3.31.91 \(\int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx\) [3091]

3.31.91.1 Optimal result

Integrand size = 49, antiderivative size = 535 \[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {4 c \sqrt {-b+a^2 x^2} \left (-416 a b^2 x+455 a^3 b x^3+260 a^5 x^5\right )+4 c \left (128 b^3-676 a^2 b^2 x^2+325 a^4 b x^4+260 a^6 x^6\right )}{715 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}+\frac {\sqrt {2+\sqrt {2}} d \arctan \left (\frac {\left (\sqrt {\frac {2}{2-\sqrt {2}}} \sqrt [8]{b}-\frac {2 \sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}\right ) \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} d \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2-\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \text {arctanh}\left (\frac {\frac {\sqrt [8]{b}}{\sqrt {2+\sqrt {2}}}+\frac {\sqrt {a x+\sqrt {-b+a^2 x^2}}}{\sqrt {2+\sqrt {2}} \sqrt [8]{b}}}{\sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}} \]

output

1/715*(4*c*(a^2*x^2-b)^(1/2)*(260*a^5*x^5+455*a^3*b*x^3-416*a*b^2*x)+4*c*( 
260*a^6*x^6+325*a^4*b*x^4-676*a^2*b^2*x^2+128*b^3))/a^4/(a*x+(a^2*x^2-b)^( 
1/2))^(13/4)+(2+2^(1/2))^(1/2)*d*arctan((2^(1/2)/(2-2^(1/2))^(1/2)*b^(1/8) 
-2*b^(1/8)/(2-2^(1/2))^(1/2))*(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(-b^(1/4)+(a*x 
+(a^2*x^2-b)^(1/2))^(1/2)))/b^(5/8)+(2-2^(1/2))^(1/2)*d*arctan((2+2^(1/2)) 
^(1/2)*b^(1/8)*(a*x+(a^2*x^2-b)^(1/2))^(1/4)/(-b^(1/4)+(a*x+(a^2*x^2-b)^(1 
/2))^(1/2)))/b^(5/8)-(2+2^(1/2))^(1/2)*d*arctanh((b^(1/8)/(2-2^(1/2))^(1/2 
)+(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(2-2^(1/2))^(1/2)/b^(1/8))/(a*x+(a^2*x^2-b 
)^(1/2))^(1/4))/b^(5/8)+(2-2^(1/2))^(1/2)*d*arctanh((b^(1/8)/(2+2^(1/2))^( 
1/2)+(a*x+(a^2*x^2-b)^(1/2))^(1/2)/(2+2^(1/2))^(1/2)/b^(1/8))/(a*x+(a^2*x^ 
2-b)^(1/2))^(1/4))/b^(5/8)
 

3.31.91.2 Mathematica [A] (verified)

Time = 1.63 (sec) , antiderivative size = 477, normalized size of antiderivative = 0.89 \[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {4 c \left (128 b^3-676 a^2 b^2 x^2+325 a^4 b x^4+260 a^6 x^6+13 a x \sqrt {-b+a^2 x^2} \left (-32 b^2+35 a^2 b x^2+20 a^4 x^4\right )\right )}{715 a^4 \left (a x+\sqrt {-b+a^2 x^2}\right )^{13/4}}+\frac {\sqrt {2+\sqrt {2}} d \arctan \left (\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{\sqrt [4]{b}-\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \arctan \left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}{-\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}+\frac {\sqrt {2-\sqrt {2}} d \text {arctanh}\left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}}-\frac {\sqrt {2+\sqrt {2}} d \text {arctanh}\left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{b}+\sqrt {a x+\sqrt {-b+a^2 x^2}}\right )}{\sqrt [8]{b} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}}\right )}{b^{5/8}} \]

input

Integrate[(d + c*x^4)/(x*Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/ 
4)),x]
 

