\(\int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{(b+a^2 x^2)^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx\) [2831]

Optimal result

Integrand size = 62, antiderivative size = 287 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\frac {a x \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}+\sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{a^2 b x \sqrt {b+a^2 x^2}+a b \left (b+a^2 x^2\right )}+\frac {\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-c \log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right )-\log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{c \text {$\#$1}-\text {$\#$1}^3}\&\right ]}{4 a b} \]

[Out]

Unintegrable

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=-\frac {-\frac {4 \left (a x+\sqrt {b+a^2 x^2}\right )^{3/2} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a x \left (a x+\sqrt {b+a^2 x^2}\right )}+\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {c \log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right )+\log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}^2}{c \text {$\#$1}-\text {$\#$1}^3}\&\right ]}{4 a b} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.00 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.17

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 35.81 (sec) , antiderivative size = 1089073, normalized size of antiderivative = 3794.68 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [N/A]

Not integrable

Time = 1.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.19 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\int \frac {\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{\sqrt {a x + \sqrt {a^{2} x^{2} + b}} \left (a^{2} x^{2} + b\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((c+(a*x+(a**2*x**2+b)**(1/2))**(1/2))**(1/2)/(a**2*x**2+b)**(3/2)/(a*x+(a**2*x**2+b)**(1/2))**(1/2),
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 = 1.41 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\int { \frac {\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{{\left (a^{2} x^{2} + b\right )}^{\frac {3}{2}} \sqrt {a x + \sqrt {a^{2} x^{2} + b}}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((c+(a*x+(a^2*x^2+b)^(1/2))^(1/2))^(1/2)/(a^2*x^2+b)^(3/2)/(a*x+(a^2*x^2+b)^(1/2))^(1/2),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 = 0.18 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\left (b+a^2 x^2\right )^{3/2} \sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx=\int \frac {\sqrt {c+\sqrt {\sqrt {a^2\,x^2+b}+a\,x}}}{\sqrt {\sqrt {a^2\,x^2+b}+a\,x}\,{\left (a^2\,x^2+b\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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