3.1.87 \(\int \frac {\sqrt {a+c x^2} (d+e x+f x^2)}{(g+h x)^6} \, dx\) [87]

3.1.87.1 Optimal result
3.1.87.2 Mathematica [A] (verified)
3.1.87.3 Rubi [A] (verified)
3.1.87.4 Maple [B] (verified)
3.1.87.5 Fricas [B] (verification not implemented)
3.1.87.6 Sympy [F]
3.1.87.7 Maxima [B] (verification not implemented)
3.1.87.8 Giac [B] (verification not implemented)
3.1.87.9 Mupad [F(-1)]

3.1.87.1 Optimal result

Integrand size = 29, antiderivative size = 433 \[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=-\frac {c \left (4 c^2 d g^3+a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )\right ) (a h-c g x) \sqrt {a+c x^2}}{8 \left (c g^2+a h^2\right )^4 (g+h x)^2}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{3/2}}{5 h \left (c g^2+a h^2\right ) (g+h x)^5}+\frac {\left (5 a h^2 (2 f g-e h)+c g \left (3 f g^2+h (2 e g-7 d h)\right )\right ) \left (a+c x^2\right )^{3/2}}{20 h \left (c g^2+a h^2\right )^2 (g+h x)^4}-\frac {\left (20 a^2 f h^4-c^2 g^2 \left (3 f g^2+h (2 e g-27 d h)\right )-a c h^2 \left (18 f g^2-h (33 e g-8 d h)\right )\right ) \left (a+c x^2\right )^{3/2}}{60 h \left (c g^2+a h^2\right )^3 (g+h x)^3}-\frac {a c^2 \left (4 c^2 d g^3+a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{8 \left (c g^2+a h^2\right )^{9/2}} \]

output
-1/5*(d*h^2-e*g*h+f*g^2)*(c*x^2+a)^(3/2)/h/(a*h^2+c*g^2)/(h*x+g)^5+1/20*(5 
*a*h^2*(-e*h+2*f*g)+c*g*(3*f*g^2+h*(-7*d*h+2*e*g)))*(c*x^2+a)^(3/2)/h/(a*h 
^2+c*g^2)^2/(h*x+g)^4-1/60*(20*a^2*f*h^4-c^2*g^2*(3*f*g^2+h*(-27*d*h+2*e*g 
))-a*c*h^2*(18*f*g^2-h*(-8*d*h+33*e*g)))*(c*x^2+a)^(3/2)/h/(a*h^2+c*g^2)^3 
/(h*x+g)^3-1/8*a*c^2*(4*c^2*d*g^3+a^2*h^2*(-e*h+6*f*g)-a*c*g*(f*g^2-3*h*(- 
d*h+2*e*g)))*arctanh((-c*g*x+a*h)/(a*h^2+c*g^2)^(1/2)/(c*x^2+a)^(1/2))/(a* 
h^2+c*g^2)^(9/2)-1/8*c*(4*c^2*d*g^3+a^2*h^2*(-e*h+6*f*g)-a*c*g*(f*g^2-3*h* 
(-d*h+2*e*g)))*(-c*g*x+a*h)*(c*x^2+a)^(1/2)/(a*h^2+c*g^2)^4/(h*x+g)^2
 
