\(\int x^4 (a+b x^2)^2 (c+d x^2)^{5/2} \, dx\) [868]

Optimal result

Integrand size = 24, antiderivative size = 322 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=-\frac {c^4 \left (24 a^2 d^2+5 b c (b c-4 a d)\right ) x \sqrt {c+d x^2}}{2048 d^4}+\frac {c^3 \left (24 a^2 d^2+5 b c (b c-4 a d)\right ) x^3 \sqrt {c+d x^2}}{3072 d^3}+\frac {c^2 \left (24 a^2 d^2+5 b c (b c-4 a d)\right ) x^5 \sqrt {c+d x^2}}{768 d^2}+\frac {c \left (24 a^2 d^2+5 b c (b c-4 a d)\right ) x^5 \left (c+d x^2\right )^{3/2}}{384 d^2}+\frac {1}{240} \left (24 a^2+\frac {5 b c (b c-4 a d)}{d^2}\right ) x^5 \left (c+d x^2\right )^{5/2}-\frac {b (b c-4 a d) x^5 \left (c+d x^2\right )^{7/2}}{24 d^2}+\frac {b^2 x^7 \left (c+d x^2\right )^{7/2}}{14 d}+\frac {c^5 \left (24 a^2 d^2+5 b c (b c-4 a d)\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2048 d^{9/2}} \] Output:

-1/2048*c^4*(24*a^2*d^2+5*b*c*(-4*a*d+b*c))*x*(d*x^2+c)^(1/2)/d^4+1/3072*c 
^3*(24*a^2*d^2+5*b*c*(-4*a*d+b*c))*x^3*(d*x^2+c)^(1/2)/d^3+1/768*c^2*(24*a 
^2*d^2+5*b*c*(-4*a*d+b*c))*x^5*(d*x^2+c)^(1/2)/d^2+1/384*c*(24*a^2*d^2+5*b 
*c*(-4*a*d+b*c))*x^5*(d*x^2+c)^(3/2)/d^2+1/240*(24*a^2+5*b*c*(-4*a*d+b*c)/ 
d^2)*x^5*(d*x^2+c)^(5/2)-1/24*b*(-4*a*d+b*c)*x^5*(d*x^2+c)^(7/2)/d^2+1/14* 
b^2*x^7*(d*x^2+c)^(7/2)/d+1/2048*c^5*(24*a^2*d^2+5*b*c*(-4*a*d+b*c))*arcta 
nh(d^(1/2)*x/(d*x^2+c)^(1/2))/d^(9/2)
 

Mathematica [A] (verified)

Time = 1.30 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.92 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {\sqrt {d} x \sqrt {c+d x^2} \left (168 a^2 d^2 \left (-15 c^4+10 c^3 d x^2+248 c^2 d^2 x^4+336 c d^3 x^6+128 d^4 x^8\right )+140 a b d \left (15 c^5-10 c^4 d x^2+8 c^3 d^2 x^4+432 c^2 d^3 x^6+640 c d^4 x^8+256 d^5 x^{10}\right )-5 b^2 \left (105 c^6-70 c^5 d x^2+56 c^4 d^2 x^4-48 c^3 d^3 x^6-4736 c^2 d^4 x^8-7424 c d^5 x^{10}-3072 d^6 x^{12}\right )\right )+4200 a b c^6 d \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c}-\sqrt {c+d x^2}}\right )+210 c^5 \left (5 b^2 c^2+24 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} x}{-\sqrt {c}+\sqrt {c+d x^2}}\right )}{215040 d^{9/2}} \] Input:

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

Output:

(Sqrt[d]*x*Sqrt[c + d*x^2]*(168*a^2*d^2*(-15*c^4 + 10*c^3*d*x^2 + 248*c^2* 
d^2*x^4 + 336*c*d^3*x^6 + 128*d^4*x^8) + 140*a*b*d*(15*c^5 - 10*c^4*d*x^2 
+ 8*c^3*d^2*x^4 + 432*c^2*d^3*x^6 + 640*c*d^4*x^8 + 256*d^5*x^10) - 5*b^2* 
(105*c^6 - 70*c^5*d*x^2 + 56*c^4*d^2*x^4 - 48*c^3*d^3*x^6 - 4736*c^2*d^4*x 
^8 - 7424*c*d^5*x^10 - 3072*d^6*x^12)) + 4200*a*b*c^6*d*ArcTanh[(Sqrt[d]*x 
)/(Sqrt[c] - Sqrt[c + d*x^2])] + 210*c^5*(5*b^2*c^2 + 24*a^2*d^2)*ArcTanh[ 
(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/(215040*d^(9/2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.75, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {367, 27, 363, 248, 248, 248, 262, 262, 224, 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.

