\(\int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx\) [1312]

Optimal result

Integrand size = 39, antiderivative size = 94 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx=\frac {\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{c^2-2 c \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{a} \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx=\int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx \]

[In]

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

[Out]

-1/2*Defer[Int][Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]]/(Sqrt[-b] - a*x), x]/Sqrt[-b] - Defer[Int][Sqrt[c + Sq
rt[a*x + Sqrt[b + a^2*x^2]]]/(Sqrt[-b] + a*x), x]/(2*Sqrt[-b])

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

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx=\frac {\text {RootSum}\left [b+c^4-4 c^3 \text {$\#$1}^2+6 c^2 \text {$\#$1}^4-4 c \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}-\text {$\#$1}\right ) \text {$\#$1}}{c^2-2 c \text {$\#$1}^2+\text {$\#$1}^4}\&\right ]}{a} \]

[In]

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

[Out]

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^2 - 2*c*#1^2 + #1^4) & ]/a

Maple [N/A] (verified)

Not integrable

Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.35

\[\int \frac {\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}}{a^{2} x^{2}+b}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.31 (sec) , antiderivative size = 829, normalized size of antiderivative = 8.82 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx=\sqrt {-\frac {a^{2} b \sqrt {-\sqrt {-\frac {1}{a^{8} b^{3}}}} + c}{a^{2} b}} \log \left (2 \, a^{5} b^{2} \sqrt {-\frac {a^{2} b \sqrt {-\sqrt {-\frac {1}{a^{8} b^{3}}}} + c}{a^{2} b}} \sqrt {-\frac {1}{a^{8} b^{3}}} + 2 \, \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right ) - \sqrt {-\frac {a^{2} b \sqrt {-\sqrt {-\frac {1}{a^{8} b^{3}}}} + c}{a^{2} b}} \log \left (-2 \, a^{5} b^{2} \sqrt {-\frac {a^{2} b \sqrt {-\sqrt {-\frac {1}{a^{8} b^{3}}}} + c}{a^{2} b}} \sqrt {-\frac {1}{a^{8} b^{3}}} + 2 \, \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right ) + \sqrt {\frac {a^{2} b \sqrt {-\sqrt {-\frac {1}{a^{8} b^{3}}}} - c}{a^{2} b}} \log \left (2 \, a^{5} b^{2} \sqrt {\frac {a^{2} b \sqrt {-\sqrt {-\frac {1}{a^{8} b^{3}}}} - c}{a^{2} b}} \sqrt {-\frac {1}{a^{8} b^{3}}} + 2 \, \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right ) - \sqrt {\frac {a^{2} b \sqrt {-\sqrt {-\frac {1}{a^{8} b^{3}}}} - c}{a^{2} b}} \log \left (-2 \, a^{5} b^{2} \sqrt {\frac {a^{2} b \sqrt {-\sqrt {-\frac {1}{a^{8} b^{3}}}} - c}{a^{2} b}} \sqrt {-\frac {1}{a^{8} b^{3}}} + 2 \, \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right ) - \sqrt {-\frac {a^{2} b \left (-\frac {1}{a^{8} b^{3}}\right )^{\frac {1}{4}} + c}{a^{2} b}} \log \left (2 \, a^{5} b^{2} \sqrt {-\frac {a^{2} b \left (-\frac {1}{a^{8} b^{3}}\right )^{\frac {1}{4}} + c}{a^{2} b}} \sqrt {-\frac {1}{a^{8} b^{3}}} + 2 \, \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right ) + \sqrt {-\frac {a^{2} b \left (-\frac {1}{a^{8} b^{3}}\right )^{\frac {1}{4}} + c}{a^{2} b}} \log \left (-2 \, a^{5} b^{2} \sqrt {-\frac {a^{2} b \left (-\frac {1}{a^{8} b^{3}}\right )^{\frac {1}{4}} + c}{a^{2} b}} \sqrt {-\frac {1}{a^{8} b^{3}}} + 2 \, \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right ) - \sqrt {\frac {a^{2} b \left (-\frac {1}{a^{8} b^{3}}\right )^{\frac {1}{4}} - c}{a^{2} b}} \log \left (2 \, a^{5} b^{2} \sqrt {\frac {a^{2} b \left (-\frac {1}{a^{8} b^{3}}\right )^{\frac {1}{4}} - c}{a^{2} b}} \sqrt {-\frac {1}{a^{8} b^{3}}} + 2 \, \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right ) + \sqrt {\frac {a^{2} b \left (-\frac {1}{a^{8} b^{3}}\right )^{\frac {1}{4}} - c}{a^{2} b}} \log \left (-2 \, a^{5} b^{2} \sqrt {\frac {a^{2} b \left (-\frac {1}{a^{8} b^{3}}\right )^{\frac {1}{4}} - c}{a^{2} b}} \sqrt {-\frac {1}{a^{8} b^{3}}} + 2 \, \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}\right ) \]

