\(\int (d x)^m (a^2+2 a b x^2+b^2 x^4)^{5/2} \, dx\) [678]

Optimal result

Integrand size = 28, antiderivative size = 313 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {a^5 (d x)^{1+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d (1+m) \left (a+b x^2\right )}+\frac {5 a^4 b (d x)^{3+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^3 (3+m) \left (a+b x^2\right )}+\frac {10 a^3 b^2 (d x)^{5+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^5 (5+m) \left (a+b x^2\right )}+\frac {10 a^2 b^3 (d x)^{7+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^7 (7+m) \left (a+b x^2\right )}+\frac {5 a b^4 (d x)^{9+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^9 (9+m) \left (a+b x^2\right )}+\frac {b^5 (d x)^{11+m} \sqrt {a^2+2 a b x^2+b^2 x^4}}{d^{11} (11+m) \left (a+b x^2\right )} \] Output:

a^5*(d*x)^(1+m)*((b*x^2+a)^2)^(1/2)/d/(1+m)/(b*x^2+a)+5*a^4*b*(d*x)^(3+m)* 
((b*x^2+a)^2)^(1/2)/d^3/(3+m)/(b*x^2+a)+10*a^3*b^2*(d*x)^(5+m)*((b*x^2+a)^ 
2)^(1/2)/d^5/(5+m)/(b*x^2+a)+10*a^2*b^3*(d*x)^(7+m)*((b*x^2+a)^2)^(1/2)/d^ 
7/(7+m)/(b*x^2+a)+5*a*b^4*(d*x)^(9+m)*((b*x^2+a)^2)^(1/2)/d^9/(9+m)/(b*x^2 
+a)+b^5*(d*x)^(11+m)*((b*x^2+a)^2)^(1/2)/d^11/(11+m)/(b*x^2+a)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.35 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {x (d x)^m \left (\left (a+b x^2\right )^2\right )^{5/2} \left (\frac {a^5}{1+m}+\frac {5 a^4 b x^2}{3+m}+\frac {10 a^3 b^2 x^4}{5+m}+\frac {10 a^2 b^3 x^6}{7+m}+\frac {5 a b^4 x^8}{9+m}+\frac {b^5 x^{10}}{11+m}\right )}{\left (a+b x^2\right )^5} \] Input:

Integrate[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

(x*(d*x)^m*((a + b*x^2)^2)^(5/2)*(a^5/(1 + m) + (5*a^4*b*x^2)/(3 + m) + (1 
0*a^3*b^2*x^4)/(5 + m) + (10*a^2*b^3*x^6)/(7 + m) + (5*a*b^4*x^8)/(9 + m) 
+ (b^5*x^10)/(11 + m)))/(a + b*x^2)^5
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.51, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1384, 27, 244, 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.

\(\displaystyle \int \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} (d x)^m \, dx\)

\(\Big \downarrow \) 1384

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int b^5 (d x)^m \left (b x^2+a\right )^5dx}{b^5 \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int (d x)^m \left (b x^2+a\right )^5dx}{a+b x^2}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \int \left (a^5 (d x)^m+\frac {5 a^4 b (d x)^{m+2}}{d^2}+\frac {10 a^3 b^2 (d x)^{m+4}}{d^4}+\frac {10 a^2 b^3 (d x)^{m+6}}{d^6}+\frac {5 a b^4 (d x)^{m+8}}{d^8}+\frac {b^5 (d x)^{m+10}}{d^{10}}\right )dx}{a+b x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {a^2+2 a b x^2+b^2 x^4} \left (\frac {a^5 (d x)^{m+1}}{d (m+1)}+\frac {5 a^4 b (d x)^{m+3}}{d^3 (m+3)}+\frac {10 a^3 b^2 (d x)^{m+5}}{d^5 (m+5)}+\frac {10 a^2 b^3 (d x)^{m+7}}{d^7 (m+7)}+\frac {5 a b^4 (d x)^{m+9}}{d^9 (m+9)}+\frac {b^5 (d x)^{m+11}}{d^{11} (m+11)}\right )}{a+b x^2}\)

Input:

Int[(d*x)^m*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
 

Output:

(Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]*((a^5*(d*x)^(1 + m))/(d*(1 + m)) + (5*a^4 
*b*(d*x)^(3 + m))/(d^3*(3 + m)) + (10*a^3*b^2*(d*x)^(5 + m))/(d^5*(5 + m)) 
 + (10*a^2*b^3*(d*x)^(7 + m))/(d^7*(7 + m)) + (5*a*b^4*(d*x)^(9 + m))/(d^9 
*(9 + m)) + (b^5*(d*x)^(11 + m))/(d^11*(11 + m))))/(a + b*x^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 1384

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S 
imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac 
Part[p]))   Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] 
&& EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n 
- 1)] && NeQ[u, x^(2*n - 1)] &&  !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.45

method	result	size
gosper	\(\frac {x \left (b^{5} m^{5} x^{10}+25 b^{5} m^{4} x^{10}+5 a \,b^{4} m^{5} x^{8}+230 b^{5} m^{3} x^{10}+135 a \,b^{4} m^{4} x^{8}+950 b^{5} m^{2} x^{10}+10 a^{2} b^{3} m^{5} x^{6}+1310 a \,b^{4} m^{3} x^{8}+1689 m \,x^{10} b^{5}+290 a^{2} b^{3} m^{4} x^{6}+5610 a \,b^{4} m^{2} x^{8}+945 x^{10} b^{5}+10 a^{3} b^{2} m^{5} x^{4}+3020 a^{2} b^{3} m^{3} x^{6}+10205 m \,x^{8} b^{4} a +310 a^{3} b^{2} m^{4} x^{4}+13660 a^{2} b^{3} m^{2} x^{6}+5775 a \,x^{8} b^{4}+5 a^{4} b \,m^{5} x^{2}+3500 a^{3} b^{2} m^{3} x^{4}+25770 m \,x^{6} a^{2} b^{3}+165 a^{4} b \,m^{4} x^{2}+17300 a^{3} b^{2} m^{2} x^{4}+14850 a^{2} x^{6} b^{3}+a^{5} m^{5}+2030 a^{4} b \,m^{3} x^{2}+34890 m \,x^{4} a^{3} b^{2}+35 a^{5} m^{4}+11310 a^{4} b \,m^{2} x^{2}+20790 a^{3} x^{4} b^{2}+470 a^{5} m^{3}+26765 m \,x^{2} b \,a^{4}+3010 a^{5} m^{2}+17325 x^{2} a^{4} b +9129 m \,a^{5}+10395 a^{5}\right ) \left (d x \right )^{m} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {5}{2}}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right ) \left (b \,x^{2}+a \right )^{5}}\)	\(453\)
risch	\(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (b^{5} m^{5} x^{10}+25 b^{5} m^{4} x^{10}+5 a \,b^{4} m^{5} x^{8}+230 b^{5} m^{3} x^{10}+135 a \,b^{4} m^{4} x^{8}+950 b^{5} m^{2} x^{10}+10 a^{2} b^{3} m^{5} x^{6}+1310 a \,b^{4} m^{3} x^{8}+1689 m \,x^{10} b^{5}+290 a^{2} b^{3} m^{4} x^{6}+5610 a \,b^{4} m^{2} x^{8}+945 x^{10} b^{5}+10 a^{3} b^{2} m^{5} x^{4}+3020 a^{2} b^{3} m^{3} x^{6}+10205 m \,x^{8} b^{4} a +310 a^{3} b^{2} m^{4} x^{4}+13660 a^{2} b^{3} m^{2} x^{6}+5775 a \,x^{8} b^{4}+5 a^{4} b \,m^{5} x^{2}+3500 a^{3} b^{2} m^{3} x^{4}+25770 m \,x^{6} a^{2} b^{3}+165 a^{4} b \,m^{4} x^{2}+17300 a^{3} b^{2} m^{2} x^{4}+14850 a^{2} x^{6} b^{3}+a^{5} m^{5}+2030 a^{4} b \,m^{3} x^{2}+34890 m \,x^{4} a^{3} b^{2}+35 a^{5} m^{4}+11310 a^{4} b \,m^{2} x^{2}+20790 a^{3} x^{4} b^{2}+470 a^{5} m^{3}+26765 m \,x^{2} b \,a^{4}+3010 a^{5} m^{2}+17325 x^{2} a^{4} b +9129 m \,a^{5}+10395 a^{5}\right ) x \left (d x \right )^{m}}{\left (b \,x^{2}+a \right ) \left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\)	\(453\)
orering	\(\frac {\left (b^{5} m^{5} x^{10}+25 b^{5} m^{4} x^{10}+5 a \,b^{4} m^{5} x^{8}+230 b^{5} m^{3} x^{10}+135 a \,b^{4} m^{4} x^{8}+950 b^{5} m^{2} x^{10}+10 a^{2} b^{3} m^{5} x^{6}+1310 a \,b^{4} m^{3} x^{8}+1689 m \,x^{10} b^{5}+290 a^{2} b^{3} m^{4} x^{6}+5610 a \,b^{4} m^{2} x^{8}+945 x^{10} b^{5}+10 a^{3} b^{2} m^{5} x^{4}+3020 a^{2} b^{3} m^{3} x^{6}+10205 m \,x^{8} b^{4} a +310 a^{3} b^{2} m^{4} x^{4}+13660 a^{2} b^{3} m^{2} x^{6}+5775 a \,x^{8} b^{4}+5 a^{4} b \,m^{5} x^{2}+3500 a^{3} b^{2} m^{3} x^{4}+25770 m \,x^{6} a^{2} b^{3}+165 a^{4} b \,m^{4} x^{2}+17300 a^{3} b^{2} m^{2} x^{4}+14850 a^{2} x^{6} b^{3}+a^{5} m^{5}+2030 a^{4} b \,m^{3} x^{2}+34890 m \,x^{4} a^{3} b^{2}+35 a^{5} m^{4}+11310 a^{4} b \,m^{2} x^{2}+20790 a^{3} x^{4} b^{2}+470 a^{5} m^{3}+26765 m \,x^{2} b \,a^{4}+3010 a^{5} m^{2}+17325 x^{2} a^{4} b +9129 m \,a^{5}+10395 a^{5}\right ) x \left (d x \right )^{m} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {5}{2}}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right ) \left (b \,x^{2}+a \right )^{5}}\)	\(462\)

