\(\int \frac {(a+b x^2)^{5/2}}{x^8 (c+d x)} \, dx\) [1138]

Optimal result

Integrand size = 22, antiderivative size = 415 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^8 (c+d x)} \, dx=-\frac {a^2 \sqrt {a+b x^2}}{7 c x^7}+\frac {a^2 d \sqrt {a+b x^2}}{6 c^2 x^6}-\frac {a \left (15 b c^2+7 a d^2\right ) \sqrt {a+b x^2}}{35 c^3 x^5}+\frac {a d \left (13 b c^2+6 a d^2\right ) \sqrt {a+b x^2}}{24 c^4 x^4}-\frac {\left (45 b^2 c^4+77 a b c^2 d^2+35 a^2 d^4\right ) \sqrt {a+b x^2}}{105 c^5 x^3}+\frac {d \left (11 b^2 c^4+18 a b c^2 d^2+8 a^2 d^4\right ) \sqrt {a+b x^2}}{16 c^6 x^2}-\frac {\left (15 b^3 c^6+161 a b^2 c^4 d^2+245 a^2 b c^2 d^4+105 a^3 d^6\right ) \sqrt {a+b x^2}}{105 a c^7 x}-\frac {d^2 \left (b c^2+a d^2\right )^{5/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^8}+\frac {d \left (5 b^3 c^6+30 a b^2 c^4 d^2+40 a^2 b c^2 d^4+16 a^3 d^6\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{16 \sqrt {a} c^8} \] Output:

-1/7*a^2*(b*x^2+a)^(1/2)/c/x^7+1/6*a^2*d*(b*x^2+a)^(1/2)/c^2/x^6-1/35*a*(7 
*a*d^2+15*b*c^2)*(b*x^2+a)^(1/2)/c^3/x^5+1/24*a*d*(6*a*d^2+13*b*c^2)*(b*x^ 
2+a)^(1/2)/c^4/x^4-1/105*(35*a^2*d^4+77*a*b*c^2*d^2+45*b^2*c^4)*(b*x^2+a)^ 
(1/2)/c^5/x^3+1/16*d*(8*a^2*d^4+18*a*b*c^2*d^2+11*b^2*c^4)*(b*x^2+a)^(1/2) 
/c^6/x^2-1/105*(105*a^3*d^6+245*a^2*b*c^2*d^4+161*a*b^2*c^4*d^2+15*b^3*c^6 
)*(b*x^2+a)^(1/2)/a/c^7/x-d^2*(a*d^2+b*c^2)^(5/2)*arctanh((-b*c*x+a*d)/(a* 
d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/c^8+1/16*d*(16*a^3*d^6+40*a^2*b*c^2*d^4+ 
30*a*b^2*c^4*d^2+5*b^3*c^6)*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(1/2)/c^8
 

Mathematica [A] (verified)

Time = 3.04 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.83 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^8 (c+d x)} \, dx=-\frac {\frac {c \sqrt {a+b x^2} \left (240 b^3 c^6 x^6+a b^2 c^4 x^4 \left (720 c^2-1155 c d x+2576 d^2 x^2\right )+2 a^2 b c^2 x^2 \left (360 c^4-455 c^3 d x+616 c^2 d^2 x^2-945 c d^3 x^3+1960 d^4 x^4\right )+4 a^3 \left (60 c^6-70 c^5 d x+84 c^4 d^2 x^2-105 c^3 d^3 x^3+140 c^2 d^4 x^4-210 c d^5 x^5+420 d^6 x^6\right )\right )}{a x^7}-3360 \sqrt {-b c^2-a d^2} \left (b c^2 d+a d^3\right )^2 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )+\frac {210 d \left (5 b^3 c^6+30 a b^2 c^4 d^2+40 a^2 b c^2 d^4+16 a^3 d^6\right ) \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}}{1680 c^8} \] Input:

Integrate[(a + b*x^2)^(5/2)/(x^8*(c + d*x)),x]
 

Output:

