\(\int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx\) [256]

Optimal result

Integrand size = 40, antiderivative size = 268 \[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) d \sqrt {\sqrt {3}-2 i x^2}}{\left (2 i c^2-\sqrt {3} d^2\right ) (c+d x)}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) d \sqrt {\sqrt {3}+2 i x^2}}{\left (2 i c^2+\sqrt {3} d^2\right ) (c+d x)}+\frac {(1+i) c \arctan \left (\frac {\sqrt {3} d+2 i c x}{\sqrt {2 i c^2-\sqrt {3} d^2} \sqrt {\sqrt {3}-2 i x^2}}\right )}{\left (2 i c^2-\sqrt {3} d^2\right )^{3/2}}+\frac {(1-i) c \text {arctanh}\left (\frac {\sqrt {3} d-2 i c x}{\sqrt {2 i c^2+\sqrt {3} d^2} \sqrt {\sqrt {3}+2 i x^2}}\right )}{\left (2 i c^2+\sqrt {3} d^2\right )^{3/2}} \] Output:

(1/2-1/2*I)*d*(3^(1/2)-2*I*x^2)^(1/2)/(2*I*c^2-3^(1/2)*d^2)/(d*x+c)-(1/2+1 
/2*I)*d*(3^(1/2)+2*I*x^2)^(1/2)/(2*I*c^2+3^(1/2)*d^2)/(d*x+c)+(1+I)*c*arct 
an((3^(1/2)*d+2*I*c*x)/(2*I*c^2-3^(1/2)*d^2)^(1/2)/(3^(1/2)-2*I*x^2)^(1/2) 
)/(2*I*c^2-3^(1/2)*d^2)^(3/2)+(1-I)*c*arctanh((3^(1/2)*d-2*I*c*x)/(2*I*c^2 
+3^(1/2)*d^2)^(1/2)/(3^(1/2)+2*I*x^2)^(1/2))/(2*I*c^2+3^(1/2)*d^2)^(3/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 5.87 (sec) , antiderivative size = 1464, normalized size of antiderivative = 5.46 \[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx =\text {Too large to display} \] Input:

Integrate[Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]/((c + d*x)^2*Sqrt[3 + 4*x^4]),x]
 

Output:

-((d*Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]*(3*d^2*(2*x^2 + Sqrt[3 + 4*x^4]) + 2*c^ 
2*(3 + 8*x^4 + 4*x^2*Sqrt[3 + 4*x^4])))/((4*c^4 + 3*d^4)*(c + d*x)*(3 + 8* 
x^4 + 4*x^2*Sqrt[3 + 4*x^4]))) + (8*c^5*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4* 
x^4]])/Sqrt[-2*c^2 - Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)^(3/2)*Sqrt[-2 
*c^2 - Sqrt[4*c^4 + 3*d^4]]) - (6*c*d^4*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4* 
x^4]])/Sqrt[-2*c^2 - Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)^(3/2)*Sqrt[-2 
*c^2 - Sqrt[4*c^4 + 3*d^4]]) + (4*c^3*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4*x^ 
4]])/Sqrt[-2*c^2 - Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)*Sqrt[-2*c^2 - S 
qrt[4*c^4 + 3*d^4]]) - (8*c^5*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4*x^4]])/Sqr 
t[-2*c^2 + Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)^(3/2)*Sqrt[-2*c^2 + Sqr 
t[4*c^4 + 3*d^4]]) + (6*c*d^4*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4*x^4]])/Sqr 
t[-2*c^2 + Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)^(3/2)*Sqrt[-2*c^2 + Sqr 
t[4*c^4 + 3*d^4]]) + (4*c^3*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4*x^4]])/Sqrt[ 
-2*c^2 + Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)*Sqrt[-2*c^2 + Sqrt[4*c^4 
+ 3*d^4]]) - RootSum[9*d^2 - 24*c^2*#1 - 6*d^2*#1^2 - 8*c^2*#1^3 + d^2*#1^ 
4 & , (128*c^4*Log[2*x^2 + Sqrt[3 + 4*x^4] + 2*x*Sqrt[2*x^2 + Sqrt[3 + 4*x 
^4]] - #1] + 3*d^4*Log[2*x^2 + Sqrt[3 + 4*x^4] + 2*x*Sqrt[2*x^2 + Sqrt[3 + 
 4*x^4]] - #1] + 16*c^2*d^2*Log[2*x^2 + Sqrt[3 + 4*x^4] + 2*x*Sqrt[2*x^2 + 
 Sqrt[3 + 4*x^4]] - #1]*#1 + d^4*Log[2*x^2 + Sqrt[3 + 4*x^4] + 2*x*Sqrt[2* 
x^2 + Sqrt[3 + 4*x^4]] - #1]*#1^2)/(6*c^2 + 3*d^2*#1 + 6*c^2*#1^2 - d^2...
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2558, 491, 488, 217, 219}

