\(\int \frac {(a+b x+c x^2)^{3/2}}{(d+e x+f x^2)^3} \, dx\) [144]

Optimal result

Integrand size = 27, antiderivative size = 669 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^3} \, dx=-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}+\frac {3 \left (4 c d e+4 a e f-b \left (e^2+4 d f\right )+2 \left (c e^2-2 b e f+4 a f^2\right ) x\right ) \sqrt {a+b x+c x^2}}{4 \left (e^2-4 d f\right )^2 \left (d+e x+f x^2\right )}+\frac {3 \left (4 b e f (c d+3 a f)-b^2 f \left (e^2+4 d f\right )-4 a f \left (c e^2+4 a f^2\right )-2 (2 c d-b e+2 a f) (c e-b f) \left (e-\sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \sqrt {c e^2-2 c d f-b e f+2 a f^2-(c e-b f) \sqrt {e^2-4 d f}}}-\frac {3 \left (4 b e f (c d+3 a f)-b^2 f \left (e^2+4 d f\right )-4 a f \left (c e^2+4 a f^2\right )-2 (2 c d-b e+2 a f) (c e-b f) \left (e+\sqrt {e^2-4 d f}\right )\right ) \text {arctanh}\left (\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{2 \sqrt {2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}} \sqrt {a+b x+c x^2}}\right )}{4 \sqrt {2} \left (e^2-4 d f\right )^{5/2} \sqrt {c e^2-2 c d f-b e f+2 a f^2+(c e-b f) \sqrt {e^2-4 d f}}} \] Output:

-1/2*(2*f*x+e)*(c*x^2+b*x+a)^(3/2)/(-4*d*f+e^2)/(f*x^2+e*x+d)^2+3/4*(4*c*d 
*e+4*a*e*f-b*(4*d*f+e^2)+2*(4*a*f^2-2*b*e*f+c*e^2)*x)*(c*x^2+b*x+a)^(1/2)/ 
(-4*d*f+e^2)^2/(f*x^2+e*x+d)+3/8*(4*b*e*f*(3*a*f+c*d)-b^2*f*(4*d*f+e^2)-4* 
a*f*(4*a*f^2+c*e^2)-2*(2*a*f-b*e+2*c*d)*(-b*f+c*e)*(e-(-4*d*f+e^2)^(1/2))) 
*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)/(-4*d*f+e^2)^(5/2)/(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)-3/8*(4*b*e*f*(3*a*f+c*d)-b^ 
2*f*(4*d*f+e^2)-4*a*f*(4*a*f^2+c*e^2)-2*(2*a*f-b*e+2*c*d)*(-b*f+c*e)*(e+(- 
4*d*f+e^2)^(1/2)))*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)/(-4*d*f+e^2)^(5/2)/( 
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 [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(4727\) vs. \(2(669)=1338\).

Time = 18.30 (sec) , antiderivative size = 4727, normalized size of antiderivative = 7.07 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{\left (d+e x+f x^2\right )^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

