\(\int \frac {(A+B x^2) \sqrt {b x^2+c x^4}}{x^{13}} \, dx\) [164]

Optimal result

Integrand size = 26, antiderivative size = 170 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{13}} \, dx=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}-\frac {(11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{99 b^2 x^{12}}+\frac {2 c (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{231 b^3 x^{10}}-\frac {8 c^2 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{1155 b^4 x^8}+\frac {16 c^3 (11 b B-8 A c) \left (b x^2+c x^4\right )^{3/2}}{3465 b^5 x^6} \] Output:

-1/11*A*(c*x^4+b*x^2)^(3/2)/b/x^14-1/99*(-8*A*c+11*B*b)*(c*x^4+b*x^2)^(3/2 
)/b^2/x^12+2/231*c*(-8*A*c+11*B*b)*(c*x^4+b*x^2)^(3/2)/b^3/x^10-8/1155*c^2 
*(-8*A*c+11*B*b)*(c*x^4+b*x^2)^(3/2)/b^4/x^8+16/3465*c^3*(-8*A*c+11*B*b)*( 
c*x^4+b*x^2)^(3/2)/b^5/x^6
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.65 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{13}} \, dx=\frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (11 b B x^2 \left (-35 b^3+30 b^2 c x^2-24 b c^2 x^4+16 c^3 x^6\right )+A \left (-315 b^4+280 b^3 c x^2-240 b^2 c^2 x^4+192 b c^3 x^6-128 c^4 x^8\right )\right )}{3465 b^5 x^{14}} \] Input:

Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^13,x]
 

Output:

((x^2*(b + c*x^2))^(3/2)*(11*b*B*x^2*(-35*b^3 + 30*b^2*c*x^2 - 24*b*c^2*x^ 
4 + 16*c^3*x^6) + A*(-315*b^4 + 280*b^3*c*x^2 - 240*b^2*c^2*x^4 + 192*b*c^ 
3*x^6 - 128*c^4*x^8)))/(3465*b^5*x^14)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1940, 1220, 1129, 1129, 1129, 1123}

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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{13}} \, dx\)

\(\Big \downarrow \) 1940

\(\displaystyle \frac {1}{2} \int \frac {\left (B x^2+A\right ) \sqrt {c x^4+b x^2}}{x^{14}}dx^2\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {1}{2} \left (\frac {(11 b B-8 A c) \int \frac {\sqrt {c x^4+b x^2}}{x^{12}}dx^2}{11 b}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}\right )\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {1}{2} \left (\frac {(11 b B-8 A c) \left (-\frac {2 c \int \frac {\sqrt {c x^4+b x^2}}{x^{10}}dx^2}{3 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}\right )}{11 b}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}\right )\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {1}{2} \left (\frac {(11 b B-8 A c) \left (-\frac {2 c \left (-\frac {4 c \int \frac {\sqrt {c x^4+b x^2}}{x^8}dx^2}{7 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}}\right )}{3 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}\right )}{11 b}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}\right )\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {1}{2} \left (\frac {(11 b B-8 A c) \left (-\frac {2 c \left (-\frac {4 c \left (-\frac {2 c \int \frac {\sqrt {c x^4+b x^2}}{x^6}dx^2}{5 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{5 b x^8}\right )}{7 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}}\right )}{3 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}\right )}{11 b}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}\right )\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {1}{2} \left (\frac {\left (-\frac {2 c \left (-\frac {4 c \left (\frac {4 c \left (b x^2+c x^4\right )^{3/2}}{15 b^2 x^6}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{5 b x^8}\right )}{7 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}}\right )}{3 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}\right ) (11 b B-8 A c)}{11 b}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{11 b x^{14}}\right )\)

Input:

Int[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^13,x]
 

Output:

