\(\int \frac {(a+b x+c x^2)^{3/2}}{x^6 (d+e x)} \, dx\) [29]

Optimal result

Integrand size = 25, antiderivative size = 577 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=-\frac {a \sqrt {a+b x+c x^2}}{5 d x^5}-\frac {(11 b d-10 a e) \sqrt {a+b x+c x^2}}{40 d^2 x^4}-\frac {\left (3 b^2 d^2+96 a c d^2-90 a b d e+80 a^2 e^2\right ) \sqrt {a+b x+c x^2}}{240 a d^3 x^3}+\frac {\left (15 b^3 d^3-84 a b c d^3+30 a b^2 d^2 e+600 a^2 c d^2 e-560 a^2 b d e^2+480 a^3 e^3\right ) \sqrt {a+b x+c x^2}}{960 a^2 d^4 x^2}-\frac {\left (45 b^4 d^4+90 a b^3 d^3 e-60 a b^2 d^2 \left (5 c d^2-4 a e^2\right )-600 a^2 b d e \left (c d^2+4 a e^2\right )+128 a^2 \left (3 c^2 d^4+20 a c d^2 e^2+15 a^2 e^4\right )\right ) \sqrt {a+b x+c x^2}}{1920 a^3 d^5 x}+\frac {\left (3 b^5 d^5+6 a b^4 d^4 e-48 a^2 b^2 d^2 e \left (c d^2-2 a e^2\right )-8 a b^3 d^3 \left (3 c d^2-2 a e^2\right )+48 a^2 b d \left (c^2 d^4-4 a c d^2 e^2-8 a^2 e^4\right )+32 a^3 e \left (3 c^2 d^4+12 a c d^2 e^2+8 a^2 e^4\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{256 a^{7/2} d^6}+\frac {e^2 \left (c d^2-b d e+a e^2\right )^{3/2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d^6} \] Output:

-1/5*a*(c*x^2+b*x+a)^(1/2)/d/x^5-1/40*(-10*a*e+11*b*d)*(c*x^2+b*x+a)^(1/2) 
/d^2/x^4-1/240*(80*a^2*e^2-90*a*b*d*e+96*a*c*d^2+3*b^2*d^2)*(c*x^2+b*x+a)^ 
(1/2)/a/d^3/x^3+1/960*(480*a^3*e^3-560*a^2*b*d*e^2+600*a^2*c*d^2*e+30*a*b^ 
2*d^2*e-84*a*b*c*d^3+15*b^3*d^3)*(c*x^2+b*x+a)^(1/2)/a^2/d^4/x^2-1/1920*(4 
5*b^4*d^4+90*a*b^3*d^3*e-60*a*b^2*d^2*(-4*a*e^2+5*c*d^2)-600*a^2*b*d*e*(4* 
a*e^2+c*d^2)+128*a^2*(15*a^2*e^4+20*a*c*d^2*e^2+3*c^2*d^4))*(c*x^2+b*x+a)^ 
(1/2)/a^3/d^5/x+1/256*(3*b^5*d^5+6*a*b^4*d^4*e-48*a^2*b^2*d^2*e*(-2*a*e^2+ 
c*d^2)-8*a*b^3*d^3*(-2*a*e^2+3*c*d^2)+48*a^2*b*d*(-8*a^2*e^4-4*a*c*d^2*e^2 
+c^2*d^4)+32*a^3*e*(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^(7/2)/d^6+e^2*(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^6
 

Mathematica [A] (verified)

Time = 9.17 (sec) , antiderivative size = 522, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=\frac {-\sqrt {a} \left (d \sqrt {a+x (b+c x)} \left (45 b^4 d^4 x^4+32 a^4 \left (12 d^4-15 d^3 e x+20 d^2 e^2 x^2-30 d e^3 x^3+60 e^4 x^4\right )-30 a b^2 d^3 x^3 (10 c d x+b (d-3 e x))+12 a^2 d^2 x^2 \left (32 c^2 d^2 x^2+2 b c d x (7 d-25 e x)+b^2 \left (2 d^2-5 d e x+20 e^2 x^2\right )\right )+16 a^3 d x \left (c d x \left (48 d^2-75 d e x+160 e^2 x^2\right )+b \left (33 d^3-45 d^2 e x+70 d e^2 x^2-150 e^3 x^3\right )\right )\right )+3840 a^3 e^2 \left (-c d^2+e (b d-a e)\right )^{3/2} x^5 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )\right )-3840 a^5 e^5 x^5 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+15 d \left (3 b^5 d^4+6 a b^4 d^3 e+8 a b^3 d^2 \left (-3 c d^2+2 a e^2\right )+48 a^2 b^2 d e \left (-c d^2+2 a e^2\right )+96 a^3 c d e \left (c d^2+4 a e^2\right )-48 a^2 b \left (-c^2 d^4+4 a c d^2 e^2+8 a^2 e^4\right )\right ) x^5 \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{1920 a^{7/2} d^6 x^5} \] Input:

Integrate[(a + b*x + c*x^2)^(3/2)/(x^6*(d + e*x)),x]
 

Output:

(-(Sqrt[a]*(d*Sqrt[a + x*(b + c*x)]*(45*b^4*d^4*x^4 + 32*a^4*(12*d^4 - 15* 
d^3*e*x + 20*d^2*e^2*x^2 - 30*d*e^3*x^3 + 60*e^4*x^4) - 30*a*b^2*d^3*x^3*( 
10*c*d*x + b*(d - 3*e*x)) + 12*a^2*d^2*x^2*(32*c^2*d^2*x^2 + 2*b*c*d*x*(7* 
d - 25*e*x) + b^2*(2*d^2 - 5*d*e*x + 20*e^2*x^2)) + 16*a^3*d*x*(c*d*x*(48* 
d^2 - 75*d*e*x + 160*e^2*x^2) + b*(33*d^3 - 45*d^2*e*x + 70*d*e^2*x^2 - 15 
0*e^3*x^3))) + 3840*a^3*e^2*(-(c*d^2) + e*(b*d - a*e))^(3/2)*x^5*ArcTan[(S 
qrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)] 
])) - 3840*a^5*e^5*x^5*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x)])/Sqrt[a] 
] + 15*d*(3*b^5*d^4 + 6*a*b^4*d^3*e + 8*a*b^3*d^2*(-3*c*d^2 + 2*a*e^2) + 4 
8*a^2*b^2*d*e*(-(c*d^2) + 2*a*e^2) + 96*a^3*c*d*e*(c*d^2 + 4*a*e^2) - 48*a 
^2*b*(-(c^2*d^4) + 4*a*c*d^2*e^2 + 8*a^2*e^4))*x^5*ArcTanh[(-(Sqrt[c]*x) + 
 Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(1920*a^(7/2)*d^6*x^5)
 

Rubi [A] (verified)

Time = 1.62 (sec) , antiderivative size = 1154, normalized size of antiderivative = 2.00, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^6 (d+e x)} \, dx\)

\(\Big \downarrow \) 1289

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

\(\Big \downarrow \) 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^5}{d^6}+\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^5}{16 c^{3/2} d^6}-\frac {\left (b^2+2 c x b+8 a c\right ) \sqrt {c x^2+b x+a} e^5}{8 c d^6}-\frac {\left (c x^2+b x+a\right )^{3/2} e^4}{d^5 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^4}{2 d^5}+\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^4}{8 \sqrt {c} d^5}+\frac {3 (3 b+2 c x) \sqrt {c x^2+b x+a} e^4}{4 d^5}+\frac {\left (c x^2+b x+a\right )^{3/2} e^3}{2 d^4 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^3}{8 \sqrt {a} d^4}-\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^3}{2 d^4}+\frac {3 (b-2 c x) \sqrt {c x^2+b x+a} e^3}{4 d^4 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^3}{8 c d^6}-\frac {\left (c x^2+b x+a\right )^{3/2} e^2}{3 d^3 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^2}{16 a^{3/2} d^3}-\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^2}{16 c^{3/2} d^6}+\frac {c^{3/2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) e^2}{d^3}+\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^2}{d^6}-\frac {\left (2 a b+\left (b^2+8 a c\right ) x\right ) \sqrt {c x^2+b x+a} e^2}{8 a d^3 x^2}+\frac {(2 a+b x) \left (c x^2+b x+a\right )^{3/2} e}{8 a d^2 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 ) e}{128 a^{5/2} d^2}-\frac {3 \left (b^2-4 a c\right ) (2 a+b x) \sqrt {c x^2+b x+a} e}{64 a^2 d^2 x^2}-\frac {\left (c x^2+b x+a\right )^{5/2}}{5 a d x^5}+\frac {b (2 a+b x) \left (c x^2+b x+a\right )^{3/2}}{16 a^2 d x^4}+\frac {3 b \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 )}{256 a^{7/2} d}-\frac {3 b \left (b^2-4 a c\right ) (2 a+b x) \sqrt {c x^2+b x+a}}{128 a^3 d x^2}\)

Input:

Int[(a + b*x + c*x^2)^(3/2)/(x^6*(d + e*x)),x]
 

Output:

(-3*b*(b^2 - 4*a*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(128*a^3*d*x^2) - ( 
3*(b^2 - 4*a*c)*e*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(64*a^2*d^2*x^2) + (3 
*e^3*(b - 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d^4*x) + (3*e^4*(3*b + 2*c*x)*S 
qrt[a + b*x + c*x^2])/(4*d^5) - (e^5*(b^2 + 8*a*c + 2*b*c*x)*Sqrt[a + b*x 
+ c*x^2])/(8*c*d^6) - (e^2*(2*a*b + (b^2 + 8*a*c)*x)*Sqrt[a + b*x + c*x^2] 
)/(8*a*d^3*x^2) + (e^3*(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^6) - (e^2*(a + b*x + c*x^ 
2)^(3/2))/(3*d^3*x^3) + (e^3*(a + b*x + c*x^2)^(3/2))/(2*d^4*x^2) - (e^4*( 
a + b*x + c*x^2)^(3/2))/(d^5*x) + (b*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/ 
(16*a^2*d*x^4) + (e*(2*a + b*x)*(a + b*x + c*x^2)^(3/2))/(8*a*d^2*x^4) - ( 
a + b*x + c*x^2)^(5/2)/(5*a*d*x^5) + (3*b*(b^2 - 4*a*c)^2*ArcTanh[(2*a + b 
*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(256*a^(7/2)*d) + (3*(b^2 - 4*a*c) 
^2*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(5/2)* 
d^2) + (b*(b^2 - 12*a*c)*e^2*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + 
 c*x^2])])/(16*a^(3/2)*d^3) + (3*(b^2 + 4*a*c)*e^3*ArcTanh[(2*a + b*x)/(2* 
Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(8*Sqrt[a]*d^4) - (3*Sqrt[a]*b*e^4*ArcTan 
h[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(2*d^5) + (a^(3/2)*e^5*A 
rcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/d^6 + (c^(3/2)*e^2* 
ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/d^3 - (3*b*Sqrt[c] 
*e^3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2*d^4) + ...
 

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 = 1.98 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.22

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (1920 a^{4} e^{4} x^{4}-2400 a^{3} b d \,e^{3} x^{4}+2560 a^{3} c \,d^{2} e^{2} x^{4}+240 a^{2} b^{2} d^{2} e^{2} x^{4}-600 a^{2} b c \,d^{3} e \,x^{4}+384 a^{2} c^{2} d^{4} x^{4}+90 a \,b^{3} d^{3} e \,x^{4}-300 a \,b^{2} c \,d^{4} x^{4}+45 b^{4} d^{4} x^{4}-960 a^{4} d \,e^{3} x^{3}+1120 a^{3} b \,d^{2} e^{2} x^{3}-1200 a^{3} c \,d^{3} e \,x^{3}-60 a^{2} b^{2} d^{3} e \,x^{3}+168 a^{2} b c \,d^{4} x^{3}-30 a \,b^{3} d^{4} x^{3}+640 a^{4} d^{2} e^{2} x^{2}-720 a^{3} b \,d^{3} e \,x^{2}+768 a^{3} c \,d^{4} x^{2}+24 a^{2} b^{2} d^{4} x^{2}-480 a^{4} d^{3} e x +528 a^{3} b \,d^{4} x +384 a^{4} d^{4}\right )}{1920 a^{3} d^{5} x^{5}}-\frac {-\frac {\left (256 e^{5} a^{5}-384 a^{4} b d \,e^{4}+384 a^{4} c \,d^{2} e^{3}+96 a^{3} b^{2} d^{2} e^{3}-192 a^{3} b c \,d^{3} e^{2}+96 a^{3} c^{2} d^{4} e +16 a^{2} b^{3} d^{3} e^{2}-48 a^{2} b^{2} c \,d^{4} e +48 a^{2} b \,c^{2} d^{5}+6 a \,b^{4} d^{4} e -24 a \,b^{3} c \,d^{5}+3 b^{5} d^{5}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}+\frac {256 a^{3} e \left (a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+d^{2} e^{2} b^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{256 a^{3} d^{5}}\)	\(702\)
default	\(\text {Expression too large to display}\)	\(5173\)

Input:

int((c*x^2+b*x+a)^(3/2)/x^6/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