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.

\(\displaystyle \int \frac {\sqrt {\sqrt {4 x^4+3}+2 x^2}}{\sqrt {4 x^4+3} (c+d x)^2} \, dx\)

\(\Big \downarrow \) 2558

\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{(c+d x)^2 \sqrt {\sqrt {3}-2 i x^2}}dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(c+d x)^2 \sqrt {2 i x^2+\sqrt {3}}}dx\)

\(\Big \downarrow \) 491

\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {2 c \int \frac {1}{(c+d x) \sqrt {\sqrt {3}-2 i x^2}}dx}{2 c^2+i \sqrt {3} d^2}+\frac {d \sqrt {\sqrt {3}-2 i x^2}}{\left (-\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2 c \int \frac {1}{(c+d x) \sqrt {2 i x^2+\sqrt {3}}}dx}{2 c^2-i \sqrt {3} d^2}-\frac {d \sqrt {\sqrt {3}+2 i x^2}}{\left (\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )\)

\(\Big \downarrow \) 488

\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {d \sqrt {\sqrt {3}-2 i x^2}}{\left (-\sqrt {3} d^2+2 i c^2\right ) (c+d x)}-\frac {2 c \int \frac {1}{-2 i c^2+\sqrt {3} d^2-\frac {\left (\sqrt {3} d+2 i c x\right )^2}{\sqrt {3}-2 i x^2}}d\frac {\sqrt {3} d+2 i c x}{\sqrt {\sqrt {3}-2 i x^2}}}{2 c^2+i \sqrt {3} d^2}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {2 c \int \frac {1}{2 i c^2+\sqrt {3} d^2-\frac {\left (\sqrt {3} d-2 i c x\right )^2}{2 i x^2+\sqrt {3}}}d\frac {\sqrt {3} d-2 i c x}{\sqrt {2 i x^2+\sqrt {3}}}}{2 c^2-i \sqrt {3} d^2}-\frac {d \sqrt {\sqrt {3}+2 i x^2}}{\left (\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {2 c \int \frac {1}{2 i c^2+\sqrt {3} d^2-\frac {\left (\sqrt {3} d-2 i c x\right )^2}{2 i x^2+\sqrt {3}}}d\frac {\sqrt {3} d-2 i c x}{\sqrt {2 i x^2+\sqrt {3}}}}{2 c^2-i \sqrt {3} d^2}-\frac {d \sqrt {\sqrt {3}+2 i x^2}}{\left (\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {2 c \arctan \left (\frac {\sqrt {3} d+2 i c x}{\sqrt {\sqrt {3}-2 i x^2} \sqrt {-\sqrt {3} d^2+2 i c^2}}\right )}{\sqrt {-\sqrt {3} d^2+2 i c^2} \left (2 c^2+i \sqrt {3} d^2\right )}+\frac {d \sqrt {\sqrt {3}-2 i x^2}}{\left (-\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {2 c \arctan \left (\frac {\sqrt {3} d+2 i c x}{\sqrt {\sqrt {3}-2 i x^2} \sqrt {-\sqrt {3} d^2+2 i c^2}}\right )}{\sqrt {-\sqrt {3} d^2+2 i c^2} \left (2 c^2+i \sqrt {3} d^2\right )}+\frac {d \sqrt {\sqrt {3}-2 i x^2}}{\left (-\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {2 c \text {arctanh}\left (\frac {\sqrt {3} d-2 i c x}{\sqrt {\sqrt {3}+2 i x^2} \sqrt {\sqrt {3} d^2+2 i c^2}}\right )}{\left (2 c^2-i \sqrt {3} d^2\right ) \sqrt {\sqrt {3} d^2+2 i c^2}}-\frac {d \sqrt {\sqrt {3}+2 i x^2}}{\left (\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )\)