3.1.87.2 Mathematica [A] (verified)

Time = 10.88 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=-\frac {\sqrt {a+c x^2} \left (24 \left (c g^2+a h^2\right )^4 \left (f g^2+h (-e g+d h)\right )-6 \left (c g^2+a h^2\right )^3 \left (11 c f g^3+c g h (-6 e g+d h)-5 a h^2 (-2 f g+e h)\right ) (g+h x)+2 \left (c g^2+a h^2\right )^2 \left (20 a^2 f h^4+c^2 \left (27 f g^4-g^2 h (2 e g+3 d h)\right )+a c h^2 \left (54 f g^2+h (-9 e g+4 d h)\right )\right ) (g+h x)^2-c \left (c g^2+a h^2\right ) \left (5 a^2 h^4 (10 f g-3 e h)+a c g h^2 \left (21 f g^2+h (24 e g-29 d h)\right )+c^2 \left (6 f g^5+2 g^3 h (2 e g+3 d h)\right )\right ) (g+h x)^3-c \left (-40 a^3 f h^6+a c^2 g^2 h^2 \left (27 f g^2+h (28 e g-83 d h)\right )+c^3 \left (6 f g^6+2 g^4 h (2 e g+3 d h)\right )+a^2 c h^4 \left (86 f g^2+h (-81 e g+16 d h)\right )\right ) (g+h x)^4\right )}{120 h^3 \left (c g^2+a h^2\right )^4 (g+h x)^5}+\frac {a c^2 \left (4 c^2 d g^3+a^2 h^2 (6 f g-e h)-a c g \left (f g^2+3 h (-2 e g+d h)\right )\right ) \log (g+h x)}{8 \left (c g^2+a h^2\right )^{9/2}}-\frac {a c^2 \left (4 c^2 d g^3+a^2 h^2 (6 f g-e h)-a c g \left (f g^2+3 h (-2 e g+d h)\right )\right ) \log \left (a h-c g x+\sqrt {c g^2+a h^2} \sqrt {a+c x^2}\right )}{8 \left (c g^2+a h^2\right )^{9/2}} \]

input
Integrate[(Sqrt[a + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^6,x]
 
output
-1/120*(Sqrt[a + c*x^2]*(24*(c*g^2 + a*h^2)^4*(f*g^2 + h*(-(e*g) + d*h)) - 
 6*(c*g^2 + a*h^2)^3*(11*c*f*g^3 + c*g*h*(-6*e*g + d*h) - 5*a*h^2*(-2*f*g 
+ e*h))*(g + h*x) + 2*(c*g^2 + a*h^2)^2*(20*a^2*f*h^4 + c^2*(27*f*g^4 - g^ 
2*h*(2*e*g + 3*d*h)) + a*c*h^2*(54*f*g^2 + h*(-9*e*g + 4*d*h)))*(g + h*x)^ 
2 - c*(c*g^2 + a*h^2)*(5*a^2*h^4*(10*f*g - 3*e*h) + a*c*g*h^2*(21*f*g^2 + 
h*(24*e*g - 29*d*h)) + c^2*(6*f*g^5 + 2*g^3*h*(2*e*g + 3*d*h)))*(g + h*x)^ 
3 - c*(-40*a^3*f*h^6 + a*c^2*g^2*h^2*(27*f*g^2 + h*(28*e*g - 83*d*h)) + c^ 
3*(6*f*g^6 + 2*g^4*h*(2*e*g + 3*d*h)) + a^2*c*h^4*(86*f*g^2 + h*(-81*e*g + 
 16*d*h)))*(g + h*x)^4))/(h^3*(c*g^2 + a*h^2)^4*(g + h*x)^5) + (a*c^2*(4*c 
^2*d*g^3 + a^2*h^2*(6*f*g - e*h) - a*c*g*(f*g^2 + 3*h*(-2*e*g + d*h)))*Log 
[g + h*x])/(8*(c*g^2 + a*h^2)^(9/2)) - (a*c^2*(4*c^2*d*g^3 + a^2*h^2*(6*f* 
g - e*h) - a*c*g*(f*g^2 + 3*h*(-2*e*g + d*h)))*Log[a*h - c*g*x + Sqrt[c*g^ 
2 + a*h^2]*Sqrt[a + c*x^2]])/(8*(c*g^2 + a*h^2)^(9/2))
 
3.1.87.3 Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2182, 25, 688, 25, 27, 679, 486, 488, 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.

