\(\int \frac {(a+b x+c x^2)^{3/2}}{x^5 (d+e x)} \, dx\) [28]

Optimal result

Integrand size = 25, antiderivative size = 417 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=-\frac {a \sqrt {a+b x+c x^2}}{4 d x^4}-\frac {(9 b d-8 a e) \sqrt {a+b x+c x^2}}{24 d^2 x^3}-\frac {\left (3 b^2 d^2+60 a c d^2-56 a b d e+48 a^2 e^2\right ) \sqrt {a+b x+c x^2}}{96 a d^3 x^2}+\frac {\left (9 b^3 d^3+24 a b^2 d^2 e+64 a^2 e \left (4 c d^2+3 a e^2\right )-60 a b d \left (c d^2+4 a e^2\right )\right ) \sqrt {a+b x+c x^2}}{192 a^2 d^4 x}-\frac {\left (3 b^4 d^4+8 a b^3 d^3 e-24 a b^2 d^2 \left (c d^2-2 a e^2\right )-96 a^2 b d e \left (c d^2+2 a e^2\right )+16 a^2 \left (3 c^2 d^4+12 a c d^2 e^2+8 a^2 e^4\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{5/2} d^5}-\frac {e \left (c d^2-b d e+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d^5} \] Output:

-1/4*a*(c*x^2+b*x+a)^(1/2)/d/x^4-1/24*(-8*a*e+9*b*d)*(c*x^2+b*x+a)^(1/2)/d 
^2/x^3-1/96*(48*a^2*e^2-56*a*b*d*e+60*a*c*d^2+3*b^2*d^2)*(c*x^2+b*x+a)^(1/ 
2)/a/d^3/x^2+1/192*(9*b^3*d^3+24*a*b^2*d^2*e+64*a^2*e*(3*a*e^2+4*c*d^2)-60 
*a*b*d*(4*a*e^2+c*d^2))*(c*x^2+b*x+a)^(1/2)/a^2/d^4/x-1/128*(3*b^4*d^4+8*a 
*b^3*d^3*e-24*a*b^2*d^2*(-2*a*e^2+c*d^2)-96*a^2*b*d*e*(2*a*e^2+c*d^2)+16*a 
^2*(8*a^2*e^4+12*a*c*d^2*e^2+3*c^2*d^4))*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c* 
x^2+b*x+a)^(1/2))/a^(5/2)/d^5-e*(a*e^2-b*d*e+c*d^2)^(3/2)*arctanh(1/2*(b*d 
-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/d^5
 

Mathematica [A] (verified)

Time = 5.22 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\frac {\sqrt {a} \left (d \sqrt {a+x (b+c x)} \left (9 b^3 d^3 x^3-16 a^3 \left (3 d^3-4 d^2 e x+6 d e^2 x^2-12 e^3 x^3\right )-6 a b d^2 x^2 (10 c d x+b (d-4 e x))-8 a^2 d x \left (c d x (15 d-32 e x)+b \left (9 d^2-14 d e x+30 e^2 x^2\right )\right )\right )+384 a^2 e \left (-c d^2+e (b d-a e)\right )^{3/2} x^4 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )\right )+384 a^4 e^4 x^4 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-3 d \left (3 b^4 d^3+8 a b^3 d^2 e+24 a b^2 d \left (-c d^2+2 a e^2\right )-96 a^2 b e \left (c d^2+2 a e^2\right )+48 a^2 c d \left (c d^2+4 a e^2\right )\right ) x^4 \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{192 a^{5/2} d^5 x^4} \] Input:

Integrate[(a + b*x + c*x^2)^(3/2)/(x^5*(d + e*x)),x]
 

Output:

(Sqrt[a]*(d*Sqrt[a + x*(b + c*x)]*(9*b^3*d^3*x^3 - 16*a^3*(3*d^3 - 4*d^2*e 
*x + 6*d*e^2*x^2 - 12*e^3*x^3) - 6*a*b*d^2*x^2*(10*c*d*x + b*(d - 4*e*x)) 
- 8*a^2*d*x*(c*d*x*(15*d - 32*e*x) + b*(9*d^2 - 14*d*e*x + 30*e^2*x^2))) + 
 384*a^2*e*(-(c*d^2) + e*(b*d - a*e))^(3/2)*x^4*ArcTan[(Sqrt[c]*(d + e*x) 
- e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]]) + 384*a^4*e^4* 
x^4*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a]] - 3*d*(3*b^4*d^3 
+ 8*a*b^3*d^2*e + 24*a*b^2*d*(-(c*d^2) + 2*a*e^2) - 96*a^2*b*e*(c*d^2 + 2* 
a*e^2) + 48*a^2*c*d*(c*d^2 + 4*a*e^2))*x^4*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a 
+ x*(b + c*x)])/Sqrt[a]])/(192*a^(5/2)*d^5*x^4)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(982\) vs. \(2(417)=834\).

