3.27.53 \(\int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [2653]

3.27.53.1 Optimal result

Integrand size = 34, antiderivative size = 236 \[ \int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {b^8 c}{72 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{9/2}}-\frac {b^6 c}{20 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}+\frac {2 d}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{12 a^4}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}{56 a^4}+\frac {2 d \arctan \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \]

output

1/72*b^8*c/a^4/(a*x+(a^2*x^2+b^2)^(1/2))^(9/2)-1/20*b^6*c/a^4/(a*x+(a^2*x^ 
2+b^2)^(1/2))^(5/2)+2*d/(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)-1/12*b^2*c*(a*x+(a 
^2*x^2+b^2)^(1/2))^(3/2)/a^4+1/56*c*(a*x+(a^2*x^2+b^2)^(1/2))^(7/2)/a^4+2* 
d*arctan((a*x+(a^2*x^2+b^2)^(1/2))^(1/2)/b^(1/2))/b^(1/2)-2*d*arctanh((a*x 
+(a^2*x^2+b^2)^(1/2))^(1/2)/b^(1/2))/b^(1/2)
 

3.27.53.2 Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00 \[ \int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {b^8 c}{72 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{9/2}}-\frac {b^6 c}{20 a^4 \left (a x+\sqrt {b^2+a^2 x^2}\right )^{5/2}}+\frac {2 d}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {b^2 c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{12 a^4}+\frac {c \left (a x+\sqrt {b^2+a^2 x^2}\right )^{7/2}}{56 a^4}+\frac {2 d \arctan \left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \]

input

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

output

(b^8*c)/(72*a^4*(a*x + Sqrt[b^2 + a^2*x^2])^(9/2)) - (b^6*c)/(20*a^4*(a*x 
+ Sqrt[b^2 + a^2*x^2])^(5/2)) + (2*d)/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - (b 
^2*c*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/(12*a^4) + (c*(a*x + Sqrt[b^2 + a^ 
2*x^2])^(7/2))/(56*a^4) + (2*d*ArcTan[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt 
[b]])/Sqrt[b] - (2*d*ArcTanh[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/Sqr 
t[b]
 

3.27.53.3 Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 d \arctan \left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {2 d}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}-\frac {b^2 c \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{12 a^4}+\frac {c \left (\sqrt {a^2 x^2+b^2}+a x\right )^{7/2}}{56 a^4}+\frac {b^8 c}{72 a^4 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{9/2}}-\frac {b^6 c}{20 a^4 \left (\sqrt {a^2 x^2+b^2}+a x\right )^{5/2}}\)

input

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

output

(b^8*c)/(72*a^4*(a*x + Sqrt[b^2 + a^2*x^2])^(9/2)) - (b^6*c)/(20*a^4*(a*x 
+ Sqrt[b^2 + a^2*x^2])^(5/2)) + (2*d)/Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - (b 
^2*c*(a*x + Sqrt[b^2 + a^2*x^2])^(3/2))/(12*a^4) + (c*(a*x + Sqrt[b^2 + a^ 
2*x^2])^(7/2))/(56*a^4) + (2*d*ArcTan[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt 
[b]])/Sqrt[b] - (2*d*ArcTanh[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]/Sqrt[b]])/Sqr 
t[b]
 

3.27.53.3.1 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] 
]
 

3.27.53.4 Maple [F]

\[\int \frac {c \,x^{4}+d}{x \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]

input

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

output

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

3.27.53.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.04 \[ \int \frac {d+c x^4}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\left [\frac {630 \, a^{4} b^{\frac {3}{2}} d \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{\sqrt {b}}\right ) + 315 \, a^{4} b^{\frac {3}{2}} d \log \left (\frac {b^{2} + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x - b\right )} \sqrt {b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {b}\right )} + \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) - 2 \, {\left (35 \, a^{5} c x^{5} + a^{3} b^{2} c x^{3} - {\left (8 \, a b^{4} c - 315 \, a^{5} d\right )} x - {\left (35 \, a^{4} c x^{4} + 6 \, a^{2} b^{2} c x^{2} - 16 \, b^{4} c + 315 \, a^{4} d\right )} \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{315 \, a^{4} b^{2}}, \frac {630 \, a^{4} \sqrt {-b} b d \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \sqrt {-b}}{b}\right ) - 315 \, a^{4} \sqrt {-b} b d \log \left (-\frac {b^{2} + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x + b\right )} \sqrt {-b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {-b}\right )} - \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) - 2 \, {\left (35 \, a^{5} c x^{5} + a^{3} b^{2} c x^{3} - {\left (8 \, a b^{4} c - 315 \, a^{5} d\right )} x - {\left (35 \, a^{4} c x^{4} + 6 \, a^{2} b^{2} c x^{2} - 16 \, b^{4} c + 315 \, a^{4} d\right )} \sqrt {a^{2} x^{2} + b^{2}}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{315 \, a^{4} b^{2}}\right ] \]

input

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

output

[1/315*(630*a^4*b^(3/2)*d*arctan(sqrt(a*x + sqrt(a^2*x^2 + b^2))/sqrt(b)) 
+ 315*a^4*b^(3/2)*d*log((b^2 + sqrt(a*x + sqrt(a^2*x^2 + b^2))*((a*x - b)* 
sqrt(b) - sqrt(a^2*x^2 + b^2)*sqrt(b)) + sqrt(a^2*x^2 + b^2)*b)/x) - 2*(35 
*a^5*c*x^5 + a^3*b^2*c*x^3 - (8*a*b^4*c - 315*a^5*d)*x - (35*a^4*c*x^4 + 6 
*a^2*b^2*c*x^2 - 16*b^4*c + 315*a^4*d)*sqrt(a^2*x^2 + b^2))*sqrt(a*x + sqr 
t(a^2*x^2 + b^2)))/(a^4*b^2), 1/315*(630*a^4*sqrt(-b)*b*d*arctan(sqrt(a*x 
+ sqrt(a^2*x^2 + b^2))*sqrt(-b)/b) - 315*a^4*sqrt(-b)*b*d*log(-(b^2 + sqrt 
(a*x + sqrt(a^2*x^2 + b^2))*((a*x + b)*sqrt(-b) - sqrt(a^2*x^2 + b^2)*sqrt 
(-b)) - sqrt(a^2*x^2 + b^2)*b)/x) - 2*(35*a^5*c*x^5 + a^3*b^2*c*x^3 - (8*a 
*b^4*c - 315*a^5*d)*x - (35*a^4*c*x^4 + 6*a^2*b^2*c*x^2 - 16*b^4*c + 315*a 
^4*d)*sqrt(a^2*x^2 + b^2))*sqrt(a*x + sqrt(a^2*x^2 + b^2)))/(a^4*b^2)]
 

3.27.53.6 Sympy [F]

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

input

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

output

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

3.27.53.7 Maxima [F]

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

input

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

output

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

3.27.53.8 Giac [F]

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

input

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

output

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

3.27.53.9 Mupad [F(-1)]

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

input

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

output

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