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

\(\Big \downarrow \) 367

\(\displaystyle \frac {\int 7 x^4 \left (d x^2+c\right )^{5/2} \left (2 a^2 d-b (b c-4 a d) x^2\right )dx}{14 d}+\frac {b^2 x^7 \left (c+d x^2\right )^{7/2}}{14 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int x^4 \left (d x^2+c\right )^{5/2} \left (2 a^2 d-b (b c-4 a d) x^2\right )dx}{2 d}+\frac {b^2 x^7 \left (c+d x^2\right )^{7/2}}{14 d}\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {\frac {\left (24 a^2 d^2+5 b c (b c-4 a d)\right ) \int x^4 \left (d x^2+c\right )^{5/2}dx}{12 d}-\frac {b x^5 \left (c+d x^2\right )^{7/2} (b c-4 a d)}{12 d}}{2 d}+\frac {b^2 x^7 \left (c+d x^2\right )^{7/2}}{14 d}\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {\frac {\left (24 a^2 d^2+5 b c (b c-4 a d)\right ) \left (\frac {1}{2} c \int x^4 \left (d x^2+c\right )^{3/2}dx+\frac {1}{10} x^5 \left (c+d x^2\right )^{5/2}\right )}{12 d}-\frac {b x^5 \left (c+d x^2\right )^{7/2} (b c-4 a d)}{12 d}}{2 d}+\frac {b^2 x^7 \left (c+d x^2\right )^{7/2}}{14 d}\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {\frac {\left (24 a^2 d^2+5 b c (b c-4 a d)\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \int x^4 \sqrt {d x^2+c}dx+\frac {1}{8} x^5 \left (c+d x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (c+d x^2\right )^{5/2}\right )}{12 d}-\frac {b x^5 \left (c+d x^2\right )^{7/2} (b c-4 a d)}{12 d}}{2 d}+\frac {b^2 x^7 \left (c+d x^2\right )^{7/2}}{14 d}\)

\(\Big \downarrow \) 248

\(\displaystyle \frac {\frac {\left (24 a^2 d^2+5 b c (b c-4 a d)\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \left (\frac {1}{6} c \int \frac {x^4}{\sqrt {d x^2+c}}dx+\frac {1}{6} x^5 \sqrt {c+d x^2}\right )+\frac {1}{8} x^5 \left (c+d x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (c+d x^2\right )^{5/2}\right )}{12 d}-\frac {b x^5 \left (c+d x^2\right )^{7/2} (b c-4 a d)}{12 d}}{2 d}+\frac {b^2 x^7 \left (c+d x^2\right )^{7/2}}{14 d}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\frac {\left (24 a^2 d^2+5 b c (b c-4 a d)\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \left (\frac {1}{6} c \left (\frac {x^3 \sqrt {c+d x^2}}{4 d}-\frac {3 c \int \frac {x^2}{\sqrt {d x^2+c}}dx}{4 d}\right )+\frac {1}{6} x^5 \sqrt {c+d x^2}\right )+\frac {1}{8} x^5 \left (c+d x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (c+d x^2\right )^{5/2}\right )}{12 d}-\frac {b x^5 \left (c+d x^2\right )^{7/2} (b c-4 a d)}{12 d}}{2 d}+\frac {b^2 x^7 \left (c+d x^2\right )^{7/2}}{14 d}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\frac {\left (24 a^2 d^2+5 b c (b c-4 a d)\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \left (\frac {1}{6} c \left (\frac {x^3 \sqrt {c+d x^2}}{4 d}-\frac {3 c \left (\frac {x \sqrt {c+d x^2}}{2 d}-\frac {c \int \frac {1}{\sqrt {d x^2+c}}dx}{2 d}\right )}{4 d}\right )+\frac {1}{6} x^5 \sqrt {c+d x^2}\right )+\frac {1}{8} x^5 \left (c+d x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (c+d x^2\right )^{5/2}\right )}{12 d}-\frac {b x^5 \left (c+d x^2\right )^{7/2} (b c-4 a d)}{12 d}}{2 d}+\frac {b^2 x^7 \left (c+d x^2\right )^{7/2}}{14 d}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\left (24 a^2 d^2+5 b c (b c-4 a d)\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \left (\frac {1}{6} c \left (\frac {x^3 \sqrt {c+d x^2}}{4 d}-\frac {3 c \left (\frac {x \sqrt {c+d x^2}}{2 d}-\frac {c \int \frac {1}{1-\frac {d x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{2 d}\right )}{4 d}\right )+\frac {1}{6} x^5 \sqrt {c+d x^2}\right )+\frac {1}{8} x^5 \left (c+d x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (c+d x^2\right )^{5/2}\right )}{12 d}-\frac {b x^5 \left (c+d x^2\right )^{7/2} (b c-4 a d)}{12 d}}{2 d}+\frac {b^2 x^7 \left (c+d x^2\right )^{7/2}}{14 d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (24 a^2 d^2+5 b c (b c-4 a d)\right ) \left (\frac {1}{2} c \left (\frac {3}{8} c \left (\frac {1}{6} c \left (\frac {x^3 \sqrt {c+d x^2}}{4 d}-\frac {3 c \left (\frac {x \sqrt {c+d x^2}}{2 d}-\frac {c \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{2 d^{3/2}}\right )}{4 d}\right )+\frac {1}{6} x^5 \sqrt {c+d x^2}\right )+\frac {1}{8} x^5 \left (c+d x^2\right )^{3/2}\right )+\frac {1}{10} x^5 \left (c+d x^2\right )^{5/2}\right )}{12 d}-\frac {b x^5 \left (c+d x^2\right )^{7/2} (b c-4 a d)}{12 d}}{2 d}+\frac {b^2 x^7 \left (c+d x^2\right )^{7/2}}{14 d}\)