-1/1680*((c*Sqrt[a + b*x^2]*(240*b^3*c^6*x^6 + a*b^2*c^4*x^4*(720*c^2 - 11 
55*c*d*x + 2576*d^2*x^2) + 2*a^2*b*c^2*x^2*(360*c^4 - 455*c^3*d*x + 616*c^ 
2*d^2*x^2 - 945*c*d^3*x^3 + 1960*d^4*x^4) + 4*a^3*(60*c^6 - 70*c^5*d*x + 8 
4*c^4*d^2*x^2 - 105*c^3*d^3*x^3 + 140*c^2*d^4*x^4 - 210*c*d^5*x^5 + 420*d^ 
6*x^6)))/(a*x^7) - 3360*Sqrt[-(b*c^2) - a*d^2]*(b*c^2*d + a*d^3)^2*ArcTan[ 
(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - a*d^2]] + (210*d*( 
5*b^3*c^6 + 30*a*b^2*c^4*d^2 + 40*a^2*b*c^2*d^4 + 16*a^3*d^6)*ArcTanh[(Sqr 
t[b]*x - Sqrt[a + b*x^2])/Sqrt[a]])/Sqrt[a])/c^8
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(953\) vs. \(2(415)=830\).

Time = 2.05 (sec) , antiderivative size = 953, normalized size of antiderivative = 2.30, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{5/2}}{x^8 (c+d x)} \, dx\)

\(\Big \downarrow \) 617

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a \left (b x^2+a\right )^{3/2} d^7}{3 c^8}+\frac {a^{5/2} \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d^7}{c^8}-\frac {a^2 \sqrt {b x^2+a} d^7}{c^8}-\frac {\left (b x^2+a\right )^{5/2} d^6}{c^7 x}+\frac {5 b x \left (b x^2+a\right )^{3/2} d^6}{4 c^7}+\frac {15 a^2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right ) d^6}{8 c^7}+\frac {15 a b x \sqrt {b x^2+a} d^6}{8 c^7}+\frac {\left (b x^2+a\right )^{5/2} d^5}{2 c^6 x^2}+\frac {\left (4 \left (b c^2+a d^2\right )-3 b c d x\right ) \left (b x^2+a\right )^{3/2} d^5}{12 c^8}-\frac {5 b \left (b x^2+a\right )^{3/2} d^5}{6 c^6}+\frac {5 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d^5}{2 c^6}-\frac {5 a b \sqrt {b x^2+a} d^5}{2 c^6}-\frac {\left (b x^2+a\right )^{5/2} d^4}{3 c^5 x^3}-\frac {5 b \left (b x^2+a\right )^{3/2} d^4}{3 c^5 x}+\frac {5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right ) d^4}{2 c^5}+\frac {5 b^2 x \sqrt {b x^2+a} d^4}{2 c^5}+\frac {\left (b x^2+a\right )^{5/2} d^3}{4 c^4 x^4}+\frac {5 b \left (b x^2+a\right )^{3/2} d^3}{8 c^4 x^2}+\frac {15 \sqrt {a} b^2 \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d^3}{8 c^4}+\frac {\left (8 \left (b c^2+a d^2\right )^2-b c d \left (4 b c^2+7 a d^2\right ) x\right ) \sqrt {b x^2+a} d^3}{8 c^8}-\frac {15 b^2 \sqrt {b x^2+a} d^3}{8 c^4}-\frac {\left (b x^2+a\right )^{5/2} d^2}{5 c^3 x^5}-\frac {b \left (b x^2+a\right )^{3/2} d^2}{3 c^3 x^3}-\frac {\sqrt {b} \left (8 b^2 c^4+20 a b d^2 c^2+15 a^2 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right ) d^2}{8 c^7}+\frac {b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a}}\right ) d^2}{c^3}-\frac {\left (b c^2+a d^2\right )^{5/2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {b x^2+a}}\right ) d^2}{c^8}-\frac {b^2 \sqrt {b x^2+a} d^2}{c^3 x}+\frac {\left (b x^2+a\right )^{5/2} d}{6 c^2 x^6}+\frac {5 b \left (b x^2+a\right )^{3/2} d}{24 c^2 x^4}+\frac {5 b^3 \text {arctanh}\left (\frac {\sqrt {b x^2+a}}{\sqrt {a}}\right ) d}{16 \sqrt {a} c^2}+\frac {5 b^2 \sqrt {b x^2+a} d}{16 c^2 x^2}-\frac {\left (b x^2+a\right )^{7/2}}{7 a c x^7}\)

Input:

Int[(a + b*x^2)^(5/2)/(x^8*(c + d*x)),x]
 

Output:

(-15*b^2*d^3*Sqrt[a + b*x^2])/(8*c^4) - (5*a*b*d^5*Sqrt[a + b*x^2])/(2*c^6 
) - (a^2*d^7*Sqrt[a + b*x^2])/c^8 + (5*b^2*d*Sqrt[a + b*x^2])/(16*c^2*x^2) 
 - (b^2*d^2*Sqrt[a + b*x^2])/(c^3*x) + (5*b^2*d^4*x*Sqrt[a + b*x^2])/(2*c^ 
5) + (15*a*b*d^6*x*Sqrt[a + b*x^2])/(8*c^7) + (d^3*(8*(b*c^2 + a*d^2)^2 - 
b*c*d*(4*b*c^2 + 7*a*d^2)*x)*Sqrt[a + b*x^2])/(8*c^8) - (5*b*d^5*(a + b*x^ 
2)^(3/2))/(6*c^6) - (a*d^7*(a + b*x^2)^(3/2))/(3*c^8) + (5*b*d*(a + b*x^2) 
^(3/2))/(24*c^2*x^4) - (b*d^2*(a + b*x^2)^(3/2))/(3*c^3*x^3) + (5*b*d^3*(a 
 + b*x^2)^(3/2))/(8*c^4*x^2) - (5*b*d^4*(a + b*x^2)^(3/2))/(3*c^5*x) + (5* 
b*d^6*x*(a + b*x^2)^(3/2))/(4*c^7) + (d^5*(4*(b*c^2 + a*d^2) - 3*b*c*d*x)* 
(a + b*x^2)^(3/2))/(12*c^8) + (d*(a + b*x^2)^(5/2))/(6*c^2*x^6) - (d^2*(a 
+ b*x^2)^(5/2))/(5*c^3*x^5) + (d^3*(a + b*x^2)^(5/2))/(4*c^4*x^4) - (d^4*( 
a + b*x^2)^(5/2))/(3*c^5*x^3) + (d^5*(a + b*x^2)^(5/2))/(2*c^6*x^2) - (d^6 
*(a + b*x^2)^(5/2))/(c^7*x) - (a + b*x^2)^(7/2)/(7*a*c*x^7) + (b^(5/2)*d^2 
*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/c^3 + (5*a*b^(3/2)*d^4*ArcTanh[(Sqr 
t[b]*x)/Sqrt[a + b*x^2]])/(2*c^5) + (15*a^2*Sqrt[b]*d^6*ArcTanh[(Sqrt[b]*x 
)/Sqrt[a + b*x^2]])/(8*c^7) - (Sqrt[b]*d^2*(8*b^2*c^4 + 20*a*b*c^2*d^2 + 1 
5*a^2*d^4)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*c^7) - (d^2*(b*c^2 + a 
*d^2)^(5/2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/ 
c^8 + (5*b^3*d*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(16*Sqrt[a]*c^2) + (15*Sq 
rt[a]*b^2*d^3*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(8*c^4) + (5*a^(3/2)*b*...
 

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 2009

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

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.13

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (1680 a^{3} d^{6} x^{6}+3920 a^{2} b \,c^{2} d^{4} x^{6}+2576 a \,b^{2} c^{4} d^{2} x^{6}+240 b^{3} c^{6} x^{6}-840 a^{3} c \,d^{5} x^{5}-1890 a^{2} b \,c^{3} d^{3} x^{5}-1155 a \,b^{2} c^{5} d \,x^{5}+560 a^{3} c^{2} d^{4} x^{4}+1232 a^{2} b \,c^{4} d^{2} x^{4}+720 a \,b^{2} c^{6} x^{4}-420 a^{3} c^{3} d^{3} x^{3}-910 a^{2} b \,c^{5} d \,x^{3}+336 a^{3} c^{4} d^{2} x^{2}+720 a^{2} b \,c^{6} x^{2}-280 a^{3} c^{5} d x +240 a^{3} c^{6}\right )}{1680 a \,c^{7} x^{7}}+\frac {d \left (\frac {\left (16 a^{3} d^{6}+40 a^{2} b \,c^{2} d^{4}+30 a \,b^{2} c^{4} d^{2}+5 b^{3} c^{6}\right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}-\frac {16 \left (a^{3} d^{6}+3 a^{2} b \,c^{2} d^{4}+3 a \,b^{2} c^{4} d^{2}+b^{3} c^{6}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{16 c^{7}}\)	\(470\)
default	\(\text {Expression too large to display}\)	\(1660\)