Input:

Int[Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]/((c + d*x)^2*Sqrt[3 + 4*x^4]),x]
 

Output:

(1/2 - I/2)*((d*Sqrt[Sqrt[3] - (2*I)*x^2])/(((2*I)*c^2 - Sqrt[3]*d^2)*(c + 
 d*x)) + (2*c*ArcTan[(Sqrt[3]*d + (2*I)*c*x)/(Sqrt[(2*I)*c^2 - Sqrt[3]*d^2 
]*Sqrt[Sqrt[3] - (2*I)*x^2])])/(Sqrt[(2*I)*c^2 - Sqrt[3]*d^2]*(2*c^2 + I*S 
qrt[3]*d^2))) + (1/2 + I/2)*(-((d*Sqrt[Sqrt[3] + (2*I)*x^2])/(((2*I)*c^2 + 
 Sqrt[3]*d^2)*(c + d*x))) - (2*c*ArcTanh[(Sqrt[3]*d - (2*I)*c*x)/(Sqrt[(2* 
I)*c^2 + Sqrt[3]*d^2]*Sqrt[Sqrt[3] + (2*I)*x^2])])/((2*c^2 - I*Sqrt[3]*d^2 
)*Sqrt[(2*I)*c^2 + Sqrt[3]*d^2]))
 

Defintions of rubi rules used

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 491

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S 
imp[b*(c/(b*c^2 + a*d^2))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, n, p}, x] && EqQ[n + 2*p + 3, 0]
 

rule 2558

Int[(((c_.) + (d_.)*(x_))^(m_.)*Sqrt[(b_.)*(x_)^2 + Sqrt[(a_) + (e_.)*(x_)^ 
4]])/Sqrt[(a_) + (e_.)*(x_)^4], x_Symbol] :> Simp[(1 - I)/2   Int[(c + d*x) 
^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Simp[(1 + I)/2   Int[(c + d*x)^m/Sqrt[ 
Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && G 
tQ[a, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\text {Timed out} \] Input:

integrate((2*x^2+(4*x^4+3)^(1/2))^(1/2)/(d*x+c)^2/(4*x^4+3)^(1/2),x, algor 
ithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\int \frac {\sqrt {2 x^{2} + \sqrt {4 x^{4} + 3}}}{\left (c + d x\right )^{2} \sqrt {4 x^{4} + 3}}\, dx \] Input:

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

Output:

Integral(sqrt(2*x**2 + sqrt(4*x**4 + 3))/((c + d*x)**2*sqrt(4*x**4 + 3)), 
x)
 

Maxima [F]

\[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\int { \frac {\sqrt {2 \, x^{2} + \sqrt {4 \, x^{4} + 3}}}{\sqrt {4 \, x^{4} + 3} {\left (d x + c\right )}^{2}} \,d x } \] Input:

integrate((2*x^2+(4*x^4+3)^(1/2))^(1/2)/(d*x+c)^2/(4*x^4+3)^(1/2),x, algor 
ithm="maxima")
 

Output:

integrate(sqrt(2*x^2 + sqrt(4*x^4 + 3))/(sqrt(4*x^4 + 3)*(d*x + c)^2), x)
 

Giac [F]

\[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\int { \frac {\sqrt {2 \, x^{2} + \sqrt {4 \, x^{4} + 3}}}{\sqrt {4 \, x^{4} + 3} {\left (d x + c\right )}^{2}} \,d x } \] Input:

integrate((2*x^2+(4*x^4+3)^(1/2))^(1/2)/(d*x+c)^2/(4*x^4+3)^(1/2),x, algor 
ithm="giac")
 