((-2*A*(b*x^2 + c*x^4)^(3/2))/(11*b*x^14) + ((11*b*B - 8*A*c)*((-2*(b*x^2 
+ c*x^4)^(3/2))/(9*b*x^12) - (2*c*((-2*(b*x^2 + c*x^4)^(3/2))/(7*b*x^10) - 
 (4*c*((-2*(b*x^2 + c*x^4)^(3/2))/(5*b*x^8) + (4*c*(b*x^2 + c*x^4)^(3/2))/ 
(15*b^2*x^6)))/(7*b)))/(3*b)))/(11*b))/2
 

Defintions of rubi rules used

rule 1123

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

rule 1129

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

rule 1220

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x 
^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e 
*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1))   Int[(d + e*x 
)^(m + 1)*(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] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0 
]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 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 = 0.44 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.61

method	result	size
pseudoelliptic	\(-\frac {\left (c \,x^{2}+b \right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (\left (\frac {11 B \,x^{2}}{9}+A \right ) b^{4}-\frac {8 c \,x^{2} \left (\frac {33 B \,x^{2}}{28}+A \right ) b^{3}}{9}+\frac {16 c^{2} x^{4} \left (\frac {11 B \,x^{2}}{10}+A \right ) b^{2}}{21}-\frac {64 c^{3} x^{6} \left (\frac {11 B \,x^{2}}{12}+A \right ) b}{105}+\frac {128 A \,c^{4} x^{8}}{315}\right )}{11 x^{12} b^{5}}\)	\(104\)
gosper	\(-\frac {\left (c \,x^{2}+b \right ) \left (128 A \,c^{4} x^{8}-176 B b \,c^{3} x^{8}-192 A b \,c^{3} x^{6}+264 B \,b^{2} c^{2} x^{6}+240 A \,b^{2} c^{2} x^{4}-330 B \,b^{3} c \,x^{4}-280 A \,b^{3} c \,x^{2}+385 B \,b^{4} x^{2}+315 A \,b^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3465 x^{12} b^{5}}\)	\(118\)
default	\(-\frac {\left (c \,x^{2}+b \right ) \left (128 A \,c^{4} x^{8}-176 B b \,c^{3} x^{8}-192 A b \,c^{3} x^{6}+264 B \,b^{2} c^{2} x^{6}+240 A \,b^{2} c^{2} x^{4}-330 B \,b^{3} c \,x^{4}-280 A \,b^{3} c \,x^{2}+385 B \,b^{4} x^{2}+315 A \,b^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3465 x^{12} b^{5}}\)	\(118\)
orering	\(-\frac {\left (c \,x^{2}+b \right ) \left (128 A \,c^{4} x^{8}-176 B b \,c^{3} x^{8}-192 A b \,c^{3} x^{6}+264 B \,b^{2} c^{2} x^{6}+240 A \,b^{2} c^{2} x^{4}-330 B \,b^{3} c \,x^{4}-280 A \,b^{3} c \,x^{2}+385 B \,b^{4} x^{2}+315 A \,b^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3465 x^{12} b^{5}}\)	\(118\)
trager	\(-\frac {\left (128 A \,c^{5} x^{10}-176 B b \,c^{4} x^{10}-64 A b \,c^{4} x^{8}+88 B \,b^{2} c^{3} x^{8}+48 A \,b^{2} c^{3} x^{6}-66 B \,b^{3} c^{2} x^{6}-40 A \,b^{3} c^{2} x^{4}+55 B \,b^{4} c \,x^{4}+35 A \,b^{4} c \,x^{2}+385 B \,b^{5} x^{2}+315 A \,b^{5}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{3465 x^{12} b^{5}}\)	\(135\)
risch	\(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (128 A \,c^{5} x^{10}-176 B b \,c^{4} x^{10}-64 A b \,c^{4} x^{8}+88 B \,b^{2} c^{3} x^{8}+48 A \,b^{2} c^{3} x^{6}-66 B \,b^{3} c^{2} x^{6}-40 A \,b^{3} c^{2} x^{4}+55 B \,b^{4} c \,x^{4}+35 A \,b^{4} c \,x^{2}+385 B \,b^{5} x^{2}+315 A \,b^{5}\right )}{3465 x^{12} b^{5}}\)	\(135\)

