #1^8 & , (c*Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]] - #1] + 5*Log[Sqrt[c + Sqrt[a*x + Sqrt[b + a^2
*x^2]]] - #1]*#1^2)/(-(c^3*#1) + 3*c^2*#1^3 - 3*c*#1^5 + #1^7) & ])/a

Maple [N/A] (verified)

Not integrable

Time = 0.00 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.21

\[\int \frac {\sqrt {a x +\sqrt {a^{2} x^{2}+b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}}{\left (a^{2} x^{2}+b \right )^{\frac {3}{2}}}d x\]

[In]

int((a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2),x)

[Out]

int((a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2),x)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 88.40 (sec) , antiderivative size = 3473933, normalized size of antiderivative = 14782.69 \[ \int \frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate((a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2),x, algorithm
="fricas")

[Out]

Too large to include

Sympy [N/A]

Not integrable

Time = 1.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.23 \[ \int \frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}{\left (a^{2} x^{2} + b\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((a*x+(a**2*x**2+b)**(1/2))**(1/2)*(c+(a*x+(a**2*x**2+b)**(1/2))**(1/2))**(1/2)/(a**2*x**2+b)**(3/2),
x)

[Out]

Integral(sqrt(c + sqrt(a*x + sqrt(a**2*x**2 + b)))*sqrt(a*x + sqrt(a**2*x**2 + b))/(a**2*x**2 + b)**(3/2), x)

Maxima [N/A]

Not integrable

Time = 1.43 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{{\left (a^{2} x^{2} + b\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2),x, algorithm
="maxima")

[Out]

integrate(sqrt(a*x + sqrt(a^2*x^2 + b))*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))/(a^2*x^2 + b)^(3/2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a*x+(a^2*x^2+b)^(1/2))^(1/2)*(c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2),x, algorithm
="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(c
onst gen &

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.22 \[ \int \frac {\sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\sqrt {a^2\,x^2+b}+a\,x}\,\sqrt {c+\sqrt {\sqrt {a^2\,x^2+b}+a\,x}}}{{\left (a^2\,x^2+b\right )}^{3/2}} \,d x \]

[In]

int((((b + a^2*x^2)^(1/2) + a*x)^(1/2)*(c + ((b + a^2*x^2)^(1/2) + a*x)^(1/2))^(1/2))/(b + a^2*x^2)^(3/2),x)

[Out]

int((((b + a^2*x^2)^(1/2) + a*x)^(1/2)*(c + ((b + a^2*x^2)^(1/2) + a*x)^(1/2))^(1/2))/(b + a^2*x^2)^(3/2), x)