\(\int x^5 (A+B x^2) (b x^2+c x^4)^{3/2} \, dx\) [171]

Optimal result

Integrand size = 26, antiderivative size = 223 \[ \int x^5 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {b^4 (9 b B-14 A c) \left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{2048 c^5}-\frac {b^2 (9 b B-14 A c) \left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{768 c^4}+\frac {b (9 b B-14 A c) \left (b x^2+c x^4\right )^{5/2}}{240 c^3}-\frac {(9 b B-14 A c) x^2 \left (b x^2+c x^4\right )^{5/2}}{168 c^2}+\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{14 c}-\frac {b^6 (9 b B-14 A c) \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{2048 c^{11/2}} \] Output:

1/2048*b^4*(-14*A*c+9*B*b)*(2*c*x^2+b)*(c*x^4+b*x^2)^(1/2)/c^5-1/768*b^2*( 
-14*A*c+9*B*b)*(2*c*x^2+b)*(c*x^4+b*x^2)^(3/2)/c^4+1/240*b*(-14*A*c+9*B*b) 
*(c*x^4+b*x^2)^(5/2)/c^3-1/168*(-14*A*c+9*B*b)*x^2*(c*x^4+b*x^2)^(5/2)/c^2 
+1/14*B*x^4*(c*x^4+b*x^2)^(5/2)/c-1/2048*b^6*(-14*A*c+9*B*b)*arctanh(c^(1/ 
2)*x^2/(c*x^4+b*x^2)^(1/2))/c^(11/2)
 

Mathematica [A] (verified)

Time = 1.22 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.11 \[ \int x^5 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (945 b^6 B-1470 A b^5 c-630 b^5 B c x^2+980 A b^4 c^2 x^2+504 b^4 B c^2 x^4-784 A b^3 c^3 x^4-432 b^3 B c^3 x^6+672 A b^2 c^4 x^6+384 b^2 B c^4 x^8+23296 A b c^5 x^8+19200 b B c^5 x^{10}+17920 A c^6 x^{10}+15360 B c^6 x^{12}\right )}{215040 c^5 x^2 \left (b+c x^2\right )}-\frac {b^6 (9 b B-14 A c) \left (x^2 \left (b+c x^2\right )\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {b}+\sqrt {b+c x^2}}\right )}{1024 c^{11/2} x^3 \left (b+c x^2\right )^{3/2}} \] Input:

Integrate[x^5*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]
 

Output:

((x^2*(b + c*x^2))^(3/2)*(945*b^6*B - 1470*A*b^5*c - 630*b^5*B*c*x^2 + 980 
*A*b^4*c^2*x^2 + 504*b^4*B*c^2*x^4 - 784*A*b^3*c^3*x^4 - 432*b^3*B*c^3*x^6 
 + 672*A*b^2*c^4*x^6 + 384*b^2*B*c^4*x^8 + 23296*A*b*c^5*x^8 + 19200*b*B*c 
^5*x^10 + 17920*A*c^6*x^10 + 15360*B*c^6*x^12))/(215040*c^5*x^2*(b + c*x^2 
)) - (b^6*(9*b*B - 14*A*c)*(x^2*(b + c*x^2))^(3/2)*ArcTanh[(Sqrt[c]*x)/(-S 
qrt[b] + Sqrt[b + c*x^2])])/(1024*c^(11/2)*x^3*(b + c*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1940, 1221, 1134, 1160, 1087, 1087, 1091, 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.

\(\displaystyle \int x^5 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx\)

\(\Big \downarrow \) 1940

\(\displaystyle \frac {1}{2} \int x^4 \left (B x^2+A\right ) \left (c x^4+b x^2\right )^{3/2}dx^2\)