(-2*f^2*(a + x*(b + c*x))^(3/2))/((e^2 - 4*d*f)^(3/2)*(e - Sqrt[e^2 - 4*d* 
f] + 2*f*x)^2) + (6*f^2*(a + x*(b + c*x))^(3/2))/((e^2 - 4*d*f)^2*(e - Sqr 
t[e^2 - 4*d*f] + 2*f*x)) + (2*f^2*(a + x*(b + c*x))^(3/2))/((e^2 - 4*d*f)^ 
(3/2)*(e + Sqrt[e^2 - 4*d*f] + 2*f*x)^2) + (6*f^2*(a + x*(b + c*x))^(3/2)) 
/((e^2 - 4*d*f)^2*(e + Sqrt[e^2 - 4*d*f] + 2*f*x)) + (9*f^2*(a + x*(b + c* 
x))^(3/2)*(((-4*b*c*f - 2*c*(b*f + 2*c*(-e + Sqrt[e^2 - 4*d*f])) - 4*c^2*f 
*x)*Sqrt[a + b*x + c*x^2])/(8*c*f^2) - ((2*Sqrt[c]*(b^2*f^2 + 4*c^2*(e^2 - 
 2*d*f - e*Sqrt[e^2 - 4*d*f]) + 4*c*f*(a*f - b*(e - Sqrt[e^2 - 4*d*f])))*A 
rcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/f + (2*Sqrt[2]*Sqrt 
[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e^2 
- 4*d*f]]*(4*c*f*(8*a*b*f^2 - 3*b^2*f*(e - Sqrt[e^2 - 4*d*f]) - 4*a*c*f*(e 
 - Sqrt[e^2 - 4*d*f]) + 4*b*c*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f])) + 4*c*( 
-e + Sqrt[e^2 - 4*d*f])*(b^2*f^2 + 4*c^2*(e^2 - 2*d*f - e*Sqrt[e^2 - 4*d*f 
]) + 4*c*f*(a*f - b*(e - Sqrt[e^2 - 4*d*f]))))*ArcTanh[(-4*a*f - b*(-e + S 
qrt[e^2 - 4*d*f]) - (2*b*f + 2*c*(-e + Sqrt[e^2 - 4*d*f]))*x)/(2*Sqrt[2]*S 
qrt[c*e^2 - 2*c*d*f - b*e*f + 2*a*f^2 - c*e*Sqrt[e^2 - 4*d*f] + b*f*Sqrt[e 
^2 - 4*d*f]]*Sqrt[a + b*x + c*x^2])])/(f*(16*a*f^2 + 8*b*f*(-e + Sqrt[e^2 
- 4*d*f]) + 4*c*(-e + Sqrt[e^2 - 4*d*f])^2)))/(16*c*f^2)))/((e^2 - 4*d*f)^ 
2*(a + b*x + c*x^2)^(3/2)) - (3*f^2*(a + x*(b + c*x))^(3/2)*(((-4*c*f*(4*a 
*f - b*(e - Sqrt[e^2 - 4*d*f])) - 2*(b*f - c*(e - Sqrt[e^2 - 4*d*f]))*(...
 

Rubi [A] (warning: unable to verify)

Time = 1.13 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1302, 27, 1346, 27, 1365, 1154, 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.

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

\(\Big \downarrow \) 1302

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1346

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1365

\(\displaystyle \frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (2 x \left (4 a f^2-2 b e f+c e^2\right )+4 a e f-b \left (4 d f+e^2\right )+4 c d e\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}-\frac {\frac {2 \left (2 \left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) (2 a f-b e+2 c d)-f \left (4 b e (3 a f+c d)-4 a \left (4 a f^2+c e^2\right )-\left (b^2 \left (4 d f+e^2\right )\right )\right )\right ) \int \frac {1}{\left (e+2 f x+\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}-\frac {2 \left (2 \left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) (2 a f-b e+2 c d)-f \left (4 b e (3 a f+c d)-4 a \left (4 a f^2+c e^2\right )-\left (b^2 \left (4 d f+e^2\right )\right )\right )\right ) \int \frac {1}{\left (e+2 f x-\sqrt {e^2-4 d f}\right ) \sqrt {c x^2+b x+a}}dx}{\sqrt {e^2-4 d f}}}{2 \left (e^2-4 d f\right )}\right )}{4 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {3 \left (\frac {\sqrt {a+b x+c x^2} \left (2 x \left (4 a f^2-2 b e f+c e^2\right )+4 a e f-b \left (4 d f+e^2\right )+4 c d e\right )}{\left (e^2-4 d f\right ) \left (d+e x+f x^2\right )}-\frac {\frac {4 \left (2 \left (e-\sqrt {e^2-4 d f}\right ) (c e-b f) (2 a f-b e+2 c d)-f \left (4 b e (3 a f+c d)-4 a \left (4 a f^2+c e^2\right )-\left (b^2 \left (4 d f+e^2\right )\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e-\sqrt {e^2-4 d f}\right ) f+c \left (e-\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e-\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e-\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}-\frac {4 \left (2 \left (\sqrt {e^2-4 d f}+e\right ) (c e-b f) (2 a f-b e+2 c d)-f \left (4 b e (3 a f+c d)-4 a \left (4 a f^2+c e^2\right )-\left (b^2 \left (4 d f+e^2\right )\right )\right )\right ) \int \frac {1}{4 \left (4 a f^2-2 b \left (e+\sqrt {e^2-4 d f}\right ) f+c \left (e+\sqrt {e^2-4 d f}\right )^2\right )-\frac {\left (4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x\right )^2}{c x^2+b x+a}}d\frac {4 a f-b \left (e+\sqrt {e^2-4 d f}\right )+2 \left (b f-c \left (e+\sqrt {e^2-4 d f}\right )\right ) x}{\sqrt {c x^2+b x+a}}}{\sqrt {e^2-4 d f}}}{2 \left (e^2-4 d f\right )}\right )}{4 \left (e^2-4 d f\right )}-\frac {(e+2 f x) \left (a+b x+c x^2\right )^{3/2}}{2 \left (e^2-4 d f\right ) \left (d+e x+f x^2\right )^2}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*((e + 2*f*x)*(a + b*x + c*x^2)^(3/2))/((e^2 - 4*d*f)*(d + e*x + f*x^2 
)^2) + (3*(((4*c*d*e + 4*a*e*f - b*(e^2 + 4*d*f) + 2*(c*e^2 - 2*b*e*f + 4* 
a*f^2)*x)*Sqrt[a + b*x + c*x^2])/((e^2 - 4*d*f)*(d + e*x + f*x^2)) - ((Sqr 
t[2]*(2*(2*c*d - b*e + 2*a*f)*(c*e - b*f)*(e - Sqrt[e^2 - 4*d*f]) - f*(4*b 
*e*(c*d + 3*a*f) - b^2*(e^2 + 4*d*f) - 4*a*(c*e^2 + 4*a*f^2)))*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[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]]) - (Sqrt[2]*(2*(2*c*d - 
b*e + 2*a*f)*(c*e - b*f)*(e + Sqrt[e^2 - 4*d*f]) - f*(4*b*e*(c*d + 3*a*f) 
- b^2*(e^2 + 4*d*f) - 4*a*(c*e^2 + 4*a*f^2)))*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[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]]))/(2*(e^2 - 4*d*f))))/(4*(e^2 - 4*d*f))
 

