\(\int (c x^2+d \sqrt {a+\frac {c^2 x^4}{d^2}})^p \, dx\) [46]

Optimal result

Integrand size = 26, antiderivative size = 250 \[ \int \left (c x^2+d \sqrt {a+\frac {c^2 x^4}{d^2}}\right )^p \, dx=\frac {x \left (c x^2+d \sqrt {a+\frac {c^2 x^4}{d^2}}\right )^p}{1-2 p}-\frac {2 \sqrt {2} p x \left (c x^2+d \sqrt {a+\frac {c^2 x^4}{d^2}}\right )^{2+p} \sqrt {1-\frac {\left (c x^2+d \sqrt {a+\frac {c^2 x^4}{d^2}}\right )^2}{a d^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} (3+2 p),\frac {1}{4} (7+2 p),\frac {\left (c x^2+d \sqrt {a+\frac {c^2 x^4}{d^2}}\right )^2}{a d^2}\right )}{\left (3-4 p-4 p^2\right ) \sqrt {c x^2 \left (c x^2+d \sqrt {a+\frac {c^2 x^4}{d^2}}\right )} \sqrt {-a d^2+\left (c x^2+d \sqrt {a+\frac {c^2 x^4}{d^2}}\right )^2}} \] Output:

x*(c*x^2+d*(a+c^2*x^4/d^2)^(1/2))^p/(1-2*p)-2*2^(1/2)*p*x*(c*x^2+d*(a+c^2* 
x^4/d^2)^(1/2))^(2+p)*(1-(c*x^2+d*(a+c^2*x^4/d^2)^(1/2))^2/a/d^2)^(1/2)*hy 
pergeom([1/2, 3/4+1/2*p],[7/4+1/2*p],(c*x^2+d*(a+c^2*x^4/d^2)^(1/2))^2/a/d 
^2)/(-4*p^2-4*p+3)/(c*x^2*(c*x^2+d*(a+c^2*x^4/d^2)^(1/2)))^(1/2)/(-a*d^2+( 
c*x^2+d*(a+c^2*x^4/d^2)^(1/2))^2)^(1/2)
 

Mathematica [F]

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

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

Output:

Integrate[(c*x^2 + d*Sqrt[a + (c^2*x^4)/d^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.

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

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (d \sqrt {a+\frac {c^2 x^4}{d^2}}+c x^2\right )^pdx\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((c*x^2 + d*sqrt((c^2*x^4 + a*d^2)/d^2))^p, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(a*d**2 + c**2*x**4) + c*x**2)**p,x)