Input:

int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

x*(b^5*m^5*x^10+25*b^5*m^4*x^10+5*a*b^4*m^5*x^8+230*b^5*m^3*x^10+135*a*b^4 
*m^4*x^8+950*b^5*m^2*x^10+10*a^2*b^3*m^5*x^6+1310*a*b^4*m^3*x^8+1689*b^5*m 
*x^10+290*a^2*b^3*m^4*x^6+5610*a*b^4*m^2*x^8+945*b^5*x^10+10*a^3*b^2*m^5*x 
^4+3020*a^2*b^3*m^3*x^6+10205*a*b^4*m*x^8+310*a^3*b^2*m^4*x^4+13660*a^2*b^ 
3*m^2*x^6+5775*a*b^4*x^8+5*a^4*b*m^5*x^2+3500*a^3*b^2*m^3*x^4+25770*a^2*b^ 
3*m*x^6+165*a^4*b*m^4*x^2+17300*a^3*b^2*m^2*x^4+14850*a^2*b^3*x^6+a^5*m^5+ 
2030*a^4*b*m^3*x^2+34890*a^3*b^2*m*x^4+35*a^5*m^4+11310*a^4*b*m^2*x^2+2079 
0*a^3*b^2*x^4+470*a^5*m^3+26765*a^4*b*m*x^2+3010*a^5*m^2+17325*a^4*b*x^2+9 
129*a^5*m+10395*a^5)*(d*x)^m*((b*x^2+a)^2)^(5/2)/(11+m)/(9+m)/(7+m)/(5+m)/ 
(3+m)/(1+m)/(b*x^2+a)^5
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.18 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {{\left ({\left (b^{5} m^{5} + 25 \, b^{5} m^{4} + 230 \, b^{5} m^{3} + 950 \, b^{5} m^{2} + 1689 \, b^{5} m + 945 \, b^{5}\right )} x^{11} + 5 \, {\left (a b^{4} m^{5} + 27 \, a b^{4} m^{4} + 262 \, a b^{4} m^{3} + 1122 \, a b^{4} m^{2} + 2041 \, a b^{4} m + 1155 \, a b^{4}\right )} x^{9} + 10 \, {\left (a^{2} b^{3} m^{5} + 29 \, a^{2} b^{3} m^{4} + 302 \, a^{2} b^{3} m^{3} + 1366 \, a^{2} b^{3} m^{2} + 2577 \, a^{2} b^{3} m + 1485 \, a^{2} b^{3}\right )} x^{7} + 10 \, {\left (a^{3} b^{2} m^{5} + 31 \, a^{3} b^{2} m^{4} + 350 \, a^{3} b^{2} m^{3} + 1730 \, a^{3} b^{2} m^{2} + 3489 \, a^{3} b^{2} m + 2079 \, a^{3} b^{2}\right )} x^{5} + 5 \, {\left (a^{4} b m^{5} + 33 \, a^{4} b m^{4} + 406 \, a^{4} b m^{3} + 2262 \, a^{4} b m^{2} + 5353 \, a^{4} b m + 3465 \, a^{4} b\right )} x^{3} + {\left (a^{5} m^{5} + 35 \, a^{5} m^{4} + 470 \, a^{5} m^{3} + 3010 \, a^{5} m^{2} + 9129 \, a^{5} m + 10395 \, a^{5}\right )} x\right )} \left (d x\right )^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \] Input:

integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="fricas")
 

Output:

((b^5*m^5 + 25*b^5*m^4 + 230*b^5*m^3 + 950*b^5*m^2 + 1689*b^5*m + 945*b^5) 
*x^11 + 5*(a*b^4*m^5 + 27*a*b^4*m^4 + 262*a*b^4*m^3 + 1122*a*b^4*m^2 + 204 
1*a*b^4*m + 1155*a*b^4)*x^9 + 10*(a^2*b^3*m^5 + 29*a^2*b^3*m^4 + 302*a^2*b 
^3*m^3 + 1366*a^2*b^3*m^2 + 2577*a^2*b^3*m + 1485*a^2*b^3)*x^7 + 10*(a^3*b 
^2*m^5 + 31*a^3*b^2*m^4 + 350*a^3*b^2*m^3 + 1730*a^3*b^2*m^2 + 3489*a^3*b^ 
2*m + 2079*a^3*b^2)*x^5 + 5*(a^4*b*m^5 + 33*a^4*b*m^4 + 406*a^4*b*m^3 + 22 
62*a^4*b*m^2 + 5353*a^4*b*m + 3465*a^4*b)*x^3 + (a^5*m^5 + 35*a^5*m^4 + 47 
0*a^5*m^3 + 3010*a^5*m^2 + 9129*a^5*m + 10395*a^5)*x)*(d*x)^m/(m^6 + 36*m^ 
5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)
 

Sympy [F]

\[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\int \left (d x\right )^{m} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}\, dx \] Input:

integrate((d*x)**m*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
 

Output:

Integral((d*x)**m*((a + b*x**2)**2)**(5/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.78 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {{\left ({\left (m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945\right )} b^{5} d^{m} x^{11} + 5 \, {\left (m^{5} + 27 \, m^{4} + 262 \, m^{3} + 1122 \, m^{2} + 2041 \, m + 1155\right )} a b^{4} d^{m} x^{9} + 10 \, {\left (m^{5} + 29 \, m^{4} + 302 \, m^{3} + 1366 \, m^{2} + 2577 \, m + 1485\right )} a^{2} b^{3} d^{m} x^{7} + 10 \, {\left (m^{5} + 31 \, m^{4} + 350 \, m^{3} + 1730 \, m^{2} + 3489 \, m + 2079\right )} a^{3} b^{2} d^{m} x^{5} + 5 \, {\left (m^{5} + 33 \, m^{4} + 406 \, m^{3} + 2262 \, m^{2} + 5353 \, m + 3465\right )} a^{4} b d^{m} x^{3} + {\left (m^{5} + 35 \, m^{4} + 470 \, m^{3} + 3010 \, m^{2} + 9129 \, m + 10395\right )} a^{5} d^{m} x\right )} x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \] Input:

integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="maxima")
 

Output:

((m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)*b^5*d^m*x^11 + 5*(m^5 + 
 27*m^4 + 262*m^3 + 1122*m^2 + 2041*m + 1155)*a*b^4*d^m*x^9 + 10*(m^5 + 29 
*m^4 + 302*m^3 + 1366*m^2 + 2577*m + 1485)*a^2*b^3*d^m*x^7 + 10*(m^5 + 31* 
m^4 + 350*m^3 + 1730*m^2 + 3489*m + 2079)*a^3*b^2*d^m*x^5 + 5*(m^5 + 33*m^ 
4 + 406*m^3 + 2262*m^2 + 5353*m + 3465)*a^4*b*d^m*x^3 + (m^5 + 35*m^4 + 47 
0*m^3 + 3010*m^2 + 9129*m + 10395)*a^5*d^m*x)*x^m/(m^6 + 36*m^5 + 505*m^4 
+ 3480*m^3 + 12139*m^2 + 19524*m + 10395)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 900 vs. \(2 (247) = 494\).

Time = 0.16 (sec) , antiderivative size = 900, normalized size of antiderivative = 2.88 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x, algorithm="giac")
 

Output:

((d*x)^m*b^5*m^5*x^11*sgn(b*x^2 + a) + 25*(d*x)^m*b^5*m^4*x^11*sgn(b*x^2 + 
 a) + 5*(d*x)^m*a*b^4*m^5*x^9*sgn(b*x^2 + a) + 230*(d*x)^m*b^5*m^3*x^11*sg 
n(b*x^2 + a) + 135*(d*x)^m*a*b^4*m^4*x^9*sgn(b*x^2 + a) + 950*(d*x)^m*b^5* 
m^2*x^11*sgn(b*x^2 + a) + 10*(d*x)^m*a^2*b^3*m^5*x^7*sgn(b*x^2 + a) + 1310 
*(d*x)^m*a*b^4*m^3*x^9*sgn(b*x^2 + a) + 1689*(d*x)^m*b^5*m*x^11*sgn(b*x^2 
+ a) + 290*(d*x)^m*a^2*b^3*m^4*x^7*sgn(b*x^2 + a) + 5610*(d*x)^m*a*b^4*m^2 
*x^9*sgn(b*x^2 + a) + 945*(d*x)^m*b^5*x^11*sgn(b*x^2 + a) + 10*(d*x)^m*a^3 
*b^2*m^5*x^5*sgn(b*x^2 + a) + 3020*(d*x)^m*a^2*b^3*m^3*x^7*sgn(b*x^2 + a) 
+ 10205*(d*x)^m*a*b^4*m*x^9*sgn(b*x^2 + a) + 310*(d*x)^m*a^3*b^2*m^4*x^5*s 
gn(b*x^2 + a) + 13660*(d*x)^m*a^2*b^3*m^2*x^7*sgn(b*x^2 + a) + 5775*(d*x)^ 
m*a*b^4*x^9*sgn(b*x^2 + a) + 5*(d*x)^m*a^4*b*m^5*x^3*sgn(b*x^2 + a) + 3500 
*(d*x)^m*a^3*b^2*m^3*x^5*sgn(b*x^2 + a) + 25770*(d*x)^m*a^2*b^3*m*x^7*sgn( 
b*x^2 + a) + 165*(d*x)^m*a^4*b*m^4*x^3*sgn(b*x^2 + a) + 17300*(d*x)^m*a^3* 
b^2*m^2*x^5*sgn(b*x^2 + a) + 14850*(d*x)^m*a^2*b^3*x^7*sgn(b*x^2 + a) + (d 
*x)^m*a^5*m^5*x*sgn(b*x^2 + a) + 2030*(d*x)^m*a^4*b*m^3*x^3*sgn(b*x^2 + a) 
 + 34890*(d*x)^m*a^3*b^2*m*x^5*sgn(b*x^2 + a) + 35*(d*x)^m*a^5*m^4*x*sgn(b 
*x^2 + a) + 11310*(d*x)^m*a^4*b*m^2*x^3*sgn(b*x^2 + a) + 20790*(d*x)^m*a^3 
*b^2*x^5*sgn(b*x^2 + a) + 470*(d*x)^m*a^5*m^3*x*sgn(b*x^2 + a) + 26765*(d* 
x)^m*a^4*b*m*x^3*sgn(b*x^2 + a) + 3010*(d*x)^m*a^5*m^2*x*sgn(b*x^2 + a) + 
17325*(d*x)^m*a^4*b*x^3*sgn(b*x^2 + a) + 9129*(d*x)^m*a^5*m*x*sgn(b*x^2...
 