Input:

int((b*x^2+a)^(5/2)/x^8/(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

-1/1680*(b*x^2+a)^(1/2)*(1680*a^3*d^6*x^6+3920*a^2*b*c^2*d^4*x^6+2576*a*b^ 
2*c^4*d^2*x^6+240*b^3*c^6*x^6-840*a^3*c*d^5*x^5-1890*a^2*b*c^3*d^3*x^5-115 
5*a*b^2*c^5*d*x^5+560*a^3*c^2*d^4*x^4+1232*a^2*b*c^4*d^2*x^4+720*a*b^2*c^6 
*x^4-420*a^3*c^3*d^3*x^3-910*a^2*b*c^5*d*x^3+336*a^3*c^4*d^2*x^2+720*a^2*b 
*c^6*x^2-280*a^3*c^5*d*x+240*a^3*c^6)/a/c^7/x^7+1/16*d/c^7*((16*a^3*d^6+40 
*a^2*b*c^2*d^4+30*a*b^2*c^4*d^2+5*b^3*c^6)/c/a^(1/2)*ln((2*a+2*a^(1/2)*(b* 
x^2+a)^(1/2))/x)-16*(a^3*d^6+3*a^2*b*c^2*d^4+3*a*b^2*c^4*d^2+b^3*c^6)/c/(( 
a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+ 
b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/( 
x+c/d)))
 

Fricas [A] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 1707, normalized size of antiderivative = 4.11 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^8 (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(5/2)/x^8/(d*x+c),x, algorithm="fricas")
 

Output:

[1/3360*(1680*(a*b^2*c^4*d^2 + 2*a^2*b*c^2*d^4 + a^3*d^6)*sqrt(b*c^2 + a*d 
^2)*x^7*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 
 - 2*sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x 
 + c^2)) + 105*(5*b^3*c^6*d + 30*a*b^2*c^4*d^3 + 40*a^2*b*c^2*d^5 + 16*a^3 
*d^7)*sqrt(a)*x^7*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2* 
(280*a^3*c^6*d*x - 240*a^3*c^7 - 16*(15*b^3*c^7 + 161*a*b^2*c^5*d^2 + 245* 
a^2*b*c^3*d^4 + 105*a^3*c*d^6)*x^6 + 105*(11*a*b^2*c^6*d + 18*a^2*b*c^4*d^ 
3 + 8*a^3*c^2*d^5)*x^5 - 16*(45*a*b^2*c^7 + 77*a^2*b*c^5*d^2 + 35*a^3*c^3* 
d^4)*x^4 + 70*(13*a^2*b*c^6*d + 6*a^3*c^4*d^3)*x^3 - 48*(15*a^2*b*c^7 + 7* 
a^3*c^5*d^2)*x^2)*sqrt(b*x^2 + a))/(a*c^8*x^7), -1/3360*(3360*(a*b^2*c^4*d 
^2 + 2*a^2*b*c^2*d^4 + a^3*d^6)*sqrt(-b*c^2 - a*d^2)*x^7*arctan(sqrt(-b*c^ 
2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a 
*b*d^2)*x^2)) - 105*(5*b^3*c^6*d + 30*a*b^2*c^4*d^3 + 40*a^2*b*c^2*d^5 + 1 
6*a^3*d^7)*sqrt(a)*x^7*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) 
 - 2*(280*a^3*c^6*d*x - 240*a^3*c^7 - 16*(15*b^3*c^7 + 161*a*b^2*c^5*d^2 + 
 245*a^2*b*c^3*d^4 + 105*a^3*c*d^6)*x^6 + 105*(11*a*b^2*c^6*d + 18*a^2*b*c 
^4*d^3 + 8*a^3*c^2*d^5)*x^5 - 16*(45*a*b^2*c^7 + 77*a^2*b*c^5*d^2 + 35*a^3 
*c^3*d^4)*x^4 + 70*(13*a^2*b*c^6*d + 6*a^3*c^4*d^3)*x^3 - 48*(15*a^2*b*c^7 
 + 7*a^3*c^5*d^2)*x^2)*sqrt(b*x^2 + a))/(a*c^8*x^7), -1/1680*(105*(5*b^3*c 
^6*d + 30*a*b^2*c^4*d^3 + 40*a^2*b*c^2*d^5 + 16*a^3*d^7)*sqrt(-a)*x^7*a...
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^8 (c+d x)} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {5}{2}}}{x^{8} \left (c + d x\right )}\, dx \] Input:

integrate((b*x**2+a)**(5/2)/x**8/(d*x+c),x)
 