output

(4*c*(128*b^3 - 676*a^2*b^2*x^2 + 325*a^4*b*x^4 + 260*a^6*x^6 + 13*a*x*Sqr 
t[-b + a^2*x^2]*(-32*b^2 + 35*a^2*b*x^2 + 20*a^4*x^4)))/(715*a^4*(a*x + Sq 
rt[-b + a^2*x^2])^(13/4)) + (Sqrt[2 + Sqrt[2]]*d*ArcTan[(Sqrt[2 - Sqrt[2]] 
*b^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(b^(1/4) - Sqrt[a*x + Sqrt[-b + 
 a^2*x^2]])])/b^(5/8) + (Sqrt[2 - Sqrt[2]]*d*ArcTan[(Sqrt[2 + Sqrt[2]]*b^( 
1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b^(1/4) + Sqrt[a*x + Sqrt[-b + a^ 
2*x^2]])])/b^(5/8) + (Sqrt[2 - Sqrt[2]]*d*ArcTanh[(Sqrt[1 - 1/Sqrt[2]]*(b^ 
(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]))/(b^(1/8)*(a*x + Sqrt[-b + a^2*x^2 
])^(1/4))])/b^(5/8) - (Sqrt[2 + Sqrt[2]]*d*ArcTanh[(Sqrt[1 + 1/Sqrt[2]]*(b 
^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]))/(b^(1/8)*(a*x + Sqrt[-b + a^2*x^ 
2])^(1/4))])/b^(5/8)
 

3.31.91.3 Rubi [A] (verified)

Time = 2.00 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.92, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {c x^3}{\sqrt {a^2 x^2-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}+\frac {d}{x \sqrt {a^2 x^2-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 d \arctan \left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}+\frac {\sqrt {2} d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {\sqrt {2} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}+1\right )}{(-b)^{5/8}}-\frac {2 d \text {arctanh}\left (\frac {\sqrt [4]{\sqrt {a^2 x^2-b}+a x}}{\sqrt [8]{-b}}\right )}{(-b)^{5/8}}-\frac {d \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}-\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{\sqrt {2} (-b)^{5/8}}+\frac {d \log \left (\sqrt {\sqrt {a^2 x^2-b}+a x}+\sqrt {2} \sqrt [8]{-b} \sqrt [4]{\sqrt {a^2 x^2-b}+a x}+\sqrt [4]{-b}\right )}{\sqrt {2} (-b)^{5/8}}-\frac {b^3 c}{26 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{13/4}}-\frac {3 b^2 c}{10 a^4 \left (\sqrt {a^2 x^2-b}+a x\right )^{5/4}}+\frac {b c \left (\sqrt {a^2 x^2-b}+a x\right )^{3/4}}{2 a^4}+\frac {c \left (\sqrt {a^2 x^2-b}+a x\right )^{11/4}}{22 a^4}\)

input

Int[(d + c*x^4)/(x*Sqrt[-b + a^2*x^2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4)),x]
 

output

-1/26*(b^3*c)/(a^4*(a*x + Sqrt[-b + a^2*x^2])^(13/4)) - (3*b^2*c)/(10*a^4* 
(a*x + Sqrt[-b + a^2*x^2])^(5/4)) + (b*c*(a*x + Sqrt[-b + a^2*x^2])^(3/4)) 
/(2*a^4) + (c*(a*x + Sqrt[-b + a^2*x^2])^(11/4))/(22*a^4) + (2*d*ArcTan[(a 
*x + Sqrt[-b + a^2*x^2])^(1/4)/(-b)^(1/8)])/(-b)^(5/8) + (Sqrt[2]*d*ArcTan 
[1 - (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b)^(1/8)])/(-b)^(5/8) - 
(Sqrt[2]*d*ArcTan[1 + (Sqrt[2]*(a*x + Sqrt[-b + a^2*x^2])^(1/4))/(-b)^(1/8 
)])/(-b)^(5/8) - (2*d*ArcTanh[(a*x + Sqrt[-b + a^2*x^2])^(1/4)/(-b)^(1/8)] 
)/(-b)^(5/8) - (d*Log[(-b)^(1/4) - Sqrt[2]*(-b)^(1/8)*(a*x + Sqrt[-b + a^2 
*x^2])^(1/4) + Sqrt[a*x + Sqrt[-b + a^2*x^2]]])/(Sqrt[2]*(-b)^(5/8)) + (d* 
Log[(-b)^(1/4) + Sqrt[2]*(-b)^(1/8)*(a*x + Sqrt[-b + a^2*x^2])^(1/4) + Sqr 
t[a*x + Sqrt[-b + a^2*x^2]]])/(Sqrt[2]*(-b)^(5/8))
 

3.31.91.3.1 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] 
]
 