\(\Big \downarrow \) 1221

\(\displaystyle \frac {1}{2} \left (\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{7 c}-\frac {(9 b B-14 A c) \int x^4 \left (c x^4+b x^2\right )^{3/2}dx^2}{14 c}\right )\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {1}{2} \left (\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{7 c}-\frac {(9 b B-14 A c) \left (\frac {x^2 \left (b x^2+c x^4\right )^{5/2}}{6 c}-\frac {7 b \int x^2 \left (c x^4+b x^2\right )^{3/2}dx^2}{12 c}\right )}{14 c}\right )\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {1}{2} \left (\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{7 c}-\frac {(9 b B-14 A c) \left (\frac {x^2 \left (b x^2+c x^4\right )^{5/2}}{6 c}-\frac {7 b \left (\frac {\left (b x^2+c x^4\right )^{5/2}}{5 c}-\frac {b \int \left (c x^4+b x^2\right )^{3/2}dx^2}{2 c}\right )}{12 c}\right )}{14 c}\right )\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{2} \left (\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{7 c}-\frac {(9 b B-14 A c) \left (\frac {x^2 \left (b x^2+c x^4\right )^{5/2}}{6 c}-\frac {7 b \left (\frac {\left (b x^2+c x^4\right )^{5/2}}{5 c}-\frac {b \left (\frac {\left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 b^2 \int \sqrt {c x^4+b x^2}dx^2}{16 c}\right )}{2 c}\right )}{12 c}\right )}{14 c}\right )\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {1}{2} \left (\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{7 c}-\frac {(9 b B-14 A c) \left (\frac {x^2 \left (b x^2+c x^4\right )^{5/2}}{6 c}-\frac {7 b \left (\frac {\left (b x^2+c x^4\right )^{5/2}}{5 c}-\frac {b \left (\frac {\left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {\left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{4 c}-\frac {b^2 \int \frac {1}{\sqrt {c x^4+b x^2}}dx^2}{8 c}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )}{14 c}\right )\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {1}{2} \left (\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{7 c}-\frac {(9 b B-14 A c) \left (\frac {x^2 \left (b x^2+c x^4\right )^{5/2}}{6 c}-\frac {7 b \left (\frac {\left (b x^2+c x^4\right )^{5/2}}{5 c}-\frac {b \left (\frac {\left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {\left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{4 c}-\frac {b^2 \int \frac {1}{1-c x^4}d\frac {x^2}{\sqrt {c x^4+b x^2}}}{4 c}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )}{14 c}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {B x^4 \left (b x^2+c x^4\right )^{5/2}}{7 c}-\frac {(9 b B-14 A c) \left (\frac {x^2 \left (b x^2+c x^4\right )^{5/2}}{6 c}-\frac {7 b \left (\frac {\left (b x^2+c x^4\right )^{5/2}}{5 c}-\frac {b \left (\frac {\left (b+2 c x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{8 c}-\frac {3 b^2 \left (\frac {\left (b+2 c x^2\right ) \sqrt {b x^2+c x^4}}{4 c}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{4 c^{3/2}}\right )}{16 c}\right )}{2 c}\right )}{12 c}\right )}{14 c}\right )\)

Input:

Int[x^5*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x]
 

Output:

((B*x^4*(b*x^2 + c*x^4)^(5/2))/(7*c) - ((9*b*B - 14*A*c)*((x^2*(b*x^2 + c* 
x^4)^(5/2))/(6*c) - (7*b*((b*x^2 + c*x^4)^(5/2)/(5*c) - (b*(((b + 2*c*x^2) 
*(b*x^2 + c*x^4)^(3/2))/(8*c) - (3*b^2*(((b + 2*c*x^2)*Sqrt[b*x^2 + c*x^4] 
)/(4*c) - (b^2*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^4]])/(4*c^(3/2))))/( 
16*c)))/(2*c)))/(12*c)))/(14*c))/2
 

Defintions of rubi rules used

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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1134

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

rule 1160

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

rule 1221

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1 
)/(c*(m + 2*p + 2))), x] + Simp[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c 
*f - b*g))/(c*e*(m + 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && NeQ[m + 2*p + 2, 0]
 

rule 1940

Int[(x_)^(m_.)*((b_.)*(x_)^(k_.) + (a_.)*(x_)^(j_))^(p_)*((c_) + (d_.)*(x_) 
^(n_))^(q_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1) 
*(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x] /; 
FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] &&  !IntegerQ[p] && NeQ[k, j] && I 
ntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 
 1)/n]] && NeQ[n^2, 1]
 