Input:

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

Output:

(b^2*x^7*(c + d*x^2)^(7/2))/(14*d) + (-1/12*(b*(b*c - 4*a*d)*x^5*(c + d*x^ 
2)^(7/2))/d + ((24*a^2*d^2 + 5*b*c*(b*c - 4*a*d))*((x^5*(c + d*x^2)^(5/2)) 
/10 + (c*((x^5*(c + d*x^2)^(3/2))/8 + (3*c*((x^5*Sqrt[c + d*x^2])/6 + (c*( 
(x^3*Sqrt[c + d*x^2])/(4*d) - (3*c*((x*Sqrt[c + d*x^2])/(2*d) - (c*ArcTanh 
[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(2*d^(3/2))))/(4*d)))/6))/8))/2))/(12*d))/( 
2*d)
 

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 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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 248

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) 
  Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ 
p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 367

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, 
x_Symbol] :> Simp[d^2*(e*x)^(m + 3)*((a + b*x^2)^(p + 1)/(b*e^3*(m + 2*p + 
5))), x] + Simp[1/(b*(m + 2*p + 5))   Int[(e*x)^m*(a + b*x^2)^p*Simp[b*c^2* 
(m + 2*p + 5) - d*(a*d*(m + 3) - 2*b*c*(m + 2*p + 5))*x^2, x], x], x] /; Fr 
eeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + 2*p + 5, 0]
 

