3.29.63 \(\int \frac {\sqrt {-b+x^2} (c+x^4) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx\) [2863]

3.29.63.1 Optimal result

Integrand size = 37, antiderivative size = 301 \[ \int \frac {\sqrt {-b+x^2} \left (c+x^4\right ) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx=\frac {2 \sqrt {-b+x^2} \left (-1368 b^4 x+10395 b^2 c x+3705 b^3 x^3-32340 b c x^3+1335 b^2 x^5+18480 c x^5-8100 b x^7+5040 x^9\right )+2 \left (304 b^5-2310 b^3 c-3078 b^4 x^2+24255 b^2 c x^2+3735 b^3 x^4-41580 b c x^4+4755 b^2 x^6+18480 c x^6-10620 b x^8+5040 x^{10}\right )}{3465 \left (x+\sqrt {-b+x^2}\right )^{9/2}}+\sqrt {2} b^{3/4} c \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}}{-\sqrt {b}+x+\sqrt {-b+x^2}}\right )+\sqrt {2} b^{3/4} c \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt {-b+x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt {x+\sqrt {-b+x^2}}}\right ) \]

output

1/3465*(2*(x^2-b)^(1/2)*(5040*x^9-8100*b*x^7+1335*b^2*x^5+3705*b^3*x^3+184 
80*c*x^5-1368*b^4*x-32340*b*c*x^3+10395*b^2*c*x)+10080*x^10-21240*b*x^8+95 
10*b^2*x^6+7470*b^3*x^4+36960*c*x^6-6156*b^4*x^2-83160*b*c*x^4+608*b^5+485 
10*b^2*c*x^2-4620*b^3*c)/(x+(x^2-b)^(1/2))^(9/2)+2^(1/2)*b^(3/4)*c*arctan( 
2^(1/2)*b^(1/4)*(x+(x^2-b)^(1/2))^(1/2)/(-b^(1/2)+x+(x^2-b)^(1/2)))+2^(1/2 
)*b^(3/4)*c*arctanh((1/2*b^(1/4)*2^(1/2)+1/2*x*2^(1/2)/b^(1/4)+1/2*(x^2-b) 
^(1/2)*2^(1/2)/b^(1/4))/(x+(x^2-b)^(1/2))^(1/2))
 

3.29.63.2 Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {-b+x^2} \left (c+x^4\right ) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx=\frac {608 b^5-4620 b^3 c-6156 b^4 x^2+48510 b^2 c x^2+7470 b^3 x^4-83160 b c x^4+9510 b^2 x^6+36960 c x^6-21240 b x^8+10080 x^{10}+6 \sqrt {-b+x^2} \left (-456 b^4 x+1235 b^3 x^3+5 b^2 \left (693 c x+89 x^5\right )-20 b \left (539 c x^3+135 x^7\right )+560 \left (11 c x^5+3 x^9\right )\right )}{3465 \left (x+\sqrt {-b+x^2}\right )^{9/2}}+\sqrt {2} b^{3/4} c \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}}{-\sqrt {b}+x+\sqrt {-b+x^2}}\right )+\sqrt {2} b^{3/4} c \text {arctanh}\left (\frac {\sqrt {b}+x+\sqrt {-b+x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt {x+\sqrt {-b+x^2}}}\right ) \]

input

Integrate[(Sqrt[-b + x^2]*(c + x^4)*Sqrt[x + Sqrt[-b + x^2]])/x,x]
 

output

(608*b^5 - 4620*b^3*c - 6156*b^4*x^2 + 48510*b^2*c*x^2 + 7470*b^3*x^4 - 83 
160*b*c*x^4 + 9510*b^2*x^6 + 36960*c*x^6 - 21240*b*x^8 + 10080*x^10 + 6*Sq 
rt[-b + x^2]*(-456*b^4*x + 1235*b^3*x^3 + 5*b^2*(693*c*x + 89*x^5) - 20*b* 
(539*c*x^3 + 135*x^7) + 560*(11*c*x^5 + 3*x^9)))/(3465*(x + Sqrt[-b + x^2] 
)^(9/2)) + Sqrt[2]*b^(3/4)*c*ArcTan[(Sqrt[2]*b^(1/4)*Sqrt[x + Sqrt[-b + x^ 
2]])/(-Sqrt[b] + x + Sqrt[-b + x^2])] + Sqrt[2]*b^(3/4)*c*ArcTanh[(Sqrt[b] 
 + x + Sqrt[-b + x^2])/(Sqrt[2]*b^(1/4)*Sqrt[x + Sqrt[-b + x^2]])]
 