Maple [A] (verified)

Time = 0.49 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {\frac {7 \left (A \,b^{6} c -\frac {9}{14} B \,b^{7}\right ) \ln \left (\frac {2 c \,x^{2}+2 \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \sqrt {c}+b}{\sqrt {c}}\right )}{2048}+\frac {7 \left (\frac {832 x^{8} \left (\frac {75 B \,x^{2}}{91}+A \right ) b \,c^{\frac {11}{2}}}{35}+\frac {128 x^{10} \left (\frac {6 B \,x^{2}}{7}+A \right ) c^{\frac {13}{2}}}{7}+\left (-\frac {3 \left (\frac {3 B \,x^{2}}{7}+A \right ) b^{3} c^{\frac {3}{2}}}{2}+b^{2} x^{2} \left (\frac {18 B \,x^{2}}{35}+A \right ) c^{\frac {5}{2}}-\frac {4 x^{4} b \left (\frac {27 B \,x^{2}}{49}+A \right ) c^{\frac {7}{2}}}{5}+\frac {24 \left (\frac {4 B \,x^{2}}{7}+A \right ) x^{6} c^{\frac {9}{2}}}{35}+\frac {27 B \sqrt {c}\, b^{4}}{28}\right ) b^{2}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{1536}-\frac {7 \ln \left (2\right ) \left (A c -\frac {9 B b}{14}\right ) b^{6}}{2048}}{c^{\frac {11}{2}}}\)	\(197\)
risch	\(-\frac {\left (-15360 B \,c^{6} x^{12}-17920 A \,c^{6} x^{10}-19200 B b \,c^{5} x^{10}-23296 A b \,c^{5} x^{8}-384 B \,b^{2} c^{4} x^{8}-672 A \,b^{2} c^{4} x^{6}+432 B \,b^{3} c^{3} x^{6}+784 A \,b^{3} c^{3} x^{4}-504 B \,b^{4} c^{2} x^{4}-980 A \,b^{4} c^{2} x^{2}+630 B \,b^{5} c \,x^{2}+1470 A \,b^{5} c -945 B \,b^{6}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{215040 c^{5}}+\frac {b^{6} \left (14 A c -9 B b \right ) \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}}{2048 c^{\frac {11}{2}} x \sqrt {c \,x^{2}+b}}\)	\(212\)
default	\(\frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (15360 B \,c^{\frac {9}{2}} \left (c \,x^{2}+b \right )^{\frac {5}{2}} x^{9}+17920 A \,c^{\frac {9}{2}} \left (c \,x^{2}+b \right )^{\frac {5}{2}} x^{7}-11520 B \,c^{\frac {7}{2}} \left (c \,x^{2}+b \right )^{\frac {5}{2}} b \,x^{7}-12544 A \,c^{\frac {7}{2}} \left (c \,x^{2}+b \right )^{\frac {5}{2}} b \,x^{5}+8064 B \,c^{\frac {5}{2}} \left (c \,x^{2}+b \right )^{\frac {5}{2}} b^{2} x^{5}+7840 A \,c^{\frac {5}{2}} \left (c \,x^{2}+b \right )^{\frac {5}{2}} b^{2} x^{3}-5040 B \,c^{\frac {3}{2}} \left (c \,x^{2}+b \right )^{\frac {5}{2}} b^{3} x^{3}-3920 A \,c^{\frac {3}{2}} \left (c \,x^{2}+b \right )^{\frac {5}{2}} b^{3} x +980 A \,c^{\frac {3}{2}} \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{4} x +2520 B \sqrt {c}\, \left (c \,x^{2}+b \right )^{\frac {5}{2}} b^{4} x +1470 A \,c^{\frac {3}{2}} \sqrt {c \,x^{2}+b}\, b^{5} x -630 B \sqrt {c}\, \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{5} x -945 B \sqrt {c}\, \sqrt {c \,x^{2}+b}\, b^{6} x +1470 A \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{6} c -945 B \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right ) b^{7}\right )}{215040 x^{3} \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{\frac {11}{2}}}\)	\(328\)