-1/1920*(c*x^2+b*x+a)^(1/2)*(1920*a^4*e^4*x^4-2400*a^3*b*d*e^3*x^4+2560*a^ 
3*c*d^2*e^2*x^4+240*a^2*b^2*d^2*e^2*x^4-600*a^2*b*c*d^3*e*x^4+384*a^2*c^2* 
d^4*x^4+90*a*b^3*d^3*e*x^4-300*a*b^2*c*d^4*x^4+45*b^4*d^4*x^4-960*a^4*d*e^ 
3*x^3+1120*a^3*b*d^2*e^2*x^3-1200*a^3*c*d^3*e*x^3-60*a^2*b^2*d^3*e*x^3+168 
*a^2*b*c*d^4*x^3-30*a*b^3*d^4*x^3+640*a^4*d^2*e^2*x^2-720*a^3*b*d^3*e*x^2+ 
768*a^3*c*d^4*x^2+24*a^2*b^2*d^4*x^2-480*a^4*d^3*e*x+528*a^3*b*d^4*x+384*a 
^4*d^4)/a^3/d^5/x^5-1/256/a^3/d^5*(-(256*a^5*e^5-384*a^4*b*d*e^4+384*a^4*c 
*d^2*e^3+96*a^3*b^2*d^2*e^3-192*a^3*b*c*d^3*e^2+96*a^3*c^2*d^4*e+16*a^2*b^ 
3*d^3*e^2-48*a^2*b^2*c*d^4*e+48*a^2*b*c^2*d^5+6*a*b^4*d^4*e-24*a*b^3*c*d^5 
+3*b^5*d^5)/d/a^(1/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+256*a^ 
3*e*(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 = 11.92 (sec) , antiderivative size = 2575, normalized size of antiderivative = 4.46 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/x^6/(e*x+d),x, algorithm="fricas")
 

Output:

[1/7680*(3840*(a^4*c*d^2*e^2 - a^4*b*d*e^3 + a^5*e^4)*sqrt(c*d^2 - b*d*e + 
 a*e^2)*x^5*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)) - 15*(384*a^4*b*d*e^4 
 - 256*a^5*e^5 - 3*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^5 - 6*(a*b^4 - 8*a^2 
*b^2*c + 16*a^3*c^2)*d^4*e - 16*(a^2*b^3 - 12*a^3*b*c)*d^3*e^2 - 96*(a^3*b 
^2 + 4*a^4*c)*d^2*e^3)*sqrt(a)*x^5*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 + 4*s 
qrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 4*(384*a^5*d^5 - 
(2400*a^4*b*d^2*e^3 - 1920*a^5*d*e^4 - 3*(15*a*b^4 - 100*a^2*b^2*c + 128*a 
^3*c^2)*d^5 - 30*(3*a^2*b^3 - 20*a^3*b*c)*d^4*e - 80*(3*a^3*b^2 + 32*a^4*c 
)*d^3*e^2)*x^4 + 2*(560*a^4*b*d^3*e^2 - 480*a^5*d^2*e^3 - 3*(5*a^2*b^3 - 2 
8*a^3*b*c)*d^5 - 30*(a^3*b^2 + 20*a^4*c)*d^4*e)*x^3 - 8*(90*a^4*b*d^4*e - 
80*a^5*d^3*e^2 - 3*(a^3*b^2 + 32*a^4*c)*d^5)*x^2 + 48*(11*a^4*b*d^5 - 10*a 
^5*d^4*e)*x)*sqrt(c*x^2 + b*x + a))/(a^4*d^6*x^5), 1/7680*(7680*(a^4*c*d^2 
*e^2 - a^4*b*d*e^3 + a^5*e^4)*sqrt(-c*d^2 + b*d*e - a*e^2)*x^5*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)) - 15*(384*a^4*b*d*e^4 - 256*a^5*e^5 - 
 3*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*d^5 - 6*(a*b^4 - 8*a^2*b^2*c + 16*a...
 

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{6} \left (d + e x\right )}\, dx \] Input:

integrate((c*x**2+b*x+a)**(3/2)/x**6/(e*x+d),x)
 

Output:

Integral((a + b*x + c*x**2)**(3/2)/(x**6*(d + e*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )} x^{6}} \,d x } \] Input:

integrate((c*x^2+b*x+a)^(3/2)/x^6/(e*x+d),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(3/2)/((e*x + d)*x^6), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3046 vs. \(2 (539) = 1078\).