Output:

Integral((a + b*x**2)**(5/2)/(x**8*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^8 (c+d x)} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}}}{{\left (d x + c\right )} x^{8}} \,d x } \] Input:

integrate((b*x^2+a)^(5/2)/x^8/(d*x+c),x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^(5/2)/((d*x + c)*x^8), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1470 vs. \(2 (374) = 748\).

Time = 0.16 (sec) , antiderivative size = 1470, normalized size of antiderivative = 3.54 \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^8 (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(5/2)/x^8/(d*x+c),x, algorithm="giac")
 

Output:

2*(b^3*c^6*d^2 + 3*a*b^2*c^4*d^4 + 3*a^2*b*c^2*d^6 + a^3*d^8)*arctan(-((sq 
rt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/(sqrt(-b*c 
^2 - a*d^2)*c^8) - 1/8*(5*b^3*c^6*d + 30*a*b^2*c^4*d^3 + 40*a^2*b*c^2*d^5 
+ 16*a^3*d^7)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*c^ 
8) - 1/840*(1155*(sqrt(b)*x - sqrt(b*x^2 + a))^13*b^3*c^5*d + 1890*(sqrt(b 
)*x - sqrt(b*x^2 + a))^13*a*b^2*c^3*d^3 + 840*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^13*a^2*b*c*d^5 - 1680*(sqrt(b)*x - sqrt(b*x^2 + a))^12*b^(7/2)*c^6 - 504 
0*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a*b^(5/2)*c^4*d^2 - 5040*(sqrt(b)*x - s 
qrt(b*x^2 + a))^12*a^2*b^(3/2)*c^2*d^4 - 1680*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^12*a^3*sqrt(b)*d^6 - 980*(sqrt(b)*x - sqrt(b*x^2 + a))^11*a*b^3*c^5*d - 
5880*(sqrt(b)*x - sqrt(b*x^2 + a))^11*a^2*b^2*c^3*d^3 - 3360*(sqrt(b)*x - 
sqrt(b*x^2 + a))^11*a^3*b*c*d^5 + 20160*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a 
^2*b^(5/2)*c^4*d^2 + 26880*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*b^(3/2)*c^ 
2*d^4 + 10080*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^4*sqrt(b)*d^6 + 2975*(sqr 
t(b)*x - sqrt(b*x^2 + a))^9*a^2*b^3*c^5*d + 6090*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^9*a^3*b^2*c^3*d^3 + 4200*(sqrt(b)*x - sqrt(b*x^2 + a))^9*a^4*b*c*d^5 
- 8400*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^2*b^(7/2)*c^6 - 40880*(sqrt(b)*x 
- sqrt(b*x^2 + a))^8*a^3*b^(5/2)*c^4*d^2 - 61040*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^8*a^4*b^(3/2)*c^2*d^4 - 25200*(sqrt(b)*x - sqrt(b*x^2 + a))^8*a^5*sqr 
t(b)*d^6 + 49280*(sqrt(b)*x - sqrt(b*x^2 + a))^6*a^4*b^(5/2)*c^4*d^2 + ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^8 (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{5/2}}{x^8\,\left (c+d\,x\right )} \,d x \] Input:

int((a + b*x^2)^(5/2)/(x^8*(c + d*x)),x)
 

Output:

int((a + b*x^2)^(5/2)/(x^8*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{5/2}}{x^8 (c+d x)} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {5}{2}}}{x^{8} \left (d x +c \right )}d x \] Input:

int((b*x^2+a)^(5/2)/x^8/(d*x+c),x)
 

Output:

int((b*x^2+a)^(5/2)/x^8/(d*x+c),x)