3.30.94 \(\int \frac {\sqrt {-b+a^2 x^2} (d+c x^4) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx\) [2994]

3.30.94.1 Optimal result

Integrand size = 49, antiderivative size = 397 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\frac {2 \sqrt {-b+a^2 x^2} \left (-1368 a b^4 c x+10395 a^5 b^2 d x+3705 a^3 b^3 c x^3-32340 a^7 b d x^3+1335 a^5 b^2 c x^5+18480 a^9 d x^5-8100 a^7 b c x^7+5040 a^9 c x^9\right )+2 \left (304 b^5 c-2310 a^4 b^3 d-3078 a^2 b^4 c x^2+24255 a^6 b^2 d x^2+3735 a^4 b^3 c x^4-41580 a^8 b d x^4+4755 a^6 b^2 c x^6+18480 a^{10} d x^6-10620 a^8 b c x^8+5040 a^{10} c x^{10}\right )}{3465 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}+\sqrt {2} b^{3/4} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{-\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}\right )+\sqrt {2} b^{3/4} d \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {a x}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt {-b+a^2 x^2}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right ) \]

output

1/3465*(2*(a^2*x^2-b)^(1/2)*(5040*a^9*c*x^9-8100*a^7*b*c*x^7+18480*a^9*d*x 
^5+1335*a^5*b^2*c*x^5-32340*a^7*b*d*x^3+3705*a^3*b^3*c*x^3+10395*a^5*b^2*d 
*x-1368*a*b^4*c*x)+10080*a^10*c*x^10-21240*a^8*b*c*x^8+36960*a^10*d*x^6+95 
10*a^6*b^2*c*x^6-83160*a^8*b*d*x^4+7470*a^4*b^3*c*x^4+48510*a^6*b^2*d*x^2- 
6156*a^2*b^4*c*x^2-4620*a^4*b^3*d+608*b^5*c)/a^4/(a*x+(a^2*x^2-b)^(1/2))^( 
9/2)+2^(1/2)*b^(3/4)*d*arctan(2^(1/2)*b^(1/4)*(a*x+(a^2*x^2-b)^(1/2))^(1/2 
)/(-b^(1/2)+a*x+(a^2*x^2-b)^(1/2)))+2^(1/2)*b^(3/4)*d*arctanh((1/2*b^(1/4) 
*2^(1/2)+1/2*a*x*2^(1/2)/b^(1/4)+1/2*(a^2*x^2-b)^(1/2)*2^(1/2)/b^(1/4))/(a 
*x+(a^2*x^2-b)^(1/2))^(1/2))
 

3.30.94.2 Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 391, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=\frac {608 b^5 c+3360 a^9 x^5 \left (11 d+3 c x^4\right ) \left (a x+\sqrt {-b+a^2 x^2}\right )-684 a b^4 c x \left (9 a x+4 \sqrt {-b+a^2 x^2}\right )+30 a^3 b^3 \left (-154 a d+249 a c x^4+247 c x^3 \sqrt {-b+a^2 x^2}\right )-120 a^7 b x^3 \left (\sqrt {-b+a^2 x^2} \left (539 d+135 c x^4\right )+3 a \left (231 d x+59 c x^5\right )\right )+30 a^5 b^2 x \left (\sqrt {-b+a^2 x^2} \left (693 d+89 c x^4\right )+a \left (1617 d x+317 c x^5\right )\right )}{3465 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{9/2}}+\sqrt {2} b^{3/4} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}{-\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}\right )+\sqrt {2} b^{3/4} d \text {arctanh}\left (\frac {\sqrt {b}+a x+\sqrt {-b+a^2 x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt {a x+\sqrt {-b+a^2 x^2}}}\right ) \]

input

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

output

(608*b^5*c + 3360*a^9*x^5*(11*d + 3*c*x^4)*(a*x + Sqrt[-b + a^2*x^2]) - 68 
4*a*b^4*c*x*(9*a*x + 4*Sqrt[-b + a^2*x^2]) + 30*a^3*b^3*(-154*a*d + 249*a* 
c*x^4 + 247*c*x^3*Sqrt[-b + a^2*x^2]) - 120*a^7*b*x^3*(Sqrt[-b + a^2*x^2]* 
(539*d + 135*c*x^4) + 3*a*(231*d*x + 59*c*x^5)) + 30*a^5*b^2*x*(Sqrt[-b + 
a^2*x^2]*(693*d + 89*c*x^4) + a*(1617*d*x + 317*c*x^5)))/(3465*a^4*(a*x + 
Sqrt[-b + a^2*x^2])^(9/2)) + Sqrt[2]*b^(3/4)*d*ArcTan[(Sqrt[2]*b^(1/4)*Sqr 
t[a*x + Sqrt[-b + a^2*x^2]])/(-Sqrt[b] + a*x + Sqrt[-b + a^2*x^2])] + Sqrt 
[2]*b^(3/4)*d*ArcTanh[(Sqrt[b] + a*x + Sqrt[-b + a^2*x^2])/(Sqrt[2]*b^(1/4 
)*Sqrt[a*x + Sqrt[-b + a^2*x^2]])]
 

3.30.94.3 Rubi [A] (verified)

Time = 1.77 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.30.94.3.1 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] 
]
 

3.30.94.4 Maple [F]

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

input

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

output

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

3.30.94.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {-b+a^2 x^2} \left (d+c x^4\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}}}{x} \, dx=-\frac {3465 \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} + \left (-b^{3} d^{4}\right )^{\frac {3}{4}}\right ) - 3465 i \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} + i \, \left (-b^{3} d^{4}\right )^{\frac {3}{4}}\right ) + 3465 i \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} - i \, \left (-b^{3} d^{4}\right )^{\frac {3}{4}}\right ) - 3465 \, \left (-b^{3} d^{4}\right )^{\frac {1}{4}} a^{4} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} - b}} b^{2} d^{3} - \left (-b^{3} d^{4}\right )^{\frac {3}{4}}\right ) + 2 \, {\left (35 \, a^{5} c x^{5} - 19 \, a^{3} b c x^{3} + {\left (1155 \, a^{5} d - 152 \, a b^{2} c\right )} x - 2 \, {\left (175 \, a^{4} c x^{4} - 57 \, a^{2} b c x^{2} + 1155 \, a^{4} d - 152 \, b^{2} c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}{3465 \, a^{4}} \]

input

integrate((a^2*x^2-b)^(1/2)*(c*x^4+d)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/x,x, a 
lgorithm="fricas")
 

output

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

3.30.94.6 Sympy [F]

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

input

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

output

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

3.30.94.7 Maxima [F]

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

input

integrate((a^2*x^2-b)^(1/2)*(c*x^4+d)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/x,x, a 
lgorithm="maxima")
 

output

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

3.30.94.8 Giac [F]

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

input

integrate((a^2*x^2-b)^(1/2)*(c*x^4+d)*(a*x+(a^2*x^2-b)^(1/2))^(1/2)/x,x, a 
lgorithm="giac")
 

output

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

3.30.94.9 Mupad [F(-1)]

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

input

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

output

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