Input:

int((B*x^2+A)*(c*x^4+b*x^2)^(1/2)/x^13,x,method=_RETURNVERBOSE)
 

Output:

-1/11*(c*x^2+b)*(x^2*(c*x^2+b))^(1/2)*((11/9*B*x^2+A)*b^4-8/9*c*x^2*(33/28 
*B*x^2+A)*b^3+16/21*c^2*x^4*(11/10*B*x^2+A)*b^2-64/105*c^3*x^6*(11/12*B*x^ 
2+A)*b+128/315*A*c^4*x^8)/x^12/b^5
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.78 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{13}} \, dx=\frac {{\left (16 \, {\left (11 \, B b c^{4} - 8 \, A c^{5}\right )} x^{10} - 8 \, {\left (11 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} x^{8} + 6 \, {\left (11 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} x^{6} - 315 \, A b^{5} - 5 \, {\left (11 \, B b^{4} c - 8 \, A b^{3} c^{2}\right )} x^{4} - 35 \, {\left (11 \, B b^{5} + A b^{4} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{3465 \, b^{5} x^{12}} \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2)^(1/2)/x^13,x, algorithm="fricas")
 

Output:

1/3465*(16*(11*B*b*c^4 - 8*A*c^5)*x^10 - 8*(11*B*b^2*c^3 - 8*A*b*c^4)*x^8 
+ 6*(11*B*b^3*c^2 - 8*A*b^2*c^3)*x^6 - 315*A*b^5 - 5*(11*B*b^4*c - 8*A*b^3 
*c^2)*x^4 - 35*(11*B*b^5 + A*b^4*c)*x^2)*sqrt(c*x^4 + b*x^2)/(b^5*x^12)
 

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{13}} \, dx=\int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{13}}\, dx \] Input:

integrate((B*x**2+A)*(c*x**4+b*x**2)**(1/2)/x**13,x)
 

Output:

Integral(sqrt(x**2*(b + c*x**2))*(A + B*x**2)/x**13, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.51 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{13}} \, dx=\frac {1}{315} \, B {\left (\frac {16 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{b^{4} x^{2}} - \frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{3} x^{4}} + \frac {6 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{2} x^{6}} - \frac {5 \, \sqrt {c x^{4} + b x^{2}} c}{b x^{8}} - \frac {35 \, \sqrt {c x^{4} + b x^{2}}}{x^{10}}\right )} - \frac {1}{3465} \, A {\left (\frac {128 \, \sqrt {c x^{4} + b x^{2}} c^{5}}{b^{5} x^{2}} - \frac {64 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{b^{4} x^{4}} + \frac {48 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{3} x^{6}} - \frac {40 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{2} x^{8}} + \frac {35 \, \sqrt {c x^{4} + b x^{2}} c}{b x^{10}} + \frac {315 \, \sqrt {c x^{4} + b x^{2}}}{x^{12}}\right )} \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2)^(1/2)/x^13,x, algorithm="maxima")
 

Output:

1/315*B*(16*sqrt(c*x^4 + b*x^2)*c^4/(b^4*x^2) - 8*sqrt(c*x^4 + b*x^2)*c^3/ 
(b^3*x^4) + 6*sqrt(c*x^4 + b*x^2)*c^2/(b^2*x^6) - 5*sqrt(c*x^4 + b*x^2)*c/ 
(b*x^8) - 35*sqrt(c*x^4 + b*x^2)/x^10) - 1/3465*A*(128*sqrt(c*x^4 + b*x^2) 
*c^5/(b^5*x^2) - 64*sqrt(c*x^4 + b*x^2)*c^4/(b^4*x^4) + 48*sqrt(c*x^4 + b* 
x^2)*c^3/(b^3*x^6) - 40*sqrt(c*x^4 + b*x^2)*c^2/(b^2*x^8) + 35*sqrt(c*x^4 
+ b*x^2)*c/(b*x^10) + 315*sqrt(c*x^4 + b*x^2)/x^12)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 430 vs. \(2 (150) = 300\).

