\(\int x^2 \sqrt {c+d \sqrt {a+b x^2}} \, dx\) [250]

Optimal result

Integrand size = 23, antiderivative size = 406 \[ \int x^2 \sqrt {c+d \sqrt {a+b x^2}} \, dx=\frac {2 \left (5 a-\frac {4 c^2}{d^2}\right ) x \sqrt {c+d \sqrt {a+b x^2}}}{105 b}+\frac {2}{7} x^3 \sqrt {c+d \sqrt {a+b x^2}}+\frac {2 c x \sqrt {a+b x^2} \sqrt {c+d \sqrt {a+b x^2}}}{35 b d}-\frac {16 \sqrt {a} c \left (c^2-2 a d^2\right ) \sqrt {-\frac {b x^2}{a}} \sqrt {c+d \sqrt {a+b x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{105 b^2 d^3 x \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}}}+\frac {4 \sqrt {a} \left (4 c^4-9 a c^2 d^2+5 a^2 d^4\right ) \sqrt {-\frac {b x^2}{a}} \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{105 b^2 d^3 x \sqrt {c+d \sqrt {a+b x^2}}} \] Output:

2/105*(5*a-4*c^2/d^2)*x*(c+d*(b*x^2+a)^(1/2))^(1/2)/b+2/7*x^3*(c+d*(b*x^2+ 
a)^(1/2))^(1/2)+2/35*c*x*(b*x^2+a)^(1/2)*(c+d*(b*x^2+a)^(1/2))^(1/2)/b/d-1 
6/105*a^(1/2)*c*(-2*a*d^2+c^2)*(-b*x^2/a)^(1/2)*(c+d*(b*x^2+a)^(1/2))^(1/2 
)*EllipticE(1/2*(1-(b*x^2+a)^(1/2)/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2) 
*d/(c+a^(1/2)*d))^(1/2))/b^2/d^3/x/((c+d*(b*x^2+a)^(1/2))/(c+a^(1/2)*d))^( 
1/2)+4/105*a^(1/2)*(5*a^2*d^4-9*a*c^2*d^2+4*c^4)*(-b*x^2/a)^(1/2)*((c+d*(b 
*x^2+a)^(1/2))/(c+a^(1/2)*d))^(1/2)*EllipticF(1/2*(1-(b*x^2+a)^(1/2)/a^(1/ 
2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(c+a^(1/2)*d))^(1/2))/b^2/d^3/x/(c+d* 
(b*x^2+a)^(1/2))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 33.88 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.36 \[ \int x^2 \sqrt {c+d \sqrt {a+b x^2}} \, dx=\frac {2 \left (8 b c d^2 \left (c^2-2 a d^2\right ) x^2+b d^2 x^2 \left (c+d \sqrt {a+b x^2}\right ) \left (-4 c^2+3 c d \sqrt {a+b x^2}+5 d^2 \left (a+3 b x^2\right )\right )-\frac {8 i c \left (c^3+\sqrt {a} c^2 d-2 a c d^2-2 a^{3/2} d^3\right ) \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right )|\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )}{\sqrt {-c-\sqrt {a} d}}+\frac {2 i \sqrt {a} d \left (4 c^3+\sqrt {a} c^2 d-8 a c d^2-5 a^{3/2} d^3\right ) \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right ),\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )}{\sqrt {-c-\sqrt {a} d}}\right )}{105 b^2 d^4 x \sqrt {c+d \sqrt {a+b x^2}}} \] Input:

Integrate[x^2*Sqrt[c + d*Sqrt[a + b*x^2]],x]
 

Output:

(2*(8*b*c*d^2*(c^2 - 2*a*d^2)*x^2 + b*d^2*x^2*(c + d*Sqrt[a + b*x^2])*(-4* 
c^2 + 3*c*d*Sqrt[a + b*x^2] + 5*d^2*(a + 3*b*x^2)) - ((8*I)*c*(c^3 + Sqrt[ 
a]*c^2*d - 2*a*c*d^2 - 2*a^(3/2)*d^3)*Sqrt[(d*(-Sqrt[a] + Sqrt[a + b*x^2]) 
)/(c + d*Sqrt[a + b*x^2])]*Sqrt[(d*(Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqr 
t[a + b*x^2])]*(c + d*Sqrt[a + b*x^2])^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c - 
 Sqrt[a]*d]/Sqrt[c + d*Sqrt[a + b*x^2]]], (c - Sqrt[a]*d)/(c + Sqrt[a]*d)] 
)/Sqrt[-c - Sqrt[a]*d] + ((2*I)*Sqrt[a]*d*(4*c^3 + Sqrt[a]*c^2*d - 8*a*c*d 
^2 - 5*a^(3/2)*d^3)*Sqrt[(d*(-Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt[a + 
b*x^2])]*Sqrt[(d*(Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*(c 
+ d*Sqrt[a + b*x^2])^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c - Sqrt[a]*d]/Sqrt[c 
 + d*Sqrt[a + b*x^2]]], (c - Sqrt[a]*d)/(c + Sqrt[a]*d)])/Sqrt[-c - Sqrt[a 
]*d]))/(105*b^2*d^4*x*Sqrt[c + d*Sqrt[a + b*x^2]])
 

Rubi [F]

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.

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

\(\Big \downarrow \) 7299

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

Input:

Int[x^2*Sqrt[c + d*Sqrt[a + b*x^2]],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(sqrt(b*x^2 + a)*d + c)*x^2, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(sqrt(a + b*x**2)*d + c)*x**2,x)