[In]

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

[Out]

sqrt(-(a^2*b*sqrt(-sqrt(-1/(a^8*b^3))) + c)/(a^2*b))*log(2*a^5*b^2*sqrt(-(a^2*b*sqrt(-sqrt(-1/(a^8*b^3))) + c)
/(a^2*b))*sqrt(-1/(a^8*b^3)) + 2*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))) - sqrt(-(a^2*b*sqrt(-sqrt(-1/(a^8*b^
3))) + c)/(a^2*b))*log(-2*a^5*b^2*sqrt(-(a^2*b*sqrt(-sqrt(-1/(a^8*b^3))) + c)/(a^2*b))*sqrt(-1/(a^8*b^3)) + 2*
sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))) + sqrt((a^2*b*sqrt(-sqrt(-1/(a^8*b^3))) - c)/(a^2*b))*log(2*a^5*b^2*s
qrt((a^2*b*sqrt(-sqrt(-1/(a^8*b^3))) - c)/(a^2*b))*sqrt(-1/(a^8*b^3)) + 2*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b
)))) - sqrt((a^2*b*sqrt(-sqrt(-1/(a^8*b^3))) - c)/(a^2*b))*log(-2*a^5*b^2*sqrt((a^2*b*sqrt(-sqrt(-1/(a^8*b^3))
) - c)/(a^2*b))*sqrt(-1/(a^8*b^3)) + 2*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))) - sqrt(-(a^2*b*(-1/(a^8*b^3))^
(1/4) + c)/(a^2*b))*log(2*a^5*b^2*sqrt(-(a^2*b*(-1/(a^8*b^3))^(1/4) + c)/(a^2*b))*sqrt(-1/(a^8*b^3)) + 2*sqrt(
c + sqrt(a*x + sqrt(a^2*x^2 + b)))) + sqrt(-(a^2*b*(-1/(a^8*b^3))^(1/4) + c)/(a^2*b))*log(-2*a^5*b^2*sqrt(-(a^
2*b*(-1/(a^8*b^3))^(1/4) + c)/(a^2*b))*sqrt(-1/(a^8*b^3)) + 2*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))) - sqrt(
(a^2*b*(-1/(a^8*b^3))^(1/4) - c)/(a^2*b))*log(2*a^5*b^2*sqrt((a^2*b*(-1/(a^8*b^3))^(1/4) - c)/(a^2*b))*sqrt(-1
/(a^8*b^3)) + 2*sqrt(c + sqrt(a*x + sqrt(a^2*x^2 + b)))) + sqrt((a^2*b*(-1/(a^8*b^3))^(1/4) - c)/(a^2*b))*log(
-2*a^5*b^2*sqrt((a^2*b*(-1/(a^8*b^3))^(1/4) - c)/(a^2*b))*sqrt(-1/(a^8*b^3)) + 2*sqrt(c + sqrt(a*x + sqrt(a^2*
x^2 + b))))

Sympy [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx=\int \frac {\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{a^{2} x^{2} + b}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.42 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx=\int { \frac {\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{a^{2} x^{2} + b} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 41.68 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx=\int { \frac {\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} + b}}}}{a^{2} x^{2} + b} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.93 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{b+a^2 x^2} \, dx=\int \frac {\sqrt {c+\sqrt {\sqrt {a^2\,x^2+b}+a\,x}}}{a^2\,x^2+b} \,d x \]

[In]

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

[Out]

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