\(\displaystyle \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx\)

\(\Big \downarrow \) 2182

\(\displaystyle -\frac {\int -\frac {\left (5 (c d g-a f g+a e h)+\left (5 a f h+c \left (\frac {3 f g^2}{h}+2 e g-2 d h\right )\right ) x\right ) \sqrt {c x^2+a}}{(g+h x)^5}dx}{5 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (5 (c d g-a f g+a e h)+\left (5 a f h+c \left (\frac {3 f g^2}{h}+2 e g-2 d h\right )\right ) x\right ) \sqrt {c x^2+a}}{(g+h x)^5}dx}{5 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {\frac {\left (a+c x^2\right )^{3/2} \left (5 a h^2 (2 f g-e h)+c g h (2 e g-7 d h)+3 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )}-\frac {\int -\frac {\left (4 h \left (5 c^2 d g^2+5 a^2 f h^2-a c \left (2 f g^2-h (7 e g-2 d h)\right )\right )+c \left (3 c f g^3+c h (2 e g-7 d h) g+5 a h^2 (2 f g-e h)\right ) x\right ) \sqrt {c x^2+a}}{h (g+h x)^4}dx}{4 \left (a h^2+c g^2\right )}}{5 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\left (4 h \left (5 c^2 d g^2+5 a^2 f h^2-a c \left (2 f g^2-h (7 e g-2 d h)\right )\right )+c \left (3 c f g^3+c h (2 e g-7 d h) g+5 a h^2 (2 f g-e h)\right ) x\right ) \sqrt {c x^2+a}}{h (g+h x)^4}dx}{4 \left (a h^2+c g^2\right )}+\frac {\left (a+c x^2\right )^{3/2} \left (5 a h^2 (2 f g-e h)+c g h (2 e g-7 d h)+3 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )}}{5 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (4 h \left (5 c^2 d g^2+5 a^2 f h^2-a c \left (2 f g^2-h (7 e g-2 d h)\right )\right )+c \left (3 c f g^3+c h (2 e g-7 d h) g+5 a h^2 (2 f g-e h)\right ) x\right ) \sqrt {c x^2+a}}{(g+h x)^4}dx}{4 h \left (a h^2+c g^2\right )}+\frac {\left (a+c x^2\right )^{3/2} \left (5 a h^2 (2 f g-e h)+c g h (2 e g-7 d h)+3 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )}}{5 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {\frac {\frac {5 c h \left (a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )+4 c^2 d g^3\right ) \int \frac {\sqrt {c x^2+a}}{(g+h x)^3}dx}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (20 a^2 f h^4-a c h^2 \left (18 f g^2-h (33 e g-8 d h)\right )-c^2 \left (g^2 h (2 e g-27 d h)+3 f g^4\right )\right )}{3 (g+h x)^3 \left (a h^2+c g^2\right )}}{4 h \left (a h^2+c g^2\right )}+\frac {\left (a+c x^2\right )^{3/2} \left (5 a h^2 (2 f g-e h)+c g h (2 e g-7 d h)+3 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )}}{5 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {\frac {\frac {5 c h \left (a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )+4 c^2 d g^3\right ) \left (\frac {a c \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{2 \left (a h^2+c g^2\right )}-\frac {\sqrt {a+c x^2} (a h-c g x)}{2 (g+h x)^2 \left (a h^2+c g^2\right )}\right )}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (20 a^2 f h^4-a c h^2 \left (18 f g^2-h (33 e g-8 d h)\right )-c^2 \left (g^2 h (2 e g-27 d h)+3 f g^4\right )\right )}{3 (g+h x)^3 \left (a h^2+c g^2\right )}}{4 h \left (a h^2+c g^2\right )}+\frac {\left (a+c x^2\right )^{3/2} \left (5 a h^2 (2 f g-e h)+c g h (2 e g-7 d h)+3 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )}}{5 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {\frac {5 c h \left (a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )+4 c^2 d g^3\right ) \left (-\frac {a c \int \frac {1}{c g^2+a h^2-\frac {(a h-c g x)^2}{c x^2+a}}d\frac {a h-c g x}{\sqrt {c x^2+a}}}{2 \left (a h^2+c g^2\right )}-\frac {\sqrt {a+c x^2} (a h-c g x)}{2 (g+h x)^2 \left (a h^2+c g^2\right )}\right )}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (20 a^2 f h^4-a c h^2 \left (18 f g^2-h (33 e g-8 d h)\right )-c^2 \left (g^2 h (2 e g-27 d h)+3 f g^4\right )\right )}{3 (g+h x)^3 \left (a h^2+c g^2\right )}}{4 h \left (a h^2+c g^2\right )}+\frac {\left (a+c x^2\right )^{3/2} \left (5 a h^2 (2 f g-e h)+c g h (2 e g-7 d h)+3 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )}}{5 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {5 c h \left (-\frac {a c \text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right )}{2 \left (a h^2+c g^2\right )^{3/2}}-\frac {\sqrt {a+c x^2} (a h-c g x)}{2 (g+h x)^2 \left (a h^2+c g^2\right )}\right ) \left (a^2 h^2 (6 f g-e h)-a c g \left (f g^2-3 h (2 e g-d h)\right )+4 c^2 d g^3\right )}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{3/2} \left (20 a^2 f h^4-a c h^2 \left (18 f g^2-h (33 e g-8 d h)\right )-c^2 \left (g^2 h (2 e g-27 d h)+3 f g^4\right )\right )}{3 (g+h x)^3 \left (a h^2+c g^2\right )}}{4 h \left (a h^2+c g^2\right )}+\frac {\left (a+c x^2\right )^{3/2} \left (5 a h^2 (2 f g-e h)+c g h (2 e g-7 d h)+3 c f g^3\right )}{4 h (g+h x)^4 \left (a h^2+c g^2\right )}}{5 \left (a h^2+c g^2\right )}-\frac {\left (a+c x^2\right )^{3/2} \left (d h^2-e g h+f g^2\right )}{5 h (g+h x)^5 \left (a h^2+c g^2\right )}\)