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\int {\left (d\,x\right )}^m\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2} \,d x \] Input:

int((d*x)^m*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2),x)
 

Output:

int((d*x)^m*(a^2 + b^2*x^4 + 2*a*b*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.38 \[ \int (d x)^m \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2} \, dx=\frac {x^{m} d^{m} x \left (b^{5} m^{5} x^{10}+25 b^{5} m^{4} x^{10}+5 a \,b^{4} m^{5} x^{8}+230 b^{5} m^{3} x^{10}+135 a \,b^{4} m^{4} x^{8}+950 b^{5} m^{2} x^{10}+10 a^{2} b^{3} m^{5} x^{6}+1310 a \,b^{4} m^{3} x^{8}+1689 b^{5} m \,x^{10}+290 a^{2} b^{3} m^{4} x^{6}+5610 a \,b^{4} m^{2} x^{8}+945 b^{5} x^{10}+10 a^{3} b^{2} m^{5} x^{4}+3020 a^{2} b^{3} m^{3} x^{6}+10205 a \,b^{4} m \,x^{8}+310 a^{3} b^{2} m^{4} x^{4}+13660 a^{2} b^{3} m^{2} x^{6}+5775 a \,b^{4} x^{8}+5 a^{4} b \,m^{5} x^{2}+3500 a^{3} b^{2} m^{3} x^{4}+25770 a^{2} b^{3} m \,x^{6}+165 a^{4} b \,m^{4} x^{2}+17300 a^{3} b^{2} m^{2} x^{4}+14850 a^{2} b^{3} x^{6}+a^{5} m^{5}+2030 a^{4} b \,m^{3} x^{2}+34890 a^{3} b^{2} m \,x^{4}+35 a^{5} m^{4}+11310 a^{4} b \,m^{2} x^{2}+20790 a^{3} b^{2} x^{4}+470 a^{5} m^{3}+26765 a^{4} b m \,x^{2}+3010 a^{5} m^{2}+17325 a^{4} b \,x^{2}+9129 a^{5} m +10395 a^{5}\right )}{m^{6}+36 m^{5}+505 m^{4}+3480 m^{3}+12139 m^{2}+19524 m +10395} \] Input:

int((d*x)^m*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
 

Output:

(x**m*d**m*x*(a**5*m**5 + 35*a**5*m**4 + 470*a**5*m**3 + 3010*a**5*m**2 + 
9129*a**5*m + 10395*a**5 + 5*a**4*b*m**5*x**2 + 165*a**4*b*m**4*x**2 + 203 
0*a**4*b*m**3*x**2 + 11310*a**4*b*m**2*x**2 + 26765*a**4*b*m*x**2 + 17325* 
a**4*b*x**2 + 10*a**3*b**2*m**5*x**4 + 310*a**3*b**2*m**4*x**4 + 3500*a**3 
*b**2*m**3*x**4 + 17300*a**3*b**2*m**2*x**4 + 34890*a**3*b**2*m*x**4 + 207 
90*a**3*b**2*x**4 + 10*a**2*b**3*m**5*x**6 + 290*a**2*b**3*m**4*x**6 + 302 
0*a**2*b**3*m**3*x**6 + 13660*a**2*b**3*m**2*x**6 + 25770*a**2*b**3*m*x**6 
 + 14850*a**2*b**3*x**6 + 5*a*b**4*m**5*x**8 + 135*a*b**4*m**4*x**8 + 1310 
*a*b**4*m**3*x**8 + 5610*a*b**4*m**2*x**8 + 10205*a*b**4*m*x**8 + 5775*a*b 
**4*x**8 + b**5*m**5*x**10 + 25*b**5*m**4*x**10 + 230*b**5*m**3*x**10 + 95 
0*b**5*m**2*x**10 + 1689*b**5*m*x**10 + 945*b**5*x**10))/(m**6 + 36*m**5 + 
 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)