3.29.92 \(\int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx\) [2892]

3.29.92.1 Optimal result

Integrand size = 31, antiderivative size = 316 \[ \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx=-\text {RootSum}\left [5-10 \text {$\#$1}+6 \text {$\#$1}^2-4 \text {$\#$1}^3+12 \text {$\#$1}^4-40 \text {$\#$1}^5+40 \text {$\#$1}^6\&,\frac {-\log \left (-1+\sqrt {-3+4 x}\right )+\log \left (-1+\sqrt {-1+2 x}+\text {$\#$1}-\sqrt {-3+4 x} \text {$\#$1}\right )+6 \log \left (-1+\sqrt {-3+4 x}\right ) \text {$\#$1}-6 \log \left (-1+\sqrt {-1+2 x}+\text {$\#$1}-\sqrt {-3+4 x} \text {$\#$1}\right ) \text {$\#$1}-12 \log \left (-1+\sqrt {-3+4 x}\right ) \text {$\#$1}^2+12 \log \left (-1+\sqrt {-1+2 x}+\text {$\#$1}-\sqrt {-3+4 x} \text {$\#$1}\right ) \text {$\#$1}^2+12 \log \left (-1+\sqrt {-3+4 x}\right ) \text {$\#$1}^3-12 \log \left (-1+\sqrt {-1+2 x}+\text {$\#$1}-\sqrt {-3+4 x} \text {$\#$1}\right ) \text {$\#$1}^3-4 \log \left (-1+\sqrt {-3+4 x}\right ) \text {$\#$1}^4+4 \log \left (-1+\sqrt {-1+2 x}+\text {$\#$1}-\sqrt {-3+4 x} \text {$\#$1}\right ) \text {$\#$1}^4}{-5+6 \text {$\#$1}-6 \text {$\#$1}^2+24 \text {$\#$1}^3-100 \text {$\#$1}^4+120 \text {$\#$1}^5}\&\right ] \]

output

Unintegrable
 

3.29.92.2 Mathematica [A] (verified)

Time = 1.53 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.56 \[ \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx=4 \text {RootSum}\left [2+3 \sqrt {2}+50 \text {$\#$1}^2+97 \sqrt {2} \text {$\#$1}^2-50 \text {$\#$1}^4+97 \sqrt {2} \text {$\#$1}^4-2 \text {$\#$1}^6+3 \sqrt {2} \text {$\#$1}^6\&,\frac {-\log \left (-\sqrt {-3+4 x}+\sqrt {-2+4 x}-\text {$\#$1}\right )+\log \left (-\sqrt {-3+4 x}+\sqrt {-2+4 x}-\text {$\#$1}\right ) \text {$\#$1}^4}{50 \text {$\#$1}+97 \sqrt {2} \text {$\#$1}-100 \text {$\#$1}^3+194 \sqrt {2} \text {$\#$1}^3-6 \text {$\#$1}^5+9 \sqrt {2} \text {$\#$1}^5}\&\right ] \]

input

Integrate[(Sqrt[-1 + 2*x]*(4 + 3*x) + (1 + x)*Sqrt[-3 + 4*x])^(-1),x]
 

output

4*RootSum[2 + 3*Sqrt[2] + 50*#1^2 + 97*Sqrt[2]*#1^2 - 50*#1^4 + 97*Sqrt[2] 
*#1^4 - 2*#1^6 + 3*Sqrt[2]*#1^6 & , (-Log[-Sqrt[-3 + 4*x] + Sqrt[-2 + 4*x] 
 - #1] + Log[-Sqrt[-3 + 4*x] + Sqrt[-2 + 4*x] - #1]*#1^4)/(50*#1 + 97*Sqrt 
[2]*#1 - 100*#1^3 + 194*Sqrt[2]*#1^3 - 6*#1^5 + 9*Sqrt[2]*#1^5) & ]
 

3.29.92.3 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.

