3.26.36 \(\int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [2536]

3.26.36.1 Optimal result

Integrand size = 48, antiderivative size = 212 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {a \log \left (a x+\sqrt {b^2+a^2 x^2}\right )}{c}-\frac {2 a \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+4 a^2 \text {$\#$1}^5+c \text {$\#$1}^8\&,\frac {b^2 c \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 d \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{2 b^2 c-4 a^2 d-5 a^2 \text {$\#$1}-2 c \text {$\#$1}^4}\&\right ]}{c} \]

output

Unintegrable
 

3.26.36.2 Mathematica [A] (verified)

Time = 3.45 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {a \left (\log \left (a x+\sqrt {b^2+a^2 x^2}\right )-2 \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+4 a^2 \text {$\#$1}^5+c \text {$\#$1}^8\&,\frac {b^2 c \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 d \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{2 b^2 c-4 a^2 d-5 a^2 \text {$\#$1}-2 c \text {$\#$1}^4}\&\right ]\right )}{c} \]

input

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

output

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

3.26.36.3 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {c^2 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}} x^4}{-c^4 x^8-4 c^3 d x^6+2 a c^2 x^5-6 c^2 d^2 x^4+4 a c d x^3-4 c d^3 x^2+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )}+\frac {2 c d \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}} x^2}{-c^4 x^8-4 c^3 d x^6+2 a c^2 x^5-6 c^2 d^2 x^4+4 a c d x^3-4 c d^3 x^2+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )}+\frac {c \left (b^2+a^2 x^2\right ) x^2}{c^4 x^8+4 c^3 d x^6-2 a c^2 x^5+6 c^2 d^2 x^4-4 a c d x^3+4 c d^3 x^2-2 a d^2 x-b^2 \left (1-\frac {d^4}{b^2}\right )}+\frac {a \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}} x}{c^4 x^8+4 c^3 d x^6-2 a c^2 x^5+6 c^2 d^2 x^4-4 a c d x^3+4 c d^3 x^2-2 a d^2 x-b^2 \left (1-\frac {d^4}{b^2}\right )}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-c^4 x^8-4 c^3 d x^6+2 a c^2 x^5-6 c^2 d^2 x^4+4 a c d x^3-4 c d^3 x^2+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )}+\frac {d^2 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-c^4 x^8-4 c^3 d x^6+2 a c^2 x^5-6 c^2 d^2 x^4+4 a c d x^3-4 c d^3 x^2+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )}+\frac {\sqrt {b^2+a^2 x^2} \left (c x^2+d\right ) \left (-c^2 x^4-2 c d x^2+a x-d^2\right )}{-c^4 x^8-4 c^3 d x^6+2 a c^2 x^5-6 c^2 d^2 x^4+4 a c d x^3-4 c d^3 x^2+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )}+\frac {d \left (b^2+a^2 x^2\right )}{c^4 x^8+4 c^3 d x^6-2 a c^2 x^5+6 c^2 d^2 x^4-4 a c d x^3+4 c d^3 x^2-2 a d^2 x-b^2 \left (1-\frac {d^4}{b^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {b^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a x \sqrt {a^2 x^2+b^2} \left (-\sqrt {\sqrt {a^2 x^2+b^2}+a x}+c x^2+d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {a^2 x^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}+\frac {\sqrt {a^2 x^2+b^2} \left (c x^2+d\right )^2 \left (\sqrt {\sqrt {a^2 x^2+b^2}+a x}-c x^2-d\right )}{2 a c^2 x^5+4 a c d x^3+2 a d^2 x+b^2 \left (1-\frac {d^4}{b^2}\right )-c^4 x^8-4 c^3 d x^6-6 c^2 d^2 x^4-4 c d^3 x^2}\right )dx\)

\(\Big \downarrow \) 7239

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

input

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

output

$Aborted
 

3.26.36.3.1 Defintions of rubi rules used

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.26.36.4 Maple [N/A] (verified)

Not integrable

Time = 0.56 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.20

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

input

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

output

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

3.26.36.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.26.36.6 Sympy [N/A]

Not integrable

Time = 0.75 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.19 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {\sqrt {a^{2} x^{2} + b^{2}}}{c x^{2} + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]

input

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

output

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

3.26.36.7 Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.21 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} + b^{2}}}{c x^{2} + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]

input

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

output

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

3.26.36.8 Giac [N/A]

Not integrable

Time = 1.45 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.21 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} + b^{2}}}{c x^{2} + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]

input

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

output

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

3.26.36.9 Mupad [N/A]

Not integrable

Time = 6.70 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.21 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {\sqrt {a^2\,x^2+b^2}}{d+\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}+c\,x^2} \,d x \]

input

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

output

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