Input:

int(x^5*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

7/1536*(3/4*(A*b^6*c-9/14*B*b^7)*ln((2*c*x^2+2*(x^2*(c*x^2+b))^(1/2)*c^(1/ 
2)+b)/c^(1/2))+(832/35*x^8*(75/91*B*x^2+A)*b*c^(11/2)+128/7*x^10*(6/7*B*x^ 
2+A)*c^(13/2)+(-3/2*(3/7*B*x^2+A)*b^3*c^(3/2)+b^2*x^2*(18/35*B*x^2+A)*c^(5 
/2)-4/5*x^4*b*(27/49*B*x^2+A)*c^(7/2)+24/35*(4/7*B*x^2+A)*x^6*c^(9/2)+27/2 
8*B*c^(1/2)*b^4)*b^2)*(x^2*(c*x^2+b))^(1/2)-3/4*ln(2)*(A*c-9/14*B*b)*b^6)/ 
c^(11/2)
 

Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.87 \[ \int x^5 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\left [-\frac {105 \, {\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, {\left (15360 \, B c^{7} x^{12} + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{10} + 128 \, {\left (3 \, B b^{2} c^{5} + 182 \, A b c^{6}\right )} x^{8} + 945 \, B b^{6} c - 1470 \, A b^{5} c^{2} - 48 \, {\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )} x^{6} + 56 \, {\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )} x^{4} - 70 \, {\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{430080 \, c^{6}}, \frac {105 \, {\left (9 \, B b^{7} - 14 \, A b^{6} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + {\left (15360 \, B c^{7} x^{12} + 1280 \, {\left (15 \, B b c^{6} + 14 \, A c^{7}\right )} x^{10} + 128 \, {\left (3 \, B b^{2} c^{5} + 182 \, A b c^{6}\right )} x^{8} + 945 \, B b^{6} c - 1470 \, A b^{5} c^{2} - 48 \, {\left (9 \, B b^{3} c^{4} - 14 \, A b^{2} c^{5}\right )} x^{6} + 56 \, {\left (9 \, B b^{4} c^{3} - 14 \, A b^{3} c^{4}\right )} x^{4} - 70 \, {\left (9 \, B b^{5} c^{2} - 14 \, A b^{4} c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{215040 \, c^{6}}\right ] \] Input:

integrate(x^5*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="fricas")
 

Output:

[-1/430080*(105*(9*B*b^7 - 14*A*b^6*c)*sqrt(c)*log(-2*c*x^2 - b - 2*sqrt(c 
*x^4 + b*x^2)*sqrt(c)) - 2*(15360*B*c^7*x^12 + 1280*(15*B*b*c^6 + 14*A*c^7 
)*x^10 + 128*(3*B*b^2*c^5 + 182*A*b*c^6)*x^8 + 945*B*b^6*c - 1470*A*b^5*c^ 
2 - 48*(9*B*b^3*c^4 - 14*A*b^2*c^5)*x^6 + 56*(9*B*b^4*c^3 - 14*A*b^3*c^4)* 
x^4 - 70*(9*B*b^5*c^2 - 14*A*b^4*c^3)*x^2)*sqrt(c*x^4 + b*x^2))/c^6, 1/215 
040*(105*(9*B*b^7 - 14*A*b^6*c)*sqrt(-c)*arctan(sqrt(c*x^4 + b*x^2)*sqrt(- 
c)/(c*x^2 + b)) + (15360*B*c^7*x^12 + 1280*(15*B*b*c^6 + 14*A*c^7)*x^10 + 
128*(3*B*b^2*c^5 + 182*A*b*c^6)*x^8 + 945*B*b^6*c - 1470*A*b^5*c^2 - 48*(9 
*B*b^3*c^4 - 14*A*b^2*c^5)*x^6 + 56*(9*B*b^4*c^3 - 14*A*b^3*c^4)*x^4 - 70* 
(9*B*b^5*c^2 - 14*A*b^4*c^3)*x^2)*sqrt(c*x^4 + b*x^2))/c^6]
 

Sympy [A] (verification not implemented)