\(\displaystyle \int \frac {1}{\sqrt {4 x-3} (x+1)+\sqrt {2 x-1} (3 x+4)} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 \sqrt {2 x-1} x}{14 x^3+34 x^2+10 x-13}-\frac {\sqrt {4 x-3} x}{14 x^3+34 x^2+10 x-13}+\frac {4 \sqrt {2 x-1}}{14 x^3+34 x^2+10 x-13}-\frac {\sqrt {4 x-3}}{14 x^3+34 x^2+10 x-13}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (\sqrt {4 x-3}-3 \sqrt {2 x-1}\right )-4 \sqrt {2 x-1}+\sqrt {4 x-3}}{-14 x^3-34 x^2-10 x+13}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x \left (3 \sqrt {2 x-1}-\sqrt {4 x-3}\right )}{14 x^3+34 x^2+10 x-13}+\frac {4 \sqrt {2 x-1}}{14 x^3+34 x^2+10 x-13}-\frac {\sqrt {4 x-3}}{14 x^3+34 x^2+10 x-13}\right )dx\)

\(\Big \downarrow \) 7296

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (\frac {3 (2 x-1)^2}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {\sqrt {2 (2 x-1)-1} (2 x-1)^{3/2}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}+\frac {11 (2 x-1)}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}-\frac {3 \sqrt {2 (2 x-1)-1} \sqrt {2 x-1}}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}\right )d\sqrt {2 x-1}\)

\(\Big \downarrow \) 7239

\(\displaystyle 2 \int \frac {\sqrt {2 x-1} \left (3 (2 x-1)^{3/2}-\sqrt {2 (2 x-1)-1} (2 x-1)+11 \sqrt {2 x-1}-3 \sqrt {2 (2 x-1)-1}\right )}{7 (2 x-1)^3+55 (2 x-1)^2+109 (2 x-1)+9}d\sqrt {2 x-1}\)

input

Int[(Sqrt[-1 + 2*x]*(4 + 3*x) + (1 + x)*Sqrt[-3 + 4*x])^(-1),x]
 

output

$Aborted
 

3.29.92.3.1 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] 
]
 

rule 7296

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst]]
 

3.29.92.4 Maple [N/A] (verified)

Time = 0.12 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.68

method	result	size
default	\(-8 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (7 \textit {\_Z}^{6}+131 \textit {\_Z}^{4}+677 \textit {\_Z}^{2}+625\right )}{\sum }\frac {\textit {\_R} \ln \left (\sqrt {-3+4 x}-\textit {\_R} \right )}{21 \textit {\_R}^{4}+262 \textit {\_R}^{2}+677}\right )+8 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (7 \textit {\_Z}^{6}+55 \textit {\_Z}^{4}+109 \textit {\_Z}^{2}+9\right )}{\sum }\frac {\textit {\_R} \ln \left (\sqrt {-1+2 x}-\textit {\_R} \right )}{21 \textit {\_R}^{4}+110 \textit {\_R}^{2}+109}\right )-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (7 \textit {\_Z}^{6}+131 \textit {\_Z}^{4}+677 \textit {\_Z}^{2}+625\right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{2}+3\right ) \ln \left (\sqrt {-3+4 x}-\textit {\_R} \right )}{21 \textit {\_R}^{4}+262 \textit {\_R}^{2}+677}\right )+3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (7 \textit {\_Z}^{6}+55 \textit {\_Z}^{4}+109 \textit {\_Z}^{2}+9\right )}{\sum }\frac {\textit {\_R} \left (\textit {\_R}^{2}+1\right ) \ln \left (\sqrt {-1+2 x}-\textit {\_R} \right )}{21 \textit {\_R}^{4}+110 \textit {\_R}^{2}+109}\right )\)	\(216\)

input

int(1/((-1+2*x)^(1/2)*(4+3*x)+(1+x)*(-3+4*x)^(1/2)),x,method=_RETURNVERBOS 
E)
 

output

-8*sum(_R*ln((-3+4*x)^(1/2)-_R)/(21*_R^4+262*_R^2+677),_R=RootOf(7*_Z^6+13 
1*_Z^4+677*_Z^2+625))+8*sum(_R*ln((-1+2*x)^(1/2)-_R)/(21*_R^4+110*_R^2+109 
),_R=RootOf(7*_Z^6+55*_Z^4+109*_Z^2+9))-2*sum(_R*(_R^2+3)*ln((-3+4*x)^(1/2 
)-_R)/(21*_R^4+262*_R^2+677),_R=RootOf(7*_Z^6+131*_Z^4+677*_Z^2+625))+3*su 
m(_R*(_R^2+1)*ln((-1+2*x)^(1/2)-_R)/(21*_R^4+110*_R^2+109),_R=RootOf(7*_Z^ 
6+55*_Z^4+109*_Z^2+9))
 