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.72

method	result	size
pseudoelliptic	\(-\frac {3 \left (\left (-a^{2} c^{5} d^{2}+\frac {5}{6} a b \,c^{6} d -\frac {5}{24} b^{2} c^{7}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}}{x \sqrt {d}}\right )+\left (-\frac {112 c \left (\frac {290}{441} b^{2} x^{4}+\frac {100}{63} a b \,x^{2}+a^{2}\right ) x^{6} d^{\frac {11}{2}}}{5}-\frac {128 \left (\frac {5}{7} b^{2} x^{4}+\frac {5}{3} a b \,x^{2}+a^{2}\right ) x^{8} d^{\frac {13}{2}}}{15}+c^{2} \left (c^{2} \left (\frac {1}{9} b^{2} x^{4}+\frac {5}{9} a b \,x^{2}+a^{2}\right ) d^{\frac {5}{2}}-\frac {2 c \left (\frac {1}{7} b^{2} x^{4}+\frac {2}{3} a b \,x^{2}+a^{2}\right ) x^{2} d^{\frac {7}{2}}}{3}+\left (-\frac {592}{63} b^{2} x^{8}-24 a b \,x^{6}-\frac {248}{15} a^{2} x^{4}\right ) d^{\frac {9}{2}}-\frac {5 c^{3} \left (\left (\frac {b \,x^{2}}{6}+a \right ) d^{\frac {3}{2}}-\frac {b c \sqrt {d}}{4}\right ) b}{6}\right )\right ) \sqrt {x^{2} d +c}\, x \right )}{256 d^{\frac {9}{2}}}\)	\(233\)
risch	\(-\frac {x \left (-15360 b^{2} d^{6} x^{12}-35840 a b \,d^{6} x^{10}-37120 b^{2} c \,d^{5} x^{10}-21504 a^{2} d^{6} x^{8}-89600 a b c \,d^{5} x^{8}-23680 b^{2} c^{2} d^{4} x^{8}-56448 a^{2} c \,d^{5} x^{6}-60480 a b \,c^{2} d^{4} x^{6}-240 b^{2} c^{3} d^{3} x^{6}-41664 a^{2} c^{2} d^{4} x^{4}-1120 a b \,c^{3} d^{3} x^{4}+280 b^{2} c^{4} d^{2} x^{4}-1680 a^{2} c^{3} d^{3} x^{2}+1400 a b \,c^{4} d^{2} x^{2}-350 b^{2} c^{5} d \,x^{2}+2520 a^{2} c^{4} d^{2}-2100 a b \,c^{5} d +525 b^{2} c^{6}\right ) \sqrt {x^{2} d +c}}{215040 d^{4}}+\frac {c^{5} \left (24 a^{2} d^{2}-20 a b c d +5 b^{2} c^{2}\right ) \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{2048 d^{\frac {9}{2}}}\)	\(280\)
default	\(a^{2} \left (\frac {x^{3} \left (x^{2} d +c \right )^{\frac {7}{2}}}{10 d}-\frac {3 c \left (\frac {x \left (x^{2} d +c \right )^{\frac {7}{2}}}{8 d}-\frac {c \left (\frac {x \left (x^{2} d +c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (x^{2} d +c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {x^{2} d +c}}{2}+\frac {c \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{8 d}\right )}{10 d}\right )+b^{2} \left (\frac {x^{7} \left (x^{2} d +c \right )^{\frac {7}{2}}}{14 d}-\frac {c \left (\frac {x^{5} \left (x^{2} d +c \right )^{\frac {7}{2}}}{12 d}-\frac {5 c \left (\frac {x^{3} \left (x^{2} d +c \right )^{\frac {7}{2}}}{10 d}-\frac {3 c \left (\frac {x \left (x^{2} d +c \right )^{\frac {7}{2}}}{8 d}-\frac {c \left (\frac {x \left (x^{2} d +c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (x^{2} d +c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {x^{2} d +c}}{2}+\frac {c \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{8 d}\right )}{10 d}\right )}{12 d}\right )}{2 d}\right )+2 a b \left (\frac {x^{5} \left (x^{2} d +c \right )^{\frac {7}{2}}}{12 d}-\frac {5 c \left (\frac {x^{3} \left (x^{2} d +c \right )^{\frac {7}{2}}}{10 d}-\frac {3 c \left (\frac {x \left (x^{2} d +c \right )^{\frac {7}{2}}}{8 d}-\frac {c \left (\frac {x \left (x^{2} d +c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (x^{2} d +c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {x^{2} d +c}}{2}+\frac {c \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{2 \sqrt {d}}\right )}{4}\right )}{6}\right )}{8 d}\right )}{10 d}\right )}{12 d}\right )\)	\(425\)

Input:

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

Output:

-3/256/d^(9/2)*((-a^2*c^5*d^2+5/6*a*b*c^6*d-5/24*b^2*c^7)*arctanh((d*x^2+c 
)^(1/2)/x/d^(1/2))+(-112/5*c*(290/441*b^2*x^4+100/63*a*b*x^2+a^2)*x^6*d^(1 
1/2)-128/15*(5/7*b^2*x^4+5/3*a*b*x^2+a^2)*x^8*d^(13/2)+c^2*(c^2*(1/9*b^2*x 
^4+5/9*a*b*x^2+a^2)*d^(5/2)-2/3*c*(1/7*b^2*x^4+2/3*a*b*x^2+a^2)*x^2*d^(7/2 
)+(-592/63*b^2*x^8-24*a*b*x^6-248/15*a^2*x^4)*d^(9/2)-5/6*c^3*((1/6*b*x^2+ 
a)*d^(3/2)-1/4*b*c*d^(1/2))*b))*(d*x^2+c)^(1/2)*x)
 