input
Int[(Sqrt[a + c*x^2]*(d + e*x + f*x^2))/(g + h*x)^6,x]
 
output
-1/5*((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(3/2))/(h*(c*g^2 + a*h^2)*(g + h 
*x)^5) + (((3*c*f*g^3 + c*g*h*(2*e*g - 7*d*h) + 5*a*h^2*(2*f*g - e*h))*(a 
+ c*x^2)^(3/2))/(4*h*(c*g^2 + a*h^2)*(g + h*x)^4) + (-1/3*((20*a^2*f*h^4 - 
 c^2*(3*f*g^4 + g^2*h*(2*e*g - 27*d*h)) - a*c*h^2*(18*f*g^2 - h*(33*e*g - 
8*d*h)))*(a + c*x^2)^(3/2))/((c*g^2 + a*h^2)*(g + h*x)^3) + (5*c*h*(4*c^2* 
d*g^3 + a^2*h^2*(6*f*g - e*h) - a*c*g*(f*g^2 - 3*h*(2*e*g - d*h)))*(-1/2*( 
(a*h - c*g*x)*Sqrt[a + c*x^2])/((c*g^2 + a*h^2)*(g + h*x)^2) - (a*c*ArcTan 
h[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2]*Sqrt[a + c*x^2])])/(2*(c*g^2 + a*h^2) 
^(3/2))))/(c*g^2 + a*h^2))/(4*h*(c*g^2 + a*h^2)))/(5*(c*g^2 + a*h^2))
 

3.1.87.3.1 Defintions of rubi rules used

rule 25
Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 486
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), 
x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2)))   Int[(c + d*x)^(n + 2)*(a + 
b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && 
GtQ[p, 0]
 

rule 488
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 679
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 688
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 2182
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 
3.1.87.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6084\) vs. \(2(409)=818\).

Time = 0.86 (sec) , antiderivative size = 6085, normalized size of antiderivative = 14.05

method result size
default \(\text {Expression too large to display}\) \(6085\)

input
int((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^6,x,method=_RETURNVERBOSE)
 
output
result too large to display
 
3.1.87.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1918 vs. \(2 (410) = 820\).