3.29.63.3 Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {7293, 2009}

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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {c \sqrt {x^2-b} \sqrt {\sqrt {x^2-b}+x}}{x}+x^3 \sqrt {x^2-b} \sqrt {\sqrt {x^2-b}+x}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \sqrt {2} b^{3/4} c \arctan \left (1-\frac {\sqrt {2} \sqrt {\sqrt {x^2-b}+x}}{\sqrt [4]{b}}\right )-\sqrt {2} b^{3/4} c \arctan \left (\frac {\sqrt {2} \sqrt {\sqrt {x^2-b}+x}}{\sqrt [4]{b}}+1\right )-\frac {b^{3/4} c \log \left (\sqrt {x^2-b}-\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {x^2-b}+x}+\sqrt {b}+x\right )}{\sqrt {2}}+\frac {b^{3/4} c \log \left (\sqrt {x^2-b}+\sqrt {2} \sqrt [4]{b} \sqrt {\sqrt {x^2-b}+x}+\sqrt {b}+x\right )}{\sqrt {2}}-\frac {b^5}{144 \left (\sqrt {x^2-b}+x\right )^{9/2}}-\frac {b^4}{80 \left (\sqrt {x^2-b}+x\right )^{5/2}}+\frac {b^3}{8 \sqrt {\sqrt {x^2-b}+x}}-\frac {1}{24} b^2 \left (\sqrt {x^2-b}+x\right )^{3/2}+\frac {1}{3} c \left (\sqrt {x^2-b}+x\right )^{3/2}-\frac {b c}{\sqrt {\sqrt {x^2-b}+x}}+\frac {1}{176} \left (\sqrt {x^2-b}+x\right )^{11/2}+\frac {1}{112} b \left (\sqrt {x^2-b}+x\right )^{7/2}\)

input

Int[(Sqrt[-b + x^2]*(c + x^4)*Sqrt[x + Sqrt[-b + x^2]])/x,x]
 

output

-1/144*b^5/(x + Sqrt[-b + x^2])^(9/2) - b^4/(80*(x + Sqrt[-b + x^2])^(5/2) 
) + b^3/(8*Sqrt[x + Sqrt[-b + x^2]]) - (b*c)/Sqrt[x + Sqrt[-b + x^2]] - (b 
^2*(x + Sqrt[-b + x^2])^(3/2))/24 + (c*(x + Sqrt[-b + x^2])^(3/2))/3 + (b* 
(x + Sqrt[-b + x^2])^(7/2))/112 + (x + Sqrt[-b + x^2])^(11/2)/176 + Sqrt[2 
]*b^(3/4)*c*ArcTan[1 - (Sqrt[2]*Sqrt[x + Sqrt[-b + x^2]])/b^(1/4)] - Sqrt[ 
2]*b^(3/4)*c*ArcTan[1 + (Sqrt[2]*Sqrt[x + Sqrt[-b + x^2]])/b^(1/4)] - (b^( 
3/4)*c*Log[Sqrt[b] + x + Sqrt[-b + x^2] - Sqrt[2]*b^(1/4)*Sqrt[x + Sqrt[-b 
 + x^2]]])/Sqrt[2] + (b^(3/4)*c*Log[Sqrt[b] + x + Sqrt[-b + x^2] + Sqrt[2] 
*b^(1/4)*Sqrt[x + Sqrt[-b + x^2]]])/Sqrt[2]
 