Fricas [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.78 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\left [\frac {105 \, {\left (5 \, b^{2} c^{7} - 20 \, a b c^{6} d + 24 \, a^{2} c^{5} d^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + 2 \, {\left (15360 \, b^{2} d^{7} x^{13} + 1280 \, {\left (29 \, b^{2} c d^{6} + 28 \, a b d^{7}\right )} x^{11} + 128 \, {\left (185 \, b^{2} c^{2} d^{5} + 700 \, a b c d^{6} + 168 \, a^{2} d^{7}\right )} x^{9} + 48 \, {\left (5 \, b^{2} c^{3} d^{4} + 1260 \, a b c^{2} d^{5} + 1176 \, a^{2} c d^{6}\right )} x^{7} - 56 \, {\left (5 \, b^{2} c^{4} d^{3} - 20 \, a b c^{3} d^{4} - 744 \, a^{2} c^{2} d^{5}\right )} x^{5} + 70 \, {\left (5 \, b^{2} c^{5} d^{2} - 20 \, a b c^{4} d^{3} + 24 \, a^{2} c^{3} d^{4}\right )} x^{3} - 105 \, {\left (5 \, b^{2} c^{6} d - 20 \, a b c^{5} d^{2} + 24 \, a^{2} c^{4} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{430080 \, d^{5}}, -\frac {105 \, {\left (5 \, b^{2} c^{7} - 20 \, a b c^{6} d + 24 \, a^{2} c^{5} d^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) - {\left (15360 \, b^{2} d^{7} x^{13} + 1280 \, {\left (29 \, b^{2} c d^{6} + 28 \, a b d^{7}\right )} x^{11} + 128 \, {\left (185 \, b^{2} c^{2} d^{5} + 700 \, a b c d^{6} + 168 \, a^{2} d^{7}\right )} x^{9} + 48 \, {\left (5 \, b^{2} c^{3} d^{4} + 1260 \, a b c^{2} d^{5} + 1176 \, a^{2} c d^{6}\right )} x^{7} - 56 \, {\left (5 \, b^{2} c^{4} d^{3} - 20 \, a b c^{3} d^{4} - 744 \, a^{2} c^{2} d^{5}\right )} x^{5} + 70 \, {\left (5 \, b^{2} c^{5} d^{2} - 20 \, a b c^{4} d^{3} + 24 \, a^{2} c^{3} d^{4}\right )} x^{3} - 105 \, {\left (5 \, b^{2} c^{6} d - 20 \, a b c^{5} d^{2} + 24 \, a^{2} c^{4} d^{3}\right )} x\right )} \sqrt {d x^{2} + c}}{215040 \, d^{5}}\right ] \] Input:

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

Output:

[1/430080*(105*(5*b^2*c^7 - 20*a*b*c^6*d + 24*a^2*c^5*d^2)*sqrt(d)*log(-2* 
d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 2*(15360*b^2*d^7*x^13 + 1280*(2 
9*b^2*c*d^6 + 28*a*b*d^7)*x^11 + 128*(185*b^2*c^2*d^5 + 700*a*b*c*d^6 + 16 
8*a^2*d^7)*x^9 + 48*(5*b^2*c^3*d^4 + 1260*a*b*c^2*d^5 + 1176*a^2*c*d^6)*x^ 
7 - 56*(5*b^2*c^4*d^3 - 20*a*b*c^3*d^4 - 744*a^2*c^2*d^5)*x^5 + 70*(5*b^2* 
c^5*d^2 - 20*a*b*c^4*d^3 + 24*a^2*c^3*d^4)*x^3 - 105*(5*b^2*c^6*d - 20*a*b 
*c^5*d^2 + 24*a^2*c^4*d^3)*x)*sqrt(d*x^2 + c))/d^5, -1/215040*(105*(5*b^2* 
c^7 - 20*a*b*c^6*d + 24*a^2*c^5*d^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 
 + c)) - (15360*b^2*d^7*x^13 + 1280*(29*b^2*c*d^6 + 28*a*b*d^7)*x^11 + 128 
*(185*b^2*c^2*d^5 + 700*a*b*c*d^6 + 168*a^2*d^7)*x^9 + 48*(5*b^2*c^3*d^4 + 
 1260*a*b*c^2*d^5 + 1176*a^2*c*d^6)*x^7 - 56*(5*b^2*c^4*d^3 - 20*a*b*c^3*d 
^4 - 744*a^2*c^2*d^5)*x^5 + 70*(5*b^2*c^5*d^2 - 20*a*b*c^4*d^3 + 24*a^2*c^ 
3*d^4)*x^3 - 105*(5*b^2*c^6*d - 20*a*b*c^5*d^2 + 24*a^2*c^4*d^3)*x)*sqrt(d 
*x^2 + c))/d^5]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 809 vs. \(2 (308) = 616\).

