\(\int x^2 (c+d \sqrt {a+b x^2})^p \, dx\) [296]

Optimal result

Integrand size = 21, antiderivative size = 184 \[ \int x^2 \left (c+d \sqrt {a+b x^2}\right )^p \, dx=\frac {1}{3} x^3 \left (c+d \sqrt {a+b x^2}\right )^p-\frac {x^3 \left (c+d \sqrt {a+b x^2}\right )^p \operatorname {AppellF1}\left (p,-\frac {3}{2},-\frac {3}{2},1+p,\frac {c+d \sqrt {a+b x^2}}{c-\sqrt {a} d},\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}\right )}{3 \left (1-\frac {c+d \sqrt {a+b x^2}}{c-\sqrt {a} d}\right )^{3/2} \left (1-\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}\right )^{3/2}} \] Output:

1/3*x^3*(c+d*(b*x^2+a)^(1/2))^p-1/3*x^3*(c+d*(b*x^2+a)^(1/2))^p*AppellF1(p 
,-3/2,-3/2,p+1,(c+d*(b*x^2+a)^(1/2))/(c-a^(1/2)*d),(c+d*(b*x^2+a)^(1/2))/( 
c+a^(1/2)*d))/(1-(c+d*(b*x^2+a)^(1/2))/(c-a^(1/2)*d))^(3/2)/(1-(c+d*(b*x^2 
+a)^(1/2))/(c+a^(1/2)*d))^(3/2)
 

Mathematica [F]

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

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

Output:

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

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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