Time = 1.29 (sec) , antiderivative size = 726, normalized size of antiderivative = 3.26 \[ \int x^5 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx =\text {Too large to display} \] Input:

integrate(x**5*(B*x**2+A)*(c*x**4+b*x**2)**(3/2),x)
 

Output:

A*b*Piecewise((7*b**5*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x**2 + c*x**4) + 
 2*c*x**2)/sqrt(c), Ne(b**2/c, 0)), ((b/(2*c) + x**2)*log(b/(2*c) + x**2)/ 
sqrt(c*(b/(2*c) + x**2)**2), True))/(256*c**4) + sqrt(b*x**2 + c*x**4)*(-7 
*b**4/(128*c**4) + 7*b**3*x**2/(192*c**3) - 7*b**2*x**4/(240*c**2) + b*x** 
6/(40*c) + x**8/5), Ne(c, 0)), (2*(b*x**2)**(9/2)/(9*b**4), Ne(b, 0)), (0, 
 True))/2 + A*c*Piecewise((-21*b**6*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x* 
*2 + c*x**4) + 2*c*x**2)/sqrt(c), Ne(b**2/c, 0)), ((b/(2*c) + x**2)*log(b/ 
(2*c) + x**2)/sqrt(c*(b/(2*c) + x**2)**2), True))/(1024*c**5) + sqrt(b*x** 
2 + c*x**4)*(21*b**5/(512*c**5) - 7*b**4*x**2/(256*c**4) + 7*b**3*x**4/(32 
0*c**3) - 3*b**2*x**6/(160*c**2) + b*x**8/(60*c) + x**10/6), Ne(c, 0)), (2 
*(b*x**2)**(11/2)/(11*b**5), Ne(b, 0)), (0, True))/2 + B*b*Piecewise((-21* 
b**6*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x**2 + c*x**4) + 2*c*x**2)/sqrt(c 
), Ne(b**2/c, 0)), ((b/(2*c) + x**2)*log(b/(2*c) + x**2)/sqrt(c*(b/(2*c) + 
 x**2)**2), True))/(1024*c**5) + sqrt(b*x**2 + c*x**4)*(21*b**5/(512*c**5) 
 - 7*b**4*x**2/(256*c**4) + 7*b**3*x**4/(320*c**3) - 3*b**2*x**6/(160*c**2 
) + b*x**8/(60*c) + x**10/6), Ne(c, 0)), (2*(b*x**2)**(11/2)/(11*b**5), Ne 
(b, 0)), (0, True))/2 + B*c*Piecewise((33*b**7*Piecewise((log(b + 2*sqrt(c 
)*sqrt(b*x**2 + c*x**4) + 2*c*x**2)/sqrt(c), Ne(b**2/c, 0)), ((b/(2*c) + x 
**2)*log(b/(2*c) + x**2)/sqrt(c*(b/(2*c) + x**2)**2), True))/(2048*c**6) + 
 sqrt(b*x**2 + c*x**4)*(-33*b**6/(1024*c**6) + 11*b**5*x**2/(512*c**5) ...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.63 \[ \int x^5 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=-\frac {1}{30720} \, {\left (\frac {420 \, \sqrt {c x^{4} + b x^{2}} b^{4} x^{2}}{c^{3}} - \frac {1120 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{2} x^{2}}{c^{2}} - \frac {2560 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} x^{2}}{c} - \frac {105 \, b^{6} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {9}{2}}} + \frac {210 \, \sqrt {c x^{4} + b x^{2}} b^{5}}{c^{4}} - \frac {560 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{3}}{c^{3}} + \frac {1792 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} b}{c^{2}}\right )} A + \frac {1}{143360} \, {\left (\frac {10240 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} x^{4}}{c} + \frac {1260 \, \sqrt {c x^{4} + b x^{2}} b^{5} x^{2}}{c^{4}} - \frac {3360 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{3} x^{2}}{c^{3}} - \frac {7680 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} b x^{2}}{c^{2}} - \frac {315 \, b^{7} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {11}{2}}} + \frac {630 \, \sqrt {c x^{4} + b x^{2}} b^{6}}{c^{5}} - \frac {1680 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} b^{4}}{c^{4}} + \frac {5376 \, {\left (c x^{4} + b x^{2}\right )}^{\frac {5}{2}} b^{2}}{c^{3}}\right )} B \] Input:

integrate(x^5*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="maxima")
 