3.31.91.4 Maple [F]

\[\int \frac {c \,x^{4}+d}{x \sqrt {a^{2} x^{2}-b}\, \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {1}{4}}}d x\]

input

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

output

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

3.31.91.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.05 \[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\frac {-\left (715 i - 715\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} + \left (i + 1\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) + \left (715 i + 715\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} - \left (i - 1\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) - \left (715 i + 715\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} + \left (i - 1\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) + \left (715 i - 715\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left (2 \, {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} - \left (i + 1\right ) \, \sqrt {2} \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) - 1430 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} + \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) + 1430 i \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} + i \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) - 1430 i \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} - i \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) + 1430 \, \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {1}{8}} a^{4} b \log \left ({\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} d^{3} - \left (-\frac {d^{8}}{b^{5}}\right )^{\frac {3}{8}} b^{2}\right ) - 8 \, {\left (55 \, a^{4} c x^{4} + 36 \, a^{2} b c x^{2} - 128 \, b^{2} c - {\left (55 \, a^{3} c x^{3} + 96 \, a b c x\right )} \sqrt {a^{2} x^{2} - b}\right )} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {3}{4}}}{1430 \, a^{4} b} \]

input

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

output

1/1430*(-(715*I - 715)*sqrt(2)*(-d^8/b^5)^(1/8)*a^4*b*log(2*(a*x + sqrt(a^ 
2*x^2 - b))^(1/4)*d^3 + (I + 1)*sqrt(2)*(-d^8/b^5)^(3/8)*b^2) + (715*I + 7 
15)*sqrt(2)*(-d^8/b^5)^(1/8)*a^4*b*log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*d 
^3 - (I - 1)*sqrt(2)*(-d^8/b^5)^(3/8)*b^2) - (715*I + 715)*sqrt(2)*(-d^8/b 
^5)^(1/8)*a^4*b*log(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*d^3 + (I - 1)*sqrt(2 
)*(-d^8/b^5)^(3/8)*b^2) + (715*I - 715)*sqrt(2)*(-d^8/b^5)^(1/8)*a^4*b*log 
(2*(a*x + sqrt(a^2*x^2 - b))^(1/4)*d^3 - (I + 1)*sqrt(2)*(-d^8/b^5)^(3/8)* 
b^2) - 1430*(-d^8/b^5)^(1/8)*a^4*b*log((a*x + sqrt(a^2*x^2 - b))^(1/4)*d^3 
 + (-d^8/b^5)^(3/8)*b^2) + 1430*I*(-d^8/b^5)^(1/8)*a^4*b*log((a*x + sqrt(a 
^2*x^2 - b))^(1/4)*d^3 + I*(-d^8/b^5)^(3/8)*b^2) - 1430*I*(-d^8/b^5)^(1/8) 
*a^4*b*log((a*x + sqrt(a^2*x^2 - b))^(1/4)*d^3 - I*(-d^8/b^5)^(3/8)*b^2) + 
 1430*(-d^8/b^5)^(1/8)*a^4*b*log((a*x + sqrt(a^2*x^2 - b))^(1/4)*d^3 - (-d 
^8/b^5)^(3/8)*b^2) - 8*(55*a^4*c*x^4 + 36*a^2*b*c*x^2 - 128*b^2*c - (55*a^ 
3*c*x^3 + 96*a*b*c*x)*sqrt(a^2*x^2 - b))*(a*x + sqrt(a^2*x^2 - b))^(3/4))/ 
(a^4*b)
 

3.31.91.6 Sympy [F]

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

input

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

output

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

3.31.91.7 Maxima [F]

\[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int { \frac {c x^{4} + d}{\sqrt {a^{2} x^{2} - b} {\left (a x + \sqrt {a^{2} x^{2} - b}\right )}^{\frac {1}{4}} x} \,d x } \]

input

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

output

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

3.31.91.8 Giac [F(-1)]

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

input

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

output

Timed out
 

3.31.91.9 Mupad [F(-1)]

Timed out. \[ \int \frac {d+c x^4}{x \sqrt {-b+a^2 x^2} \sqrt [4]{a x+\sqrt {-b+a^2 x^2}}} \, dx=\int \frac {c\,x^4+d}{x\,{\left (a\,x+\sqrt {a^2\,x^2-b}\right )}^{1/4}\,\sqrt {a^2\,x^2-b}} \,d x \]

input

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

output

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