3.29.92.5 Fricas [C] (verification not implemented)

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

Time = 0.93 (sec) , antiderivative size = 4787, normalized size of antiderivative = 15.15 \[ \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx=\text {Too large to display} \]

input

integrate(1/((-1+2*x)^(1/2)*(4+3*x)+(1+x)*(-3+4*x)^(1/2)),x, algorithm="fr 
icas")
 

output

1/43470*sqrt(805)*sqrt(1/1610)*sqrt(944820450*(3841807/119954404332000*I*s 
qrt(15) - 3044497079/7557127472916000)^(1/3)*(-I*sqrt(3) + 1) + 782460*sqr 
t(5)*sqrt(-1/1020406086000*(944820450*(3841807/119954404332000*I*sqrt(15) 
- 3044497079/7557127472916000)^(1/3)*(-I*sqrt(3) + 1) + 53117*(I*sqrt(3) + 
 1)/(3841807/119954404332000*I*sqrt(15) - 3044497079/7557127472916000)^(1/ 
3) + 18431280)^2 + 34132*(3841807/119954404332000*I*sqrt(15) - 3044497079/ 
7557127472916000)^(1/3)*(-I*sqrt(3) + 1) + 5630402/2934225*(I*sqrt(3) + 1) 
/(3841807/119954404332000*I*sqrt(15) - 3044497079/7557127472916000)^(1/3) 
+ 151178/135) + 53117*(I*sqrt(3) + 1)/(3841807/119954404332000*I*sqrt(15) 
- 3044497079/7557127472916000)^(1/3) - 36862560)*log(1/698756341500*sqrt(1 
/1610)*(1051*sqrt(805)*(944820450*(3841807/119954404332000*I*sqrt(15) - 30 
44497079/7557127472916000)^(1/3)*(-I*sqrt(3) + 1) + 53117*(I*sqrt(3) + 1)/ 
(3841807/119954404332000*I*sqrt(15) - 3044497079/7557127472916000)^(1/3) + 
 18431280)^2 - 782460*(1051*sqrt(805)*sqrt(5)*(944820450*(3841807/11995440 
4332000*I*sqrt(15) - 3044497079/7557127472916000)^(1/3)*(-I*sqrt(3) + 1) + 
 53117*(I*sqrt(3) + 1)/(3841807/119954404332000*I*sqrt(15) - 3044497079/75 
57127472916000)^(1/3) + 18431280) - 4397155875*sqrt(805)*sqrt(5))*sqrt(-1/ 
1020406086000*(944820450*(3841807/119954404332000*I*sqrt(15) - 3044497079/ 
7557127472916000)^(1/3)*(-I*sqrt(3) + 1) + 53117*(I*sqrt(3) + 1)/(3841807/ 
119954404332000*I*sqrt(15) - 3044497079/7557127472916000)^(1/3) + 18431...
 

3.29.92.6 Sympy [N/A]

Not integrable

Time = 1.89 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx=\int \frac {1}{\left (x + 1\right ) \sqrt {4 x - 3} + \sqrt {2 x - 1} \cdot \left (3 x + 4\right )}\, dx \]

input

integrate(1/((-1+2*x)**(1/2)*(4+3*x)+(1+x)*(-3+4*x)**(1/2)),x)
 

output

Integral(1/((x + 1)*sqrt(4*x - 3) + sqrt(2*x - 1)*(3*x + 4)), x)
 

3.29.92.7 Maxima [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx=\int { \frac {1}{{\left (3 \, x + 4\right )} \sqrt {2 \, x - 1} + \sqrt {4 \, x - 3} {\left (x + 1\right )}} \,d x } \]

input

integrate(1/((-1+2*x)^(1/2)*(4+3*x)+(1+x)*(-3+4*x)^(1/2)),x, algorithm="ma 
xima")
 

output

integrate(1/((3*x + 4)*sqrt(2*x - 1) + sqrt(4*x - 3)*(x + 1)), x)
 

3.29.92.8 Giac [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.09 \[ \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx=\int { \frac {1}{{\left (3 \, x + 4\right )} \sqrt {2 \, x - 1} + \sqrt {4 \, x - 3} {\left (x + 1\right )}} \,d x } \]

input

integrate(1/((-1+2*x)^(1/2)*(4+3*x)+(1+x)*(-3+4*x)^(1/2)),x, algorithm="gi 
ac")
 

output

integrate(1/((3*x + 4)*sqrt(2*x - 1) + sqrt(4*x - 3)*(x + 1)), x)
 

3.29.92.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {-1+2 x} (4+3 x)+(1+x) \sqrt {-3+4 x}} \, dx=\text {Hanged} \]

input

int(1/((2*x - 1)^(1/2)*(3*x + 4) + (4*x - 3)^(1/2)*(x + 1)),x)
 

output

\text{Hanged}