Time = 1.85 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.53 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{13}} \, dx=\frac {32 \, {\left (3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{14} B c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 4851 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} B b c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 11088 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} A c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + 231 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} B b^{2} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 7392 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} A b c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 165 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B b^{3} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 2640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} A b^{2} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + 1815 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b^{4} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 1320 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A b^{3} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 605 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{5} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 440 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b^{4} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) + 121 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{6} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 88 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{5} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right ) - 11 \, B b^{7} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 8 \, A b^{6} c^{\frac {11}{2}} \mathrm {sgn}\left (x\right )\right )}}{3465 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{11}} \] Input:

integrate((B*x^2+A)*(c*x^4+b*x^2)^(1/2)/x^13,x, algorithm="giac")
 

Output:

32/3465*(3465*(sqrt(c)*x - sqrt(c*x^2 + b))^14*B*c^(9/2)*sgn(x) - 4851*(sq 
rt(c)*x - sqrt(c*x^2 + b))^12*B*b*c^(9/2)*sgn(x) + 11088*(sqrt(c)*x - sqrt 
(c*x^2 + b))^12*A*c^(11/2)*sgn(x) + 231*(sqrt(c)*x - sqrt(c*x^2 + b))^10*B 
*b^2*c^(9/2)*sgn(x) + 7392*(sqrt(c)*x - sqrt(c*x^2 + b))^10*A*b*c^(11/2)*s 
gn(x) - 165*(sqrt(c)*x - sqrt(c*x^2 + b))^8*B*b^3*c^(9/2)*sgn(x) + 2640*(s 
qrt(c)*x - sqrt(c*x^2 + b))^8*A*b^2*c^(11/2)*sgn(x) + 1815*(sqrt(c)*x - sq 
rt(c*x^2 + b))^6*B*b^4*c^(9/2)*sgn(x) - 1320*(sqrt(c)*x - sqrt(c*x^2 + b)) 
^6*A*b^3*c^(11/2)*sgn(x) - 605*(sqrt(c)*x - sqrt(c*x^2 + b))^4*B*b^5*c^(9/ 
2)*sgn(x) + 440*(sqrt(c)*x - sqrt(c*x^2 + b))^4*A*b^4*c^(11/2)*sgn(x) + 12 
1*(sqrt(c)*x - sqrt(c*x^2 + b))^2*B*b^6*c^(9/2)*sgn(x) - 88*(sqrt(c)*x - s 
qrt(c*x^2 + b))^2*A*b^5*c^(11/2)*sgn(x) - 11*B*b^7*c^(9/2)*sgn(x) + 8*A*b^ 
6*c^(11/2)*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^11
 

Mupad [B] (verification not implemented)

Time = 10.94 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.53 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{13}} \, dx=\frac {8\,A\,c^2\,\sqrt {c\,x^4+b\,x^2}}{693\,b^2\,x^8}-\frac {B\,\sqrt {c\,x^4+b\,x^2}}{9\,x^{10}}-\frac {A\,c\,\sqrt {c\,x^4+b\,x^2}}{99\,b\,x^{10}}-\frac {B\,c\,\sqrt {c\,x^4+b\,x^2}}{63\,b\,x^8}-\frac {A\,\sqrt {c\,x^4+b\,x^2}}{11\,x^{12}}-\frac {16\,A\,c^3\,\sqrt {c\,x^4+b\,x^2}}{1155\,b^3\,x^6}+\frac {64\,A\,c^4\,\sqrt {c\,x^4+b\,x^2}}{3465\,b^4\,x^4}-\frac {128\,A\,c^5\,\sqrt {c\,x^4+b\,x^2}}{3465\,b^5\,x^2}+\frac {2\,B\,c^2\,\sqrt {c\,x^4+b\,x^2}}{105\,b^2\,x^6}-\frac {8\,B\,c^3\,\sqrt {c\,x^4+b\,x^2}}{315\,b^3\,x^4}+\frac {16\,B\,c^4\,\sqrt {c\,x^4+b\,x^2}}{315\,b^4\,x^2} \] Input:

int(((A + B*x^2)*(b*x^2 + c*x^4)^(1/2))/x^13,x)
 