Time = 1.44 (sec) , antiderivative size = 982, normalized size of antiderivative = 2.35, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1289, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5 (d+e x)} \, dx\)

\(\Big \downarrow \) 1289

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^{3/2} \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {c x^2+b x+a}}\right ) e^4}{d^5}-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^4}{16 c^{3/2} d^5}+\frac {\left (b^2+2 c x b+8 a c\right ) \sqrt {c x^2+b x+a} e^4}{8 c d^5}+\frac {\left (c x^2+b x+a\right )^{3/2} e^3}{d^4 x}+\frac {3 \sqrt {a} b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {c x^2+b x+a}}\right ) e^3}{2 d^4}-\frac {3 \left (b^2+4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^3}{8 \sqrt {c} d^4}-\frac {3 (3 b+2 c x) \sqrt {c x^2+b x+a} e^3}{4 d^4}-\frac {\left (c x^2+b x+a\right )^{3/2} e^2}{2 d^3 x^2}-\frac {3 \left (b^2+4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {c x^2+b x+a}}\right ) e^2}{8 \sqrt {a} d^3}+\frac {3 b \sqrt {c} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^2}{2 d^3}-\frac {3 (b-2 c x) \sqrt {c x^2+b x+a} e^2}{4 d^3 x}-\frac {\left (8 c^2 d^2+b^2 e^2-2 c e (5 b d-4 a e)-2 c e (2 c d-b e) x\right ) \sqrt {c x^2+b x+a} e^2}{8 c d^5}+\frac {\left (c x^2+b x+a\right )^{3/2} e}{3 d^2 x^3}-\frac {b \left (b^2-12 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {c x^2+b x+a}}\right ) e}{16 a^{3/2} d^2}+\frac {(2 c d-b e) \left (8 c^2 d^2-b^2 e^2-4 c e (2 b d-3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e}{16 c^{3/2} d^5}-\frac {c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e}{d^2}-\frac {\left (c d^2-b e d+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b e d+a e^2} \sqrt {c x^2+b x+a}}\right ) e}{d^5}+\frac {\left (2 a b+\left (b^2+8 a c\right ) x\right ) \sqrt {c x^2+b x+a} e}{8 a d^2 x^2}-\frac {(2 a+b x) \left (c x^2+b x+a\right )^{3/2}}{8 a d x^4}-\frac {3 \left (b^2-4 a c\right )^2 \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {c x^2+b x+a}}\right )}{128 a^{5/2} d}+\frac {3 \left (b^2-4 a c\right ) (2 a+b x) \sqrt {c x^2+b x+a}}{64 a^2 d x^2}\)

Input:

Int[(a + b*x + c*x^2)^(3/2)/(x^5*(d + e*x)),x]
 

Output:

(3*(b^2 - 4*a*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(64*a^2*d*x^2) - (3*e^ 
2*(b - 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d^3*x) - (3*e^3*(3*b + 2*c*x)*Sqrt 
[a + b*x + c*x^2])/(4*d^4) + (e^4*(b^2 + 8*a*c + 2*b*c*x)*Sqrt[a + b*x + c 
*x^2])/(8*c*d^5) + (e*(2*a*b + (b^2 + 8*a*c)*x)*Sqrt[a + b*x + c*x^2])/(8* 
a*d^2*x^2) - (e^2*(8*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 4*a*e) - 2*c*e*(2* 
c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*c*d^5) + (e*(a + b*x + c*x^2)^(3/2 
))/(3*d^2*x^3) - (e^2*(a + b*x + c*x^2)^(3/2))/(2*d^3*x^2) + (e^3*(a + b*x 
 + c*x^2)^(3/2))/(d^4*x) - ((2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(8*a*d*x^ 
4) - (3*(b^2 - 4*a*c)^2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^ 
2])])/(128*a^(5/2)*d) - (b*(b^2 - 12*a*c)*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a] 
*Sqrt[a + b*x + c*x^2])])/(16*a^(3/2)*d^2) - (3*(b^2 + 4*a*c)*e^2*ArcTanh[ 
(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[a]*d^3) + (3*Sqrt[ 
a]*b*e^3*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*d^4) - 
 (a^(3/2)*e^4*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d^5 
- (c^(3/2)*e*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/d^2 + 
 (3*b*Sqrt[c]*e^2*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/ 
(2*d^3) - (3*(b^2 + 4*a*c)*e^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x 
 + c*x^2])])/(8*Sqrt[c]*d^4) - (b*(b^2 - 12*a*c)*e^4*ArcTanh[(b + 2*c*x)/( 
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^5) + (e*(2*c*d - b*e)*(8* 
c^2*d^2 - b^2*e^2 - 4*c*e*(2*b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[...
 

