\(\int \frac {(a+b x+c x^2)^{3/2}}{x^2 (d+e x+f x^2)} \, dx\) [139]

Optimal result

Integrand size = 30, antiderivative size = 988 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=-\frac {\left (10 b c d-b^2 e-8 a c e\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {(2 c d-b e) x \sqrt {a+b x+c x^2}}{4 d^2}+\frac {3 (3 b+2 c x) \sqrt {a+b x+c x^2}}{4 d}-\frac {e \left (b^2+8 a c+2 b c x\right ) \sqrt {a+b x+c x^2}}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{d x}-\frac {3 \sqrt {a} b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 d}+\frac {a^{3/2} e \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{d^2}+\frac {3 \left (b^2+4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} d}+\frac {b \left (b^2-12 a c\right ) e \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2}+\frac {\left (16 c^3 d^2-24 a c^2 d f-b^3 e f-6 b c (b d-2 a e) f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{3/2} d^2 f}-\frac {\left ((c d-a f) (c d e-2 b d f+a e f) \left (e-\sqrt {e^2-4 d f}\right )-2 f \left (c^2 d^3-2 a c d^2 f-f \left (b^2 d^2-2 a b d e+a^2 \left (e^2-d f\right )\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}+\frac {\left ((c d-a f) (c d e-2 b d f+a e f) \left (e+\sqrt {e^2-4 d f}\right )-2 f \left (c^2 d^3-2 a c d^2 f-f \left (b^2 d^2-2 a b d e+a^2 \left (e^2-d f\right )\right )\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \] Output:

-1/8*(-8*a*c*e-b^2*e+10*b*c*d)*(c*x^2+b*x+a)^(1/2)/c/d^2-1/4*(-b*e+2*c*d)* 
x*(c*x^2+b*x+a)^(1/2)/d^2+3/4*(2*c*x+3*b)*(c*x^2+b*x+a)^(1/2)/d-1/8*e*(2*b 
*c*x+8*a*c+b^2)*(c*x^2+b*x+a)^(1/2)/c/d^2-(c*x^2+b*x+a)^(3/2)/d/x-3/2*a^(1 
/2)*b*arctanh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/d+a^(3/2)*e*arcta 
nh(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/d^2+3/8*(4*a*c+b^2)*arctanh( 
1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(1/2)/d+1/16*b*(-12*a*c+b^2)* 
e*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/d^2+1/16*(16* 
c^3*d^2-24*a*c^2*d*f-b^3*e*f-6*b*c*(-2*a*e+b*d)*f)*arctanh(1/2*(2*c*x+b)/c 
^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(3/2)/d^2/f-1/2*((-a*f+c*d)*(a*e*f-2*b*d*f+c 
*d*e)*(e-(-4*d*f+e^2)^(1/2))-2*f*(c^2*d^3-2*a*c*d^2*f-f*(b^2*d^2-2*a*b*d*e 
+a^2*(-d*f+e^2))))*arctanh(1/4*(4*a*f-b*(e-(-4*d*f+e^2)^(1/2))+2*(b*f-c*(e 
-(-4*d*f+e^2)^(1/2)))*x)*2^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*( 
-4*d*f+e^2)^(1/2))^(1/2)/(c*x^2+b*x+a)^(1/2))*2^(1/2)/d^2/f/(-4*d*f+e^2)^( 
1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2-(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)+1/2 
*((-a*f+c*d)*(a*e*f-2*b*d*f+c*d*e)*(e+(-4*d*f+e^2)^(1/2))-2*f*(c^2*d^3-2*a 
*c*d^2*f-f*(b^2*d^2-2*a*b*d*e+a^2*(-d*f+e^2))))*arctanh(1/4*(4*a*f-b*(e+(- 
4*d*f+e^2)^(1/2))+2*(b*f-c*(e+(-4*d*f+e^2)^(1/2)))*x)*2^(1/2)/(c*e^2-2*c*d 
*f-b*e*f+2*a*f^2+(-b*f+c*e)*(-4*d*f+e^2)^(1/2))^(1/2)/(c*x^2+b*x+a)^(1/2)) 
*2^(1/2)/d^2/f/(-4*d*f+e^2)^(1/2)/(c*e^2-2*c*d*f-b*e*f+2*a*f^2+(-b*f+c*e)* 
(-4*d*f+e^2)^(1/2))^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.63 (sec) , antiderivative size = 934, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