Output:

integrate(sqrt(2*x^2 + sqrt(4*x^4 + 3))/(sqrt(4*x^4 + 3)*(d*x + c)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\int \frac {\sqrt {2\,x^2+\sqrt {4\,x^4+3}}}{\sqrt {4\,x^4+3}\,{\left (c+d\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\text {too large to display} \] Input:

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

Output:

(16*sqrt(4*c**4 + 3*d**4)*sqrt(sqrt(4*c**4 + 3*d**4) - 2*c**2)*atan((2*sqr 
t(4*x**4 + 3)*sqrt(4*c**4 + 3*d**4)*sqrt(sqrt(4*x**4 + 3) + 2*x**2)*sqrt(s 
qrt(4*c**4 + 3*d**4) - 2*c**2)*c*x - 4*sqrt(4*c**4 + 3*d**4)*sqrt(sqrt(4*x 
**4 + 3) + 2*x**2)*sqrt(sqrt(4*c**4 + 3*d**4) - 2*c**2)*c*x**3 - 3*sqrt(4* 
c**4 + 3*d**4)*sqrt(sqrt(4*x**4 + 3) + 2*x**2)*sqrt(sqrt(4*c**4 + 3*d**4) 
- 2*c**2)*d + 4*sqrt(4*x**4 + 3)*sqrt(sqrt(4*x**4 + 3) + 2*x**2)*sqrt(sqrt 
(4*c**4 + 3*d**4) - 2*c**2)*c**3*x + 3*sqrt(4*x**4 + 3)*sqrt(sqrt(4*x**4 + 
 3) + 2*x**2)*sqrt(sqrt(4*c**4 + 3*d**4) - 2*c**2)*d**3 - 8*sqrt(sqrt(4*x* 
*4 + 3) + 2*x**2)*sqrt(sqrt(4*c**4 + 3*d**4) - 2*c**2)*c**3*x**3 - 6*sqrt( 
sqrt(4*x**4 + 3) + 2*x**2)*sqrt(sqrt(4*c**4 + 3*d**4) - 2*c**2)*c**2*d + 6 
*sqrt(sqrt(4*x**4 + 3) + 2*x**2)*sqrt(sqrt(4*c**4 + 3*d**4) - 2*c**2)*c*d* 
*2*x - 6*sqrt(sqrt(4*x**4 + 3) + 2*x**2)*sqrt(sqrt(4*c**4 + 3*d**4) - 2*c* 
*2)*d**3*x**2)/(24*c**2*d**2*x**2 - 18*d**4))*c**7 + 12*sqrt(4*c**4 + 3*d* 
*4)*sqrt(sqrt(4*c**4 + 3*d**4) - 2*c**2)*atan((2*sqrt(4*x**4 + 3)*sqrt(4*c 
**4 + 3*d**4)*sqrt(sqrt(4*x**4 + 3) + 2*x**2)*sqrt(sqrt(4*c**4 + 3*d**4) - 
 2*c**2)*c*x - 4*sqrt(4*c**4 + 3*d**4)*sqrt(sqrt(4*x**4 + 3) + 2*x**2)*sqr 
t(sqrt(4*c**4 + 3*d**4) - 2*c**2)*c*x**3 - 3*sqrt(4*c**4 + 3*d**4)*sqrt(sq 
rt(4*x**4 + 3) + 2*x**2)*sqrt(sqrt(4*c**4 + 3*d**4) - 2*c**2)*d + 4*sqrt(4 
*x**4 + 3)*sqrt(sqrt(4*x**4 + 3) + 2*x**2)*sqrt(sqrt(4*c**4 + 3*d**4) - 2* 
c**2)*c**3*x + 3*sqrt(4*x**4 + 3)*sqrt(sqrt(4*x**4 + 3) + 2*x**2)*sqrt(...