Output:

(8*A*c^2*(b*x^2 + c*x^4)^(1/2))/(693*b^2*x^8) - (B*(b*x^2 + c*x^4)^(1/2))/ 
(9*x^10) - (A*c*(b*x^2 + c*x^4)^(1/2))/(99*b*x^10) - (B*c*(b*x^2 + c*x^4)^ 
(1/2))/(63*b*x^8) - (A*(b*x^2 + c*x^4)^(1/2))/(11*x^12) - (16*A*c^3*(b*x^2 
 + c*x^4)^(1/2))/(1155*b^3*x^6) + (64*A*c^4*(b*x^2 + c*x^4)^(1/2))/(3465*b 
^4*x^4) - (128*A*c^5*(b*x^2 + c*x^4)^(1/2))/(3465*b^5*x^2) + (2*B*c^2*(b*x 
^2 + c*x^4)^(1/2))/(105*b^2*x^6) - (8*B*c^3*(b*x^2 + c*x^4)^(1/2))/(315*b^ 
3*x^4) + (16*B*c^4*(b*x^2 + c*x^4)^(1/2))/(315*b^4*x^2)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.35 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{13}} \, dx=\frac {-315 \sqrt {c \,x^{2}+b}\, a \,b^{5}-35 \sqrt {c \,x^{2}+b}\, a \,b^{4} c \,x^{2}+40 \sqrt {c \,x^{2}+b}\, a \,b^{3} c^{2} x^{4}-48 \sqrt {c \,x^{2}+b}\, a \,b^{2} c^{3} x^{6}+64 \sqrt {c \,x^{2}+b}\, a b \,c^{4} x^{8}-128 \sqrt {c \,x^{2}+b}\, a \,c^{5} x^{10}-385 \sqrt {c \,x^{2}+b}\, b^{6} x^{2}-55 \sqrt {c \,x^{2}+b}\, b^{5} c \,x^{4}+66 \sqrt {c \,x^{2}+b}\, b^{4} c^{2} x^{6}-88 \sqrt {c \,x^{2}+b}\, b^{3} c^{3} x^{8}+176 \sqrt {c \,x^{2}+b}\, b^{2} c^{4} x^{10}+128 \sqrt {c}\, a \,c^{5} x^{11}-176 \sqrt {c}\, b^{2} c^{4} x^{11}}{3465 b^{5} x^{11}} \] Input:

int((B*x^2+A)*(c*x^4+b*x^2)^(1/2)/x^13,x)
 

Output:

( - 315*sqrt(b + c*x**2)*a*b**5 - 35*sqrt(b + c*x**2)*a*b**4*c*x**2 + 40*s 
qrt(b + c*x**2)*a*b**3*c**2*x**4 - 48*sqrt(b + c*x**2)*a*b**2*c**3*x**6 + 
64*sqrt(b + c*x**2)*a*b*c**4*x**8 - 128*sqrt(b + c*x**2)*a*c**5*x**10 - 38 
5*sqrt(b + c*x**2)*b**6*x**2 - 55*sqrt(b + c*x**2)*b**5*c*x**4 + 66*sqrt(b 
 + c*x**2)*b**4*c**2*x**6 - 88*sqrt(b + c*x**2)*b**3*c**3*x**8 + 176*sqrt( 
b + c*x**2)*b**2*c**4*x**10 + 128*sqrt(c)*a*c**5*x**11 - 176*sqrt(c)*b**2* 
c**4*x**11)/(3465*b**5*x**11)