3.29.63.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

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

3.29.63.4 Maple [F]

\[\int \frac {\sqrt {x^{2}-b}\, \left (x^{4}+c \right ) \sqrt {x +\sqrt {x^{2}-b}}}{x}d x\]

input

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

output

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

3.29.63.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {-b+x^2} \left (c+x^4\right ) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx=-\frac {2}{3465} \, {\left (35 \, x^{5} - 19 \, b x^{3} - {\left (152 \, b^{2} - 1155 \, c\right )} x - 2 \, {\left (175 \, x^{4} - 57 \, b x^{2} - 152 \, b^{2} + 1155 \, c\right )} \sqrt {x^{2} - b}\right )} \sqrt {x + \sqrt {x^{2} - b}} - \left (-b^{3} c^{4}\right )^{\frac {1}{4}} \log \left (b^{2} c^{3} \sqrt {x + \sqrt {x^{2} - b}} + \left (-b^{3} c^{4}\right )^{\frac {3}{4}}\right ) + i \, \left (-b^{3} c^{4}\right )^{\frac {1}{4}} \log \left (b^{2} c^{3} \sqrt {x + \sqrt {x^{2} - b}} + i \, \left (-b^{3} c^{4}\right )^{\frac {3}{4}}\right ) - i \, \left (-b^{3} c^{4}\right )^{\frac {1}{4}} \log \left (b^{2} c^{3} \sqrt {x + \sqrt {x^{2} - b}} - i \, \left (-b^{3} c^{4}\right )^{\frac {3}{4}}\right ) + \left (-b^{3} c^{4}\right )^{\frac {1}{4}} \log \left (b^{2} c^{3} \sqrt {x + \sqrt {x^{2} - b}} - \left (-b^{3} c^{4}\right )^{\frac {3}{4}}\right ) \]

input

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

output

-2/3465*(35*x^5 - 19*b*x^3 - (152*b^2 - 1155*c)*x - 2*(175*x^4 - 57*b*x^2 
- 152*b^2 + 1155*c)*sqrt(x^2 - b))*sqrt(x + sqrt(x^2 - b)) - (-b^3*c^4)^(1 
/4)*log(b^2*c^3*sqrt(x + sqrt(x^2 - b)) + (-b^3*c^4)^(3/4)) + I*(-b^3*c^4) 
^(1/4)*log(b^2*c^3*sqrt(x + sqrt(x^2 - b)) + I*(-b^3*c^4)^(3/4)) - I*(-b^3 
*c^4)^(1/4)*log(b^2*c^3*sqrt(x + sqrt(x^2 - b)) - I*(-b^3*c^4)^(3/4)) + (- 
b^3*c^4)^(1/4)*log(b^2*c^3*sqrt(x + sqrt(x^2 - b)) - (-b^3*c^4)^(3/4))
 

3.29.63.6 Sympy [F]

\[ \int \frac {\sqrt {-b+x^2} \left (c+x^4\right ) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx=\int \frac {\sqrt {- b + x^{2}} \left (c + x^{4}\right ) \sqrt {x + \sqrt {- b + x^{2}}}}{x}\, dx \]

input

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

output

Integral(sqrt(-b + x**2)*(c + x**4)*sqrt(x + sqrt(-b + x**2))/x, x)
 

3.29.63.7 Maxima [F]

\[ \int \frac {\sqrt {-b+x^2} \left (c+x^4\right ) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx=\int { \frac {{\left (x^{4} + c\right )} \sqrt {x^{2} - b} \sqrt {x + \sqrt {x^{2} - b}}}{x} \,d x } \]

input

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

output

integrate((x^4 + c)*sqrt(x^2 - b)*sqrt(x + sqrt(x^2 - b))/x, x)
 

3.29.63.8 Giac [F]

\[ \int \frac {\sqrt {-b+x^2} \left (c+x^4\right ) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx=\int { \frac {{\left (x^{4} + c\right )} \sqrt {x^{2} - b} \sqrt {x + \sqrt {x^{2} - b}}}{x} \,d x } \]

input

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

output

integrate((x^4 + c)*sqrt(x^2 - b)*sqrt(x + sqrt(x^2 - b))/x, x)
 

3.29.63.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-b+x^2} \left (c+x^4\right ) \sqrt {x+\sqrt {-b+x^2}}}{x} \, dx=\int \frac {\sqrt {x+\sqrt {x^2-b}}\,\left (x^4+c\right )\,\sqrt {x^2-b}}{x} \,d x \]

input

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

output

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