Time = 0.59 (sec) , antiderivative size = 809, normalized size of antiderivative = 2.51 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((3*c**2*(a**2*c**3 - 5*c*(3*a**2*c**2*d + 2*a*b*c**3 - 7*c*(3*a* 
*2*c*d**2 + 6*a*b*c**2*d + b**2*c**3 - 9*c*(a**2*d**3 + 6*a*b*c*d**2 + 3*b 
**2*c**2*d - 11*c*(2*a*b*d**3 + 29*b**2*c*d**2/14)/(12*d))/(10*d))/(8*d))/ 
(6*d))*Piecewise((log(2*sqrt(d)*sqrt(c + d*x**2) + 2*d*x)/sqrt(d), Ne(c, 0 
)), (x*log(x)/sqrt(d*x**2), True))/(8*d**2) + sqrt(c + d*x**2)*(b**2*d**2* 
x**13/14 - 3*c*x*(a**2*c**3 - 5*c*(3*a**2*c**2*d + 2*a*b*c**3 - 7*c*(3*a** 
2*c*d**2 + 6*a*b*c**2*d + b**2*c**3 - 9*c*(a**2*d**3 + 6*a*b*c*d**2 + 3*b* 
*2*c**2*d - 11*c*(2*a*b*d**3 + 29*b**2*c*d**2/14)/(12*d))/(10*d))/(8*d))/( 
6*d))/(8*d**2) + x**11*(2*a*b*d**3 + 29*b**2*c*d**2/14)/(12*d) + x**9*(a** 
2*d**3 + 6*a*b*c*d**2 + 3*b**2*c**2*d - 11*c*(2*a*b*d**3 + 29*b**2*c*d**2/ 
14)/(12*d))/(10*d) + x**7*(3*a**2*c*d**2 + 6*a*b*c**2*d + b**2*c**3 - 9*c* 
(a**2*d**3 + 6*a*b*c*d**2 + 3*b**2*c**2*d - 11*c*(2*a*b*d**3 + 29*b**2*c*d 
**2/14)/(12*d))/(10*d))/(8*d) + x**5*(3*a**2*c**2*d + 2*a*b*c**3 - 7*c*(3* 
a**2*c*d**2 + 6*a*b*c**2*d + b**2*c**3 - 9*c*(a**2*d**3 + 6*a*b*c*d**2 + 3 
*b**2*c**2*d - 11*c*(2*a*b*d**3 + 29*b**2*c*d**2/14)/(12*d))/(10*d))/(8*d) 
)/(6*d) + x**3*(a**2*c**3 - 5*c*(3*a**2*c**2*d + 2*a*b*c**3 - 7*c*(3*a**2* 
c*d**2 + 6*a*b*c**2*d + b**2*c**3 - 9*c*(a**2*d**3 + 6*a*b*c*d**2 + 3*b**2 
*c**2*d - 11*c*(2*a*b*d**3 + 29*b**2*c*d**2/14)/(12*d))/(10*d))/(8*d))/(6* 
d))/(4*d)), Ne(d, 0)), (c**(5/2)*(a**2*x**5/5 + 2*a*b*x**7/7 + b**2*x**9/9 
), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.33 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} x^{7}}{14 \, d} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c x^{5}}{24 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a b x^{5}}{6 \, d} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{2} x^{3}}{48 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c x^{3}}{12 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} x^{3}}{10 \, d} - \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} b^{2} c^{3} x}{128 \, d^{4}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{2} c^{4} x}{768 \, d^{4}} + \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} b^{2} c^{5} x}{3072 \, d^{4}} + \frac {5 \, \sqrt {d x^{2} + c} b^{2} c^{6} x}{2048 \, d^{4}} + \frac {{\left (d x^{2} + c\right )}^{\frac {7}{2}} a b c^{2} x}{32 \, d^{3}} - \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a b c^{3} x}{192 \, d^{3}} - \frac {5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b c^{4} x}{768 \, d^{3}} - \frac {5 \, \sqrt {d x^{2} + c} a b c^{5} x}{512 \, d^{3}} - \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}} a^{2} c x}{80 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {5}{2}} a^{2} c^{2} x}{160 \, d^{2}} + \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} a^{2} c^{3} x}{128 \, d^{2}} + \frac {3 \, \sqrt {d x^{2} + c} a^{2} c^{4} x}{256 \, d^{2}} + \frac {5 \, b^{2} c^{7} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{2048 \, d^{\frac {9}{2}}} - \frac {5 \, a b c^{6} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{512 \, d^{\frac {7}{2}}} + \frac {3 \, a^{2} c^{5} \operatorname {arsinh}\left (\frac {d x}{\sqrt {c d}}\right )}{256 \, d^{\frac {5}{2}}} \] Input:

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

Output:

1/14*(d*x^2 + c)^(7/2)*b^2*x^7/d - 1/24*(d*x^2 + c)^(7/2)*b^2*c*x^5/d^2 + 
1/6*(d*x^2 + c)^(7/2)*a*b*x^5/d + 1/48*(d*x^2 + c)^(7/2)*b^2*c^2*x^3/d^3 - 
 1/12*(d*x^2 + c)^(7/2)*a*b*c*x^3/d^2 + 1/10*(d*x^2 + c)^(7/2)*a^2*x^3/d - 
 1/128*(d*x^2 + c)^(7/2)*b^2*c^3*x/d^4 + 1/768*(d*x^2 + c)^(5/2)*b^2*c^4*x 
/d^4 + 5/3072*(d*x^2 + c)^(3/2)*b^2*c^5*x/d^4 + 5/2048*sqrt(d*x^2 + c)*b^2 
*c^6*x/d^4 + 1/32*(d*x^2 + c)^(7/2)*a*b*c^2*x/d^3 - 1/192*(d*x^2 + c)^(5/2 
)*a*b*c^3*x/d^3 - 5/768*(d*x^2 + c)^(3/2)*a*b*c^4*x/d^3 - 5/512*sqrt(d*x^2 
 + c)*a*b*c^5*x/d^3 - 3/80*(d*x^2 + c)^(7/2)*a^2*c*x/d^2 + 1/160*(d*x^2 + 
c)^(5/2)*a^2*c^2*x/d^2 + 1/128*(d*x^2 + c)^(3/2)*a^2*c^3*x/d^2 + 3/256*sqr 
t(d*x^2 + c)*a^2*c^4*x/d^2 + 5/2048*b^2*c^7*arcsinh(d*x/sqrt(c*d))/d^(9/2) 
 - 5/512*a*b*c^6*arcsinh(d*x/sqrt(c*d))/d^(7/2) + 3/256*a^2*c^5*arcsinh(d* 
x/sqrt(c*d))/d^(5/2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.96 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {1}{215040} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, b^{2} d^{2} x^{2} + \frac {29 \, b^{2} c d^{13} + 28 \, a b d^{14}}{d^{12}}\right )} x^{2} + \frac {185 \, b^{2} c^{2} d^{12} + 700 \, a b c d^{13} + 168 \, a^{2} d^{14}}{d^{12}}\right )} x^{2} + \frac {3 \, {\left (5 \, b^{2} c^{3} d^{11} + 1260 \, a b c^{2} d^{12} + 1176 \, a^{2} c d^{13}\right )}}{d^{12}}\right )} x^{2} - \frac {7 \, {\left (5 \, b^{2} c^{4} d^{10} - 20 \, a b c^{3} d^{11} - 744 \, a^{2} c^{2} d^{12}\right )}}{d^{12}}\right )} x^{2} + \frac {35 \, {\left (5 \, b^{2} c^{5} d^{9} - 20 \, a b c^{4} d^{10} + 24 \, a^{2} c^{3} d^{11}\right )}}{d^{12}}\right )} x^{2} - \frac {105 \, {\left (5 \, b^{2} c^{6} d^{8} - 20 \, a b c^{5} d^{9} + 24 \, a^{2} c^{4} d^{10}\right )}}{d^{12}}\right )} \sqrt {d x^{2} + c} x - \frac {{\left (5 \, b^{2} c^{7} - 20 \, a b c^{6} d + 24 \, a^{2} c^{5} d^{2}\right )} \log \left ({\left | -\sqrt {d} x + \sqrt {d x^{2} + c} \right |}\right )}{2048 \, d^{\frac {9}{2}}} \] Input:

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

Output:

1/215040*(2*(4*(2*(8*(10*(12*b^2*d^2*x^2 + (29*b^2*c*d^13 + 28*a*b*d^14)/d 
^12)*x^2 + (185*b^2*c^2*d^12 + 700*a*b*c*d^13 + 168*a^2*d^14)/d^12)*x^2 + 
3*(5*b^2*c^3*d^11 + 1260*a*b*c^2*d^12 + 1176*a^2*c*d^13)/d^12)*x^2 - 7*(5* 
b^2*c^4*d^10 - 20*a*b*c^3*d^11 - 744*a^2*c^2*d^12)/d^12)*x^2 + 35*(5*b^2*c 
^5*d^9 - 20*a*b*c^4*d^10 + 24*a^2*c^3*d^11)/d^12)*x^2 - 105*(5*b^2*c^6*d^8 
 - 20*a*b*c^5*d^9 + 24*a^2*c^4*d^10)/d^12)*sqrt(d*x^2 + c)*x - 1/2048*(5*b 
^2*c^7 - 20*a*b*c^6*d + 24*a^2*c^5*d^2)*log(abs(-sqrt(d)*x + sqrt(d*x^2 + 
c)))/d^(9/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.44 \[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2} \, dx=\frac {-2520 \sqrt {d \,x^{2}+c}\, a^{2} c^{4} d^{3} x +1680 \sqrt {d \,x^{2}+c}\, a^{2} c^{3} d^{4} x^{3}+41664 \sqrt {d \,x^{2}+c}\, a^{2} c^{2} d^{5} x^{5}+56448 \sqrt {d \,x^{2}+c}\, a^{2} c \,d^{6} x^{7}+21504 \sqrt {d \,x^{2}+c}\, a^{2} d^{7} x^{9}+2100 \sqrt {d \,x^{2}+c}\, a b \,c^{5} d^{2} x -1400 \sqrt {d \,x^{2}+c}\, a b \,c^{4} d^{3} x^{3}+1120 \sqrt {d \,x^{2}+c}\, a b \,c^{3} d^{4} x^{5}+60480 \sqrt {d \,x^{2}+c}\, a b \,c^{2} d^{5} x^{7}+89600 \sqrt {d \,x^{2}+c}\, a b c \,d^{6} x^{9}+35840 \sqrt {d \,x^{2}+c}\, a b \,d^{7} x^{11}-525 \sqrt {d \,x^{2}+c}\, b^{2} c^{6} d x +350 \sqrt {d \,x^{2}+c}\, b^{2} c^{5} d^{2} x^{3}-280 \sqrt {d \,x^{2}+c}\, b^{2} c^{4} d^{3} x^{5}+240 \sqrt {d \,x^{2}+c}\, b^{2} c^{3} d^{4} x^{7}+23680 \sqrt {d \,x^{2}+c}\, b^{2} c^{2} d^{5} x^{9}+37120 \sqrt {d \,x^{2}+c}\, b^{2} c \,d^{6} x^{11}+15360 \sqrt {d \,x^{2}+c}\, b^{2} d^{7} x^{13}+2520 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a^{2} c^{5} d^{2}-2100 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) a b \,c^{6} d +525 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {d \,x^{2}+c}+\sqrt {d}\, x}{\sqrt {c}}\right ) b^{2} c^{7}}{215040 d^{5}} \] Input:

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

Output:

( - 2520*sqrt(c + d*x**2)*a**2*c**4*d**3*x + 1680*sqrt(c + d*x**2)*a**2*c* 
*3*d**4*x**3 + 41664*sqrt(c + d*x**2)*a**2*c**2*d**5*x**5 + 56448*sqrt(c + 
 d*x**2)*a**2*c*d**6*x**7 + 21504*sqrt(c + d*x**2)*a**2*d**7*x**9 + 2100*s 
qrt(c + d*x**2)*a*b*c**5*d**2*x - 1400*sqrt(c + d*x**2)*a*b*c**4*d**3*x**3 
 + 1120*sqrt(c + d*x**2)*a*b*c**3*d**4*x**5 + 60480*sqrt(c + d*x**2)*a*b*c 
**2*d**5*x**7 + 89600*sqrt(c + d*x**2)*a*b*c*d**6*x**9 + 35840*sqrt(c + d* 
x**2)*a*b*d**7*x**11 - 525*sqrt(c + d*x**2)*b**2*c**6*d*x + 350*sqrt(c + d 
*x**2)*b**2*c**5*d**2*x**3 - 280*sqrt(c + d*x**2)*b**2*c**4*d**3*x**5 + 24 
0*sqrt(c + d*x**2)*b**2*c**3*d**4*x**7 + 23680*sqrt(c + d*x**2)*b**2*c**2* 
d**5*x**9 + 37120*sqrt(c + d*x**2)*b**2*c*d**6*x**11 + 15360*sqrt(c + d*x* 
*2)*b**2*d**7*x**13 + 2520*sqrt(d)*log((sqrt(c + d*x**2) + sqrt(d)*x)/sqrt 
(c))*a**2*c**5*d**2 - 2100*sqrt(d)*log((sqrt(c + d*x**2) + sqrt(d)*x)/sqrt 
(c))*a*b*c**6*d + 525*sqrt(d)*log((sqrt(c + d*x**2) + sqrt(d)*x)/sqrt(c))* 
b**2*c**7)/(215040*d**5)