Defintions of rubi rules used

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 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1302

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x 
_)^2)^(q_), x_Symbol] :> Simp[(b + 2*c*x)*(a + b*x + c*x^2)^(p + 1)*((d + e 
*x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(p + 1)) 
   Int[(a + b*x + c*x^2)^(p + 1)*(d + e*x + f*x^2)^(q - 1)*Simp[2*c*d*(2*p 
+ 3) + b*e*q + (2*b*f*q + 2*c*e*(2*p + q + 3))*x + 2*c*f*(2*p + 2*q + 3)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ 
[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0] &&  !IGtQ[q, 0]
 

rule 1346

Int[((g_.) + (h_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e 
_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp[(g*b - 2*a*h - (b*h - 2*g* 
c)*x)*(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^q/((b^2 - 4*a*c)*(p + 1) 
)), x] - Simp[1/((b^2 - 4*a*c)*(p + 1))   Int[(a + b*x + c*x^2)^(p + 1)*(d 
+ e*x + f*x^2)^(q - 1)*Simp[e*q*(g*b - 2*a*h) - d*(b*h - 2*g*c)*(2*p + 3) + 
 (2*f*q*(g*b - 2*a*h) - e*(b*h - 2*g*c)*(2*p + q + 3))*x - f*(b*h - 2*g*c)* 
(2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Ne 
Q[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && LtQ[p, -1] && GtQ[q, 0]
 

rule 1365

Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + ( 
e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim 
p[(2*c*g - h*(b - q))/q   Int[1/((b - q + 2*c*x)*Sqrt[d + e*x + f*x^2]), x] 
, x] - Simp[(2*c*g - h*(b + q))/q   Int[1/((b + q + 2*c*x)*Sqrt[d + e*x + f 
*x^2]), x], x]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0 
] && NeQ[e^2 - 4*d*f, 0] && PosQ[b^2 - 4*a*c]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(16308\) vs. \(2(609)=1218\).

Time = 3.82 (sec) , antiderivative size = 16309, normalized size of antiderivative = 24.38

method	result	size
default	\(\text {Expression too large to display}\)	\(16309\)

Input:

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

Output:

result too large to display
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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