Time = 165.64 (sec) , antiderivative size = 3862, normalized size of antiderivative = 8.92 \[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\text {Too large to display} \]

input
integrate((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^6,x, algorithm="fricas")
 
output
[-1/240*(15*(6*a^2*c^3*e*g^7*h - a^3*c^2*e*g^5*h^3 + (4*a*c^4*d - a^2*c^3* 
f)*g^8 - 3*(a^2*c^3*d - 2*a^3*c^2*f)*g^6*h^2 + (6*a^2*c^3*e*g^2*h^6 - a^3* 
c^2*e*h^8 + (4*a*c^4*d - a^2*c^3*f)*g^3*h^5 - 3*(a^2*c^3*d - 2*a^3*c^2*f)* 
g*h^7)*x^5 + 5*(6*a^2*c^3*e*g^3*h^5 - a^3*c^2*e*g*h^7 + (4*a*c^4*d - a^2*c 
^3*f)*g^4*h^4 - 3*(a^2*c^3*d - 2*a^3*c^2*f)*g^2*h^6)*x^4 + 10*(6*a^2*c^3*e 
*g^4*h^4 - a^3*c^2*e*g^2*h^6 + (4*a*c^4*d - a^2*c^3*f)*g^5*h^3 - 3*(a^2*c^ 
3*d - 2*a^3*c^2*f)*g^3*h^5)*x^3 + 10*(6*a^2*c^3*e*g^5*h^3 - a^3*c^2*e*g^3* 
h^5 + (4*a*c^4*d - a^2*c^3*f)*g^6*h^2 - 3*(a^2*c^3*d - 2*a^3*c^2*f)*g^4*h^ 
4)*x^2 + 5*(6*a^2*c^3*e*g^6*h^2 - a^3*c^2*e*g^4*h^4 + (4*a*c^4*d - a^2*c^3 
*f)*g^7*h - 3*(a^2*c^3*d - 2*a^3*c^2*f)*g^5*h^3)*x)*sqrt(c*g^2 + a*h^2)*lo 
g((2*a*c*g*h*x - a*c*g^2 - 2*a^2*h^2 - (2*c^2*g^2 + a*c*h^2)*x^2 + 2*sqrt( 
c*g^2 + a*h^2)*(c*g*x - a*h)*sqrt(c*x^2 + a))/(h^2*x^2 + 2*g*h*x + g^2)) - 
 2*(40*a*c^4*e*g^9 - 46*a^2*c^3*e*g^7*h^2 - 113*a^3*c^2*e*g^5*h^4 - 33*a^4 
*c*e*g^3*h^6 - 6*a^5*e*g*h^8 - 24*a^5*d*h^9 - 9*(20*a*c^4*d - 9*a^2*c^3*f) 
*g^8*h - (329*a^2*c^3*d - 53*a^3*c^2*f)*g^6*h^3 - (247*a^3*c^2*d + 32*a^4* 
c*f)*g^4*h^5 - 2*(61*a^4*c*d + 2*a^5*f)*g^2*h^7 + (6*c^5*f*g^8*h + 4*c^5*e 
*g^7*h^2 + 32*a*c^4*e*g^5*h^4 - 53*a^2*c^3*e*g^3*h^6 - 81*a^3*c^2*e*g*h^8 
+ 3*(2*c^5*d + 11*a*c^4*f)*g^6*h^3 - (77*a*c^4*d - 113*a^2*c^3*f)*g^4*h^5 
- (67*a^2*c^3*d - 46*a^3*c^2*f)*g^2*h^7 + 8*(2*a^3*c^2*d - 5*a^4*c*f)*h^9) 
*x^4 + 5*(6*c^5*f*g^9 + 4*c^5*e*g^8*h + 32*a*c^4*e*g^6*h^3 - 35*a^2*c^3...
 