Output:

-1/30720*(420*sqrt(c*x^4 + b*x^2)*b^4*x^2/c^3 - 1120*(c*x^4 + b*x^2)^(3/2) 
*b^2*x^2/c^2 - 2560*(c*x^4 + b*x^2)^(5/2)*x^2/c - 105*b^6*log(2*c*x^2 + b 
+ 2*sqrt(c*x^4 + b*x^2)*sqrt(c))/c^(9/2) + 210*sqrt(c*x^4 + b*x^2)*b^5/c^4 
 - 560*(c*x^4 + b*x^2)^(3/2)*b^3/c^3 + 1792*(c*x^4 + b*x^2)^(5/2)*b/c^2)*A 
 + 1/143360*(10240*(c*x^4 + b*x^2)^(5/2)*x^4/c + 1260*sqrt(c*x^4 + b*x^2)* 
b^5*x^2/c^4 - 3360*(c*x^4 + b*x^2)^(3/2)*b^3*x^2/c^3 - 7680*(c*x^4 + b*x^2 
)^(5/2)*b*x^2/c^2 - 315*b^7*log(2*c*x^2 + b + 2*sqrt(c*x^4 + b*x^2)*sqrt(c 
))/c^(11/2) + 630*sqrt(c*x^4 + b*x^2)*b^6/c^5 - 1680*(c*x^4 + b*x^2)^(3/2) 
*b^4/c^4 + 5376*(c*x^4 + b*x^2)^(5/2)*b^2/c^3)*B
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.26 \[ \int x^5 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {1}{215040} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, B c x^{2} \mathrm {sgn}\left (x\right ) + \frac {15 \, B b c^{12} \mathrm {sgn}\left (x\right ) + 14 \, A c^{13} \mathrm {sgn}\left (x\right )}{c^{12}}\right )} x^{2} + \frac {3 \, B b^{2} c^{11} \mathrm {sgn}\left (x\right ) + 182 \, A b c^{12} \mathrm {sgn}\left (x\right )}{c^{12}}\right )} x^{2} - \frac {3 \, {\left (9 \, B b^{3} c^{10} \mathrm {sgn}\left (x\right ) - 14 \, A b^{2} c^{11} \mathrm {sgn}\left (x\right )\right )}}{c^{12}}\right )} x^{2} + \frac {7 \, {\left (9 \, B b^{4} c^{9} \mathrm {sgn}\left (x\right ) - 14 \, A b^{3} c^{10} \mathrm {sgn}\left (x\right )\right )}}{c^{12}}\right )} x^{2} - \frac {35 \, {\left (9 \, B b^{5} c^{8} \mathrm {sgn}\left (x\right ) - 14 \, A b^{4} c^{9} \mathrm {sgn}\left (x\right )\right )}}{c^{12}}\right )} x^{2} + \frac {105 \, {\left (9 \, B b^{6} c^{7} \mathrm {sgn}\left (x\right ) - 14 \, A b^{5} c^{8} \mathrm {sgn}\left (x\right )\right )}}{c^{12}}\right )} \sqrt {c x^{2} + b} x + \frac {{\left (9 \, B b^{7} \mathrm {sgn}\left (x\right ) - 14 \, A b^{6} c \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{2048 \, c^{\frac {11}{2}}} - \frac {{\left (9 \, B b^{7} \log \left ({\left | b \right |}\right ) - 14 \, A b^{6} c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\left (x\right )}{4096 \, c^{\frac {11}{2}}} \] Input:

integrate(x^5*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
 

Output:

1/215040*(2*(4*(2*(8*(10*(12*B*c*x^2*sgn(x) + (15*B*b*c^12*sgn(x) + 14*A*c 
^13*sgn(x))/c^12)*x^2 + (3*B*b^2*c^11*sgn(x) + 182*A*b*c^12*sgn(x))/c^12)* 
x^2 - 3*(9*B*b^3*c^10*sgn(x) - 14*A*b^2*c^11*sgn(x))/c^12)*x^2 + 7*(9*B*b^ 
4*c^9*sgn(x) - 14*A*b^3*c^10*sgn(x))/c^12)*x^2 - 35*(9*B*b^5*c^8*sgn(x) - 
14*A*b^4*c^9*sgn(x))/c^12)*x^2 + 105*(9*B*b^6*c^7*sgn(x) - 14*A*b^5*c^8*sg 
n(x))/c^12)*sqrt(c*x^2 + b)*x + 1/2048*(9*B*b^7*sgn(x) - 14*A*b^6*c*sgn(x) 
)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + b)))/c^(11/2) - 1/4096*(9*B*b^7*log(ab 
s(b)) - 14*A*b^6*c*log(abs(b)))*sgn(x)/c^(11/2)
 

Mupad [F(-1)]

Timed out. \[ \int x^5 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\int x^5\,\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2} \,d x \] Input:

int(x^5*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2),x)
 

Output:

int(x^5*(A + B*x^2)*(b*x^2 + c*x^4)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.35 \[ \int x^5 \left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2} \, dx=\frac {-1470 \sqrt {c \,x^{2}+b}\, a \,b^{5} c^{2} x +980 \sqrt {c \,x^{2}+b}\, a \,b^{4} c^{3} x^{3}-784 \sqrt {c \,x^{2}+b}\, a \,b^{3} c^{4} x^{5}+672 \sqrt {c \,x^{2}+b}\, a \,b^{2} c^{5} x^{7}+23296 \sqrt {c \,x^{2}+b}\, a b \,c^{6} x^{9}+17920 \sqrt {c \,x^{2}+b}\, a \,c^{7} x^{11}+945 \sqrt {c \,x^{2}+b}\, b^{7} c x -630 \sqrt {c \,x^{2}+b}\, b^{6} c^{2} x^{3}+504 \sqrt {c \,x^{2}+b}\, b^{5} c^{3} x^{5}-432 \sqrt {c \,x^{2}+b}\, b^{4} c^{4} x^{7}+384 \sqrt {c \,x^{2}+b}\, b^{3} c^{5} x^{9}+19200 \sqrt {c \,x^{2}+b}\, b^{2} c^{6} x^{11}+15360 \sqrt {c \,x^{2}+b}\, b \,c^{7} x^{13}+1470 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+b}+\sqrt {c}\, x}{\sqrt {b}}\right ) a \,b^{6} c -945 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+b}+\sqrt {c}\, x}{\sqrt {b}}\right ) b^{8}}{215040 c^{6}} \] Input:

int(x^5*(B*x^2+A)*(c*x^4+b*x^2)^(3/2),x)
 

Output:

( - 1470*sqrt(b + c*x**2)*a*b**5*c**2*x + 980*sqrt(b + c*x**2)*a*b**4*c**3 
*x**3 - 784*sqrt(b + c*x**2)*a*b**3*c**4*x**5 + 672*sqrt(b + c*x**2)*a*b** 
2*c**5*x**7 + 23296*sqrt(b + c*x**2)*a*b*c**6*x**9 + 17920*sqrt(b + c*x**2 
)*a*c**7*x**11 + 945*sqrt(b + c*x**2)*b**7*c*x - 630*sqrt(b + c*x**2)*b**6 
*c**2*x**3 + 504*sqrt(b + c*x**2)*b**5*c**3*x**5 - 432*sqrt(b + c*x**2)*b* 
*4*c**4*x**7 + 384*sqrt(b + c*x**2)*b**3*c**5*x**9 + 19200*sqrt(b + c*x**2 
)*b**2*c**6*x**11 + 15360*sqrt(b + c*x**2)*b*c**7*x**13 + 1470*sqrt(c)*log 
((sqrt(b + c*x**2) + sqrt(c)*x)/sqrt(b))*a*b**6*c - 945*sqrt(c)*log((sqrt( 
b + c*x**2) + sqrt(c)*x)/sqrt(b))*b**8)/(215040*c**6)