(Sqrt[a]*(-3*b*d + 2*a*e)*f*x*ArcTanh[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x) 
])/Sqrt[a]] - d*(a*f*Sqrt[a + x*(b + c*x)] + c^(3/2)*d*x*Log[f*(b + 2*c*x 
- 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])]) - x*RootSum[b^2*d - a*b*e + a^2*f - 4 
*b*Sqrt[c]*d*#1 + 2*a*Sqrt[c]*e*#1 + 4*c*d*#1^2 + b*e*#1^2 - 2*a*f*#1^2 - 
2*Sqrt[c]*e*#1^3 + f*#1^4 & , (-(b*c^2*d^3*Log[-(Sqrt[c]*x) + Sqrt[a + b*x 
 + c*x^2] - #1]) + a*c^2*d^2*e*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - 
#1] + b^3*d^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] - 2*a*b^2*d 
*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + a^2*b*e^2*f*Log[-(Sq 
rt[c]*x) + Sqrt[a + b*x + c*x^2] - #1] + a^2*b*d*f^2*Log[-(Sqrt[c]*x) + Sq 
rt[a + b*x + c*x^2] - #1] - a^3*e*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c* 
x^2] - #1] + 2*c^(5/2)*d^3*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]* 
#1 - 2*b^2*Sqrt[c]*d^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 
 - 4*a*c^(3/2)*d^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + 4 
*a*b*Sqrt[c]*d*e*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - 2*a 
^2*Sqrt[c]*e^2*f*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 + 2*a^2 
*Sqrt[c]*d*f^2*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1 - c^2*d^2 
*e*Log[-(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 + 2*b*c*d^2*f*Log[- 
(Sqrt[c]*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 - 2*a*b*d*f^2*Log[-(Sqrt[c] 
*x) + Sqrt[a + b*x + c*x^2] - #1]*#1^2 + a^2*e*f^2*Log[-(Sqrt[c]*x) + Sqrt 
[a + b*x + c*x^2] - #1]*#1^2)/(2*b*Sqrt[c]*d - a*Sqrt[c]*e - 4*c*d*#1 -...
 

Rubi [A] (verified)

Time = 5.24 (sec) , antiderivative size = 1050, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {7279, 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.

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

\(\Big \downarrow \) 7279

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {c x^2+b x+a}}\right ) a^{3/2}}{d^2}-\frac {3 b \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {c x^2+b x+a}}\right ) \sqrt {a}}{2 d}-\frac {\left (c x^2+b x+a\right )^{3/2}}{d x}+\frac {b \left (b^2-12 a c\right ) e \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{16 c^{3/2} d^2}+\frac {\left (-e f b^3-6 c (b d-2 a e) f b+16 c^3 d^2-24 a c^2 d f\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{16 c^{3/2} d^2 f}+\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 )}{8 \sqrt {c} d}-\frac {\left (c^2 \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right ) d^2+2 c f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right ) d^2+f^2 \left (\left (e^2+\sqrt {e^2-4 d f} e-2 d f\right ) a^2-2 b d \left (e+\sqrt {e^2-4 d f}\right ) a+2 b^2 d^2\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {c \left (e^2-\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e-\sqrt {e^2-4 d f}\right )\right )}}+\frac {\left (c^2 \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right ) d^2+2 c f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right ) d^2+f^2 \left (\left (e^2-\sqrt {e^2-4 d f} e-2 d f\right ) a^2-2 b d \left (e-\sqrt {e^2-4 d f}\right ) a+2 b^2 d^2\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-b f e+2 a f^2-2 c d f+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {c x^2+b x+a}}\right )}{\sqrt {2} d^2 f \sqrt {e^2-4 d f} \sqrt {c \left (e^2+\sqrt {e^2-4 d f} e-2 d f\right )+f \left (2 a f-b \left (e+\sqrt {e^2-4 d f}\right )\right )}}+\frac {3 (3 b+2 c x) \sqrt {c x^2+b x+a}}{4 d}-\frac {e \left (b^2+2 c x b+8 a c\right ) \sqrt {c x^2+b x+a}}{8 c d^2}-\frac {\left (-e b^2+10 c d b-8 a c e+2 c (2 c d-b e) x\right ) \sqrt {c x^2+b x+a}}{8 c d^2}\)

Input:

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

Output:

(3*(3*b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*d) - (e*(b^2 + 8*a*c + 2*b*c*x) 
*Sqrt[a + b*x + c*x^2])/(8*c*d^2) - ((10*b*c*d - b^2*e - 8*a*c*e + 2*c*(2* 
c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(8*c*d^2) - (a + b*x + c*x^2)^(3/2)/( 
d*x) - (3*Sqrt[a]*b*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])] 
)/(2*d) + (a^(3/2)*e*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2]) 
])/d^2 + (3*(b^2 + 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c* 
x^2])])/(8*Sqrt[c]*d) + (b*(b^2 - 12*a*c)*e*ArcTanh[(b + 2*c*x)/(2*Sqrt[c] 
*Sqrt[a + b*x + c*x^2])])/(16*c^(3/2)*d^2) + ((16*c^3*d^2 - 24*a*c^2*d*f - 
 b^3*e*f - 6*b*c*(b*d - 2*a*e)*f)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + 
b*x + c*x^2])])/(16*c^(3/2)*d^2*f) - ((c^2*d^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 
 4*d*f]) + 2*c*d^2*f*(2*a*f - b*(e - Sqrt[e^2 - 4*d*f])) + f^2*(2*b^2*d^2 
- 2*a*b*d*(e + Sqrt[e^2 - 4*d*f]) + a^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f] 
)))*ArcTanh[(4*a*f - b*(e - Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e - Sqrt[e^2 
- 4*d*f]))*x)/(2*Sqrt[2]*Sqrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - (c*e - b 
*f)*Sqrt[e^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(Sqrt[2]*d^2*f*Sqrt[e^2 - 
4*d*f]*Sqrt[c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f]) + f*(2*a*f - b*(e - Sqrt 
[e^2 - 4*d*f]))]) + ((c^2*d^2*(e^2 - 2*d*f + e*Sqrt[e^2 - 4*d*f]) + 2*c*d^ 
2*f*(2*a*f - b*(e + Sqrt[e^2 - 4*d*f])) + f^2*(2*b^2*d^2 - 2*a*b*d*(e - Sq 
rt[e^2 - 4*d*f]) + a^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])))*ArcTanh[(4*a* 
f - b*(e + Sqrt[e^2 - 4*d*f]) + 2*(b*f - c*(e + Sqrt[e^2 - 4*d*f]))*x)/...
 

Defintions of rubi rules used

rule 2009

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

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 2.93 (sec) , antiderivative size = 1296, normalized size of antiderivative = 1.31

method	result	size
risch	\(\text {Expression too large to display}\)	\(1296\)
default	\(\text {Expression too large to display}\)	\(3426\)

Input:

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

Output:

-a*(c*x^2+b*x+a)^(1/2)/d/x+1/2/d*(2/f^2*(a^2*f^3*(-4*d*f+e^2)^(1/2)-2*a*c* 
d*f^2*(-4*d*f+e^2)^(1/2)-b^2*d*f^2*(-4*d*f+e^2)^(1/2)+2*(-4*d*f+e^2)^(1/2) 
*b*c*d*e*f+c^2*d^2*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c^2*d*e^2+a^2*e 
*f^3-4*d*f^3*a*b+2*a*c*d*f^2*e+b^2*d*e*f^2+4*d^2*f^2*b*c-2*b*c*d*e^2*f-3*c 
^2*d^2*e*f+c^2*d*e^3)/(-4*d*f+e^2)^(1/2)/(-e+(-4*d*f+e^2)^(1/2))*2^(1/2)/( 
(f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2 
)/f^2)^(1/2)*ln(((f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b* 
e*f-2*d*f*c+c*e^2)/f^2+(c*(-4*d*f+e^2)^(1/2)+f*b-c*e)/f*(x-1/2/f*(-e+(-4*d 
*f+e^2)^(1/2)))+1/2*2^(1/2)*((f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c* 
e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2)^(1/2)*(4*c*(x-1/2/f*(-e+(-4*d*f+e^2)^( 
1/2)))^2+4*(c*(-4*d*f+e^2)^(1/2)+f*b-c*e)/f*(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2 
)))+2*(f*b*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c 
+c*e^2)/f^2)^(1/2))/(x-1/2/f*(-e+(-4*d*f+e^2)^(1/2))))-2/f^2*(a^2*f^3*(-4* 
d*f+e^2)^(1/2)-2*a*c*d*f^2*(-4*d*f+e^2)^(1/2)-b^2*d*f^2*(-4*d*f+e^2)^(1/2) 
+2*(-4*d*f+e^2)^(1/2)*b*c*d*e*f+c^2*d^2*f*(-4*d*f+e^2)^(1/2)-(-4*d*f+e^2)^ 
(1/2)*c^2*d*e^2-a^2*e*f^3+4*d*f^3*a*b-2*a*c*d*f^2*e-b^2*d*e*f^2-4*d^2*f^2* 
b*c+2*b*c*d*e^2*f+3*c^2*d^2*e*f-c^2*d*e^3)/(-4*d*f+e^2)^(1/2)/(e+(-4*d*f+e 
^2)^(1/2))*2^(1/2)/((-f*b*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2)^(1/2)*c*e+2*a*f^ 
2-b*e*f-2*d*f*c+c*e^2)/f^2)^(1/2)*ln(((-f*b*(-4*d*f+e^2)^(1/2)+(-4*d*f+e^2 
)^(1/2)*c*e+2*a*f^2-b*e*f-2*d*f*c+c*e^2)/f^2+1/f*(-c*(-4*d*f+e^2)^(1/2)...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{x^2 \left (d+e x+f x^2\right )} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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