3.1.87.6 Sympy [F]

\[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\int \frac {\sqrt {a + c x^{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{6}}\, dx \]

input
integrate((f*x**2+e*x+d)*(c*x**2+a)**(1/2)/(h*x+g)**6,x)
 
output
Integral(sqrt(a + c*x**2)*(d + e*x + f*x**2)/(g + h*x)**6, x)
 
3.1.87.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5793 vs. \(2 (410) = 820\).

Time = 0.42 (sec) , antiderivative size = 5793, normalized size of antiderivative = 13.38 \[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\text {Too large to display} \]

input
integrate((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^6,x, algorithm="maxima")
 
output
-7/8*sqrt(c*x^2 + a)*c^4*f*g^6/(c^4*g^8*h^4*x + 4*a*c^3*g^6*h^6*x + 6*a^2* 
c^2*g^4*h^8*x + 4*a^3*c*g^2*h^10*x + a^4*h^12*x + c^4*g^9*h^3 + 4*a*c^3*g^ 
7*h^5 + 6*a^2*c^2*g^5*h^7 + 4*a^3*c*g^3*h^9 + a^4*g*h^11) + 7/8*sqrt(c*x^2 
 + a)*c^4*e*g^5/(c^4*g^8*h^3*x + 4*a*c^3*g^6*h^5*x + 6*a^2*c^2*g^4*h^7*x + 
 4*a^3*c*g^2*h^9*x + a^4*h^11*x + c^4*g^9*h^2 + 4*a*c^3*g^7*h^4 + 6*a^2*c^ 
2*g^5*h^6 + 4*a^3*c*g^3*h^8 + a^4*g*h^10) - 7/8*(c*x^2 + a)^(3/2)*c^3*f*g^ 
5/(c^4*g^8*h^3*x^2 + 4*a*c^3*g^6*h^5*x^2 + 6*a^2*c^2*g^4*h^7*x^2 + 4*a^3*c 
*g^2*h^9*x^2 + a^4*h^11*x^2 + 2*c^4*g^9*h^2*x + 8*a*c^3*g^7*h^4*x + 12*a^2 
*c^2*g^5*h^6*x + 8*a^3*c*g^3*h^8*x + 2*a^4*g*h^10*x + c^4*g^10*h + 4*a*c^3 
*g^8*h^3 + 6*a^2*c^2*g^6*h^5 + 4*a^3*c*g^4*h^7 + a^4*g^2*h^9) + 7/8*sqrt(c 
*x^2 + a)*c^4*f*g^5/(c^4*g^8*h^3 + 4*a*c^3*g^6*h^5 + 6*a^2*c^2*g^4*h^7 + 4 
*a^3*c*g^2*h^9 + a^4*h^11) - 7/8*sqrt(c*x^2 + a)*c^4*d*g^4/(c^4*g^8*h^2*x 
+ 4*a*c^3*g^6*h^4*x + 6*a^2*c^2*g^4*h^6*x + 4*a^3*c*g^2*h^8*x + a^4*h^10*x 
 + c^4*g^9*h + 4*a*c^3*g^7*h^3 + 6*a^2*c^2*g^5*h^5 + 4*a^3*c*g^3*h^7 + a^4 
*g*h^9) + 7/8*(c*x^2 + a)^(3/2)*c^3*e*g^4/(c^4*g^8*h^2*x^2 + 4*a*c^3*g^6*h 
^4*x^2 + 6*a^2*c^2*g^4*h^6*x^2 + 4*a^3*c*g^2*h^8*x^2 + a^4*h^10*x^2 + 2*c^ 
4*g^9*h*x + 8*a*c^3*g^7*h^3*x + 12*a^2*c^2*g^5*h^5*x + 8*a^3*c*g^3*h^7*x + 
 2*a^4*g*h^9*x + c^4*g^10 + 4*a*c^3*g^8*h^2 + 6*a^2*c^2*g^6*h^4 + 4*a^3*c* 
g^4*h^6 + a^4*g^2*h^8) - 7/8*sqrt(c*x^2 + a)*c^4*e*g^4/(c^4*g^8*h^2 + 4*a* 
c^3*g^6*h^4 + 6*a^2*c^2*g^4*h^6 + 4*a^3*c*g^2*h^8 + a^4*h^10) - 7/8*(c*...
 