Defintions of rubi rules used

rule 1289

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( 
IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
 

rule 2009

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

Maple [A] (verified)

Time = 1.69 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.26

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-192 x^{3} e^{3} a^{3}+240 x^{3} a^{2} b d \,e^{2}-256 a^{2} c \,d^{2} e \,x^{3}-24 x^{3} a \,b^{2} d^{2} e +60 a b c \,d^{3} x^{3}-9 x^{3} b^{3} d^{3}+96 a^{3} d \,e^{2} x^{2}-112 a^{2} b \,d^{2} e \,x^{2}+120 a^{2} c \,d^{3} x^{2}+6 a \,b^{2} d^{3} x^{2}-64 x \,d^{2} e \,a^{3}+72 a^{2} b \,d^{3} x +48 a^{3} d^{3}\right )}{192 a^{2} d^{4} x^{4}}+\frac {-\frac {\left (128 a^{4} e^{4}-192 a^{3} b d \,e^{3}+192 a^{3} c \,d^{2} e^{2}+48 a^{2} b^{2} d^{2} e^{2}-96 a^{2} b c \,d^{3} e +48 a^{2} c^{2} d^{4}+8 a \,b^{3} d^{3} e -24 b^{2} c \,d^{4} a +3 b^{4} d^{4}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}+\frac {128 a^{2} \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{128 a^{2} d^{4}}\)	\(526\)
default	\(\text {Expression too large to display}\)	\(3789\)

Input:

int((c*x^2+b*x+a)^(3/2)/x^5/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

-1/192*(c*x^2+b*x+a)^(1/2)*(-192*a^3*e^3*x^3+240*a^2*b*d*e^2*x^3-256*a^2*c 
*d^2*e*x^3-24*a*b^2*d^2*e*x^3+60*a*b*c*d^3*x^3-9*b^3*d^3*x^3+96*a^3*d*e^2* 
x^2-112*a^2*b*d^2*e*x^2+120*a^2*c*d^3*x^2+6*a*b^2*d^3*x^2-64*a^3*d^2*e*x+7 
2*a^2*b*d^3*x+48*a^3*d^3)/a^2/d^4/x^4+1/128/a^2/d^4*(-(128*a^4*e^4-192*a^3 
*b*d*e^3+192*a^3*c*d^2*e^2+48*a^2*b^2*d^2*e^2-96*a^2*b*c*d^3*e+48*a^2*c^2* 
d^4+8*a*b^3*d^3*e-24*a*b^2*c*d^4+3*b^4*d^4)/d/a^(1/2)*ln((2*a+b*x+2*a^(1/2 
)*(c*x^2+b*x+a)^(1/2))/x)+128*a^2*(a^2*e^4-2*a*b*d*e^3+2*a*c*d^2*e^2+b^2*d 
^2*e^2-2*b*c*d^3*e+c^2*d^4)/d/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2 
-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)* 
(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e) 
))
 

Fricas [A] (verification not implemented)

Time = 5.22 (sec) , antiderivative size = 2011, normalized size of antiderivative = 4.82 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/x^5/(e*x+d),x, algorithm="fricas")
 

Output:

[1/768*(384*(a^3*c*d^2*e - a^3*b*d*e^2 + a^4*e^3)*sqrt(c*d^2 - b*d*e + a*e 
^2)*x^4*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b* 
c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 
+ b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3 
*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - 3*(192*a^3*b*d*e^3 - 12 
8*a^4*e^4 - 3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^4 - 8*(a*b^3 - 12*a^2*b*c)* 
d^3*e - 48*(a^2*b^2 + 4*a^3*c)*d^2*e^2)*sqrt(a)*x^4*log(-(8*a*b*x + (b^2 + 
 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 
4*(48*a^4*d^4 + (240*a^3*b*d^2*e^2 - 192*a^4*d*e^3 - 3*(3*a*b^3 - 20*a^2*b 
*c)*d^4 - 8*(3*a^2*b^2 + 32*a^3*c)*d^3*e)*x^3 - 2*(56*a^3*b*d^3*e - 48*a^4 
*d^2*e^2 - 3*(a^2*b^2 + 20*a^3*c)*d^4)*x^2 + 8*(9*a^3*b*d^4 - 8*a^4*d^3*e) 
*x)*sqrt(c*x^2 + b*x + a))/(a^3*d^5*x^4), -1/768*(768*(a^3*c*d^2*e - a^3*b 
*d*e^2 + a^4*e^3)*sqrt(-c*d^2 + b*d*e - a*e^2)*x^4*arctan(-1/2*sqrt(-c*d^2 
 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/(a 
*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 
- b^2*d*e + a*b*e^2)*x)) + 3*(192*a^3*b*d*e^3 - 128*a^4*e^4 - 3*(b^4 - 8*a 
*b^2*c + 16*a^2*c^2)*d^4 - 8*(a*b^3 - 12*a^2*b*c)*d^3*e - 48*(a^2*b^2 + 4* 
a^3*c)*d^2*e^2)*sqrt(a)*x^4*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x 
^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) + 4*(48*a^4*d^4 + (240*a^3 
*b*d^2*e^2 - 192*a^4*d*e^3 - 3*(3*a*b^3 - 20*a^2*b*c)*d^4 - 8*(3*a^2*b^...
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{5} \left (d + e x\right )}\, dx \] Input:

integrate((c*x**2+b*x+a)**(3/2)/x**5/(e*x+d),x)
 

Output:

Integral((a + b*x + c*x**2)**(3/2)/(x**5*(d + e*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )} x^{5}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(3/2)/x^5/(e*x+d),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(3/2)/((e*x + d)*x^5), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1853 vs. \(2 (383) = 766\).

Time = 0.41 (sec) , antiderivative size = 1853, normalized size of antiderivative = 4.44 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/x^5/(e*x+d),x, algorithm="giac")
 

Output:

-2*(c^2*d^4*e - 2*b*c*d^3*e^2 + b^2*d^2*e^3 + 2*a*c*d^2*e^3 - 2*a*b*d*e^4 
+ a^2*e^5)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqr 
t(-c*d^2 + b*d*e - a*e^2))/(sqrt(-c*d^2 + b*d*e - a*e^2)*d^5) + 1/64*(3*b^ 
4*d^4 - 24*a*b^2*c*d^4 + 48*a^2*c^2*d^4 + 8*a*b^3*d^3*e - 96*a^2*b*c*d^3*e 
 + 48*a^2*b^2*d^2*e^2 + 192*a^3*c*d^2*e^2 - 192*a^3*b*d*e^3 + 128*a^4*e^4) 
*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^2*d^5) 
- 1/192*(9*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*b^4*d^3 - 72*(sqrt(c)*x - 
 sqrt(c*x^2 + b*x + a))^7*a*b^2*c*d^3 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x 
+ a))^7*a^2*c^2*d^3 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a*b^3*d^2*e 
 + 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^2*b*c*d^2*e - 240*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^7*a^2*b^2*d*e^2 - 192*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x + a))^7*a^3*c*d*e^2 + 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*a^3*b 
*e^3 - 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b*c^(3/2)*d^3 + 384*( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))^6*a^2*b^2*sqrt(c)*d^2*e + 768*(sqrt(c)* 
x - sqrt(c*x^2 + b*x + a))^6*a^3*c^(3/2)*d^2*e - 768*(sqrt(c)*x - sqrt(c*x 
^2 + b*x + a))^6*a^3*b*sqrt(c)*d*e^2 + 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
 a))^6*a^4*sqrt(c)*e^3 - 33*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^4*d^ 
3 - 504*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*b^2*c*d^3 - 144*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^5*a^3*c^2*d^3 + 40*(sqrt(c)*x - sqrt(c*x^2 + b 
*x + a))^5*a^2*b^3*d^2*e - 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^...
 

Mupad [F(-1)]

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

int((a + b*x + c*x^2)^(3/2)/(x^5*(d + e*x)),x)
 

Output:

int((a + b*x + c*x^2)^(3/2)/(x^5*(d + e*x)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^5 (d+e x)} \, dx=\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{x^{5} \left (e x +d \right )}d x \] Input:

int((c*x^2+b*x+a)^(3/2)/x^5/(e*x+d),x)
 

Output:

int((c*x^2+b*x+a)^(3/2)/x^5/(e*x+d),x)