Time = 0.56 (sec) , antiderivative size = 3046, normalized size of antiderivative = 5.28 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(3/2)/x^6/(e*x+d),x, algorithm="giac")
 

Output:

2*(c^2*d^4*e^2 - 2*b*c*d^3*e^3 + b^2*d^2*e^4 + 2*a*c*d^2*e^4 - 2*a*b*d*e^5 
 + a^2*e^6)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)*d)/sq 
rt(-c*d^2 + b*d*e - a*e^2))/(sqrt(-c*d^2 + b*d*e - a*e^2)*d^6) - 1/128*(3* 
b^5*d^5 - 24*a*b^3*c*d^5 + 48*a^2*b*c^2*d^5 + 6*a*b^4*d^4*e - 48*a^2*b^2*c 
*d^4*e + 96*a^3*c^2*d^4*e + 16*a^2*b^3*d^3*e^2 - 192*a^3*b*c*d^3*e^2 + 96* 
a^3*b^2*d^2*e^3 + 384*a^4*c*d^2*e^3 - 384*a^4*b*d*e^4 + 256*a^5*e^5)*arcta 
n(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^3*d^6) + 1/19 
20*(45*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*b^5*d^4 - 360*(sqrt(c)*x - sq 
rt(c*x^2 + b*x + a))^9*a*b^3*c*d^4 + 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a 
))^9*a^2*b*c^2*d^4 + 90*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a*b^4*d^3*e 
- 720*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^2*b^2*c*d^3*e - 2400*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^9*a^3*c^2*d^3*e + 240*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^9*a^2*b^3*d^2*e^2 + 4800*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
9*a^3*b*c*d^2*e^2 - 2400*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^3*b^2*d*e 
^3 - 1920*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^9*a^4*c*d*e^3 + 1920*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^9*a^4*b*e^4 + 3840*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))^8*a^3*c^(5/2)*d^4 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a 
^3*b*c^(3/2)*d^3*e + 3840*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^3*b^2*sq 
rt(c)*d^2*e^2 + 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^4*c^(3/2)*d^2 
*e^2 - 7680*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^8*a^4*b*sqrt(c)*d*e^3 +...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^6 (d+e x)} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^6\,\left (d+e\,x\right )} \,d x \] Input:

int((a + b*x + c*x^2)^(3/2)/(x^6*(d + e*x)),x)
 

Output:

int((a + b*x + c*x^2)^(3/2)/(x^6*(d + e*x)), x)
 

Reduce [B] (verification not implemented)

Time = 3.18 (sec) , antiderivative size = 1597, normalized size of antiderivative = 2.77 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^6 (d+e x)} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(3/2)/x^6/(e*x+d),x)
 

Output:

(3840*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e* 
*2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**5*e**4*x**5 - 384 
0*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - 
 b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**4*b*d*e**3*x**5 + 384 
0*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - 
 b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**4*c*d**2*e**2*x**5 - 
3840*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*a**5*e**4*x**5 + 3840*sqrt 
(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*a**4*b*d*e**3*x**5 - 3840*sqrt(a*e* 
*2 - b*d*e + c*d**2)*log(d + e*x)*a**4*c*d**2*e**2*x**5 - 768*sqrt(a + b*x 
 + c*x**2)*a**5*d**5 + 960*sqrt(a + b*x + c*x**2)*a**5*d**4*e*x - 1280*sqr 
t(a + b*x + c*x**2)*a**5*d**3*e**2*x**2 + 1920*sqrt(a + b*x + c*x**2)*a**5 
*d**2*e**3*x**3 - 3840*sqrt(a + b*x + c*x**2)*a**5*d*e**4*x**4 - 1056*sqrt 
(a + b*x + c*x**2)*a**4*b*d**5*x + 1440*sqrt(a + b*x + c*x**2)*a**4*b*d**4 
*e*x**2 - 2240*sqrt(a + b*x + c*x**2)*a**4*b*d**3*e**2*x**3 + 4800*sqrt(a 
+ b*x + c*x**2)*a**4*b*d**2*e**3*x**4 - 1536*sqrt(a + b*x + c*x**2)*a**4*c 
*d**5*x**2 + 2400*sqrt(a + b*x + c*x**2)*a**4*c*d**4*e*x**3 - 5120*sqrt(a 
+ b*x + c*x**2)*a**4*c*d**3*e**2*x**4 - 48*sqrt(a + b*x + c*x**2)*a**3*b** 
2*d**5*x**2 + 120*sqrt(a + b*x + c*x**2)*a**3*b**2*d**4*e*x**3 - 480*sqrt( 
a + b*x + c*x**2)*a**3*b**2*d**3*e**2*x**4 - 336*sqrt(a + b*x + c*x**2)*a* 
*3*b*c*d**5*x**3 + 1200*sqrt(a + b*x + c*x**2)*a**3*b*c*d**4*e*x**4 - 7...