3.1.87.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4170 vs. \(2 (410) = 820\).

Time = 0.40 (sec) , antiderivative size = 4170, normalized size of antiderivative = 9.63 \[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\text {Too large to display} \]

input
integrate((f*x^2+e*x+d)*(c*x^2+a)^(1/2)/(h*x+g)^6,x, algorithm="giac")
 
output
-1/4*(4*a*c^4*d*g^3 - a^2*c^3*f*g^3 + 6*a^2*c^3*e*g^2*h - 3*a^2*c^3*d*g*h^ 
2 + 6*a^3*c^2*f*g*h^2 - a^3*c^2*e*h^3)*arctan(((sqrt(c)*x - sqrt(c*x^2 + a 
))*h + sqrt(c)*g)/sqrt(-c*g^2 - a*h^2))/((c^4*g^8 + 4*a*c^3*g^6*h^2 + 6*a^ 
2*c^2*g^4*h^4 + 4*a^3*c*g^2*h^6 + a^4*h^8)*sqrt(-c*g^2 - a*h^2)) - 1/60*(6 
0*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a*c^4*d*g^3*h^8 - 15*(sqrt(c)*x - sqrt(c 
*x^2 + a))^9*a^2*c^3*f*g^3*h^8 + 90*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^ 
3*e*g^2*h^9 - 45*(sqrt(c)*x - sqrt(c*x^2 + a))^9*a^2*c^3*d*g*h^10 + 90*(sq 
rt(c)*x - sqrt(c*x^2 + a))^9*a^3*c^2*f*g*h^10 - 15*(sqrt(c)*x - sqrt(c*x^2 
 + a))^9*a^3*c^2*e*h^11 - 120*(sqrt(c)*x - sqrt(c*x^2 + a))^8*c^(11/2)*f*g 
^8*h^3 - 480*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a*c^(9/2)*f*g^6*h^5 + 540*(sq 
rt(c)*x - sqrt(c*x^2 + a))^8*a*c^(9/2)*d*g^4*h^7 - 855*(sqrt(c)*x - sqrt(c 
*x^2 + a))^8*a^2*c^(7/2)*f*g^4*h^7 + 810*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a 
^2*c^(7/2)*e*g^3*h^8 - 405*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^2*c^(7/2)*d*g 
^2*h^9 + 330*(sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*f*g^2*h^9 - 135*( 
sqrt(c)*x - sqrt(c*x^2 + a))^8*a^3*c^(5/2)*e*g*h^10 - 120*(sqrt(c)*x - sqr 
t(c*x^2 + a))^8*a^4*c^(3/2)*f*h^11 - 240*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c 
^6*f*g^9*h^2 - 160*(sqrt(c)*x - sqrt(c*x^2 + a))^7*c^6*e*g^8*h^3 - 960*(sq 
rt(c)*x - sqrt(c*x^2 + a))^7*a*c^5*f*g^7*h^4 - 640*(sqrt(c)*x - sqrt(c*x^2 
 + a))^7*a*c^5*e*g^6*h^5 + 1880*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a*c^5*d*g^ 
5*h^6 - 1910*(sqrt(c)*x - sqrt(c*x^2 + a))^7*a^2*c^4*f*g^5*h^6 + 1860*(...
 
3.1.87.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+c x^2} \left (d+e x+f x^2\right )}{(g+h x)^6} \, dx=\int \frac {\sqrt {c\,x^2+a}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^6} \,d x \]

input
int(((a + c*x^2)^(1/2)*(d + e*x + f*x^2))/(g + h*x)^6,x)
 
output
int(((a + c*x^2)^(1/2)*(d + e*x + f*x^2))/(g + h*x)^6, x)