\(\int \frac {f+g x}{(d+e x)^3 (a+b x+c x^2)^{3/2}} \, dx\) [986]

Optimal result
Mathematica [A] (verified)
Rubi [A] (verified)
Maple [B] (verified)
Fricas [B] (verification not implemented)
Sympy [F(-1)]
Maxima [F(-2)]
Giac [B] (verification not implemented)
Mupad [F(-1)]
Reduce [B] (verification not implemented)

Optimal result

Integrand size = 27, antiderivative size = 556 \[ \int \frac {f+g x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {e f-d g}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt {a+b x+c x^2}}-\frac {2 c d (5 e f-3 d g)-e (5 b e f-b d g-4 a e g)}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}-\frac {4 a c (2 c d-b e) (2 c d (5 e f-3 d g)-e (5 b e f-b d g-4 a e g))+\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2 f+3 b e (5 b e f-b d g-4 a e g)-4 c (a e (3 e f-5 d g)+b d (7 e f-2 d g))\right )+c \left (16 c^3 d^3 f-3 b^2 e^2 (5 b e f-b d g-4 a e g)-2 c e \left (16 a^2 e^2 g-2 a b e (13 e f-9 d g)-b^2 d (19 e f-5 d g)\right )-8 c^2 d (a e (13 e f-11 d g)+b d (3 e f+d g))\right ) x}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+b x+c x^2}}+\frac {3 e \left (8 c^2 d^2 (2 e f-d g)+b e^2 (5 b e f-b d g-4 a e g)-4 c e (a e (e f-3 d g)+b d (4 e f-d g))\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{7/2}} \] Output:

-1/2*(-d*g+e*f)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2/(c*x^2+b*x+a)^(1/2)-1/4*(2*c 
*d*(-3*d*g+5*e*f)-e*(-4*a*e*g-b*d*g+5*b*e*f))/(a*e^2-b*d*e+c*d^2)^2/(e*x+d 
)/(c*x^2+b*x+a)^(1/2)-1/4*(4*a*c*(-b*e+2*c*d)*(2*c*d*(-3*d*g+5*e*f)-e*(-4* 
a*e*g-b*d*g+5*b*e*f))+(2*a*c*e-b^2*e+b*c*d)*(8*c^2*d^2*f+3*b*e*(-4*a*e*g-b 
*d*g+5*b*e*f)-4*c*(a*e*(-5*d*g+3*e*f)+b*d*(-2*d*g+7*e*f)))+c*(16*c^3*d^3*f 
-3*b^2*e^2*(-4*a*e*g-b*d*g+5*b*e*f)-2*c*e*(16*a^2*e^2*g-2*a*b*e*(-9*d*g+13 
*e*f)-b^2*d*(-5*d*g+19*e*f))-8*c^2*d*(a*e*(-11*d*g+13*e*f)+b*d*(d*g+3*e*f) 
))*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^(1/2)+3/8*e*(8*c^2* 
d^2*(-d*g+2*e*f)+b*e^2*(-4*a*e*g-b*d*g+5*b*e*f)-4*c*e*(a*e*(-3*d*g+e*f)+b* 
d*(-d*g+4*e*f)))*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))/(a*e^2-b*d*e+c*d^2)^(7/2)
 

Mathematica [A] (verified)

Time = 13.29 (sec) , antiderivative size = 504, normalized size of antiderivative = 0.91 \[ \int \frac {f+g x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (\frac {e \left (-8 c^2 d^2 f+b e (-5 b e f+b d g+4 a e g)+4 c (a e (3 e f-5 d g)+b d (2 e f+d g))\right ) \sqrt {a+x (b+c x)}}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}-\frac {e \left (16 c^3 d^3 f+3 b^2 e^2 (-5 b e f+b d g+4 a e g)-8 c^2 d (a e (13 e f-11 d g)+b d (3 e f+d g))-2 c e \left (16 a^2 e^2 g+b^2 d (-19 e f+5 d g)+2 a b e (-13 e f+9 d g)\right )\right ) \sqrt {a+x (b+c x)}}{8 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)}+\frac {b^2 e f-2 c (-a d g+c d f x+a e (f+g x))+b (-a e g+c (-d f+e f x+d g x))}{(d+e x)^2 \sqrt {a+x (b+c x)}}+\frac {3 \left (b^2-4 a c\right ) e \left (8 c^2 d^2 (-2 e f+d g)+b e^2 (-5 b e f+b d g+4 a e g)-4 c e (b d (-4 e f+d g)+a e (-e f+3 d g))\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{16 \left (c d^2+e (-b d+a e)\right )^{5/2}}\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \] Input:

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

Output:

(2*((e*(-8*c^2*d^2*f + b*e*(-5*b*e*f + b*d*g + 4*a*e*g) + 4*c*(a*e*(3*e*f 
- 5*d*g) + b*d*(2*e*f + d*g)))*Sqrt[a + x*(b + c*x)])/(4*(c*d^2 + e*(-(b*d 
) + a*e))*(d + e*x)^2) - (e*(16*c^3*d^3*f + 3*b^2*e^2*(-5*b*e*f + b*d*g + 
4*a*e*g) - 8*c^2*d*(a*e*(13*e*f - 11*d*g) + b*d*(3*e*f + d*g)) - 2*c*e*(16 
*a^2*e^2*g + b^2*d*(-19*e*f + 5*d*g) + 2*a*b*e*(-13*e*f + 9*d*g)))*Sqrt[a 
+ x*(b + c*x)])/(8*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) + (b^2*e*f - 2* 
c*(-(a*d*g) + c*d*f*x + a*e*(f + g*x)) + b*(-(a*e*g) + c*(-(d*f) + e*f*x + 
 d*g*x)))/((d + e*x)^2*Sqrt[a + x*(b + c*x)]) + (3*(b^2 - 4*a*c)*e*(8*c^2* 
d^2*(-2*e*f + d*g) + b*e^2*(-5*b*e*f + b*d*g + 4*a*e*g) - 4*c*e*(b*d*(-4*e 
*f + d*g) + a*e*(-(e*f) + 3*d*g)))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e 
*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(16*(c*d^2 
+ e*(-(b*d) + a*e))^(5/2))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))
 

Rubi [A] (verified)

Time = 1.87 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1235, 27, 1237, 27, 1228, 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.

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

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {2 \int \frac {e \left (-\left ((5 e f-d g) b^2\right )+4 (c d f+a e g) b+12 a c (e f-d g)+4 c (2 c d f+2 a e g-b (e f+d g)) x\right )}{2 (d+e x)^3 \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {e \int \frac {-\left ((5 e f-d g) b^2\right )+4 (c d f+a e g) b+12 a c (e f-d g)+4 c (2 c d f+2 a e g-b (e f+d g)) x}{(d+e x)^3 \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (-4 c (a e (3 e f-5 d g)+b d (d g+2 e f))+b e (-4 a e g-b d g+5 b e f)+8 c^2 d^2 f\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {3 e (5 e f-d g) b^3-4 \left (3 a g e^2+c d (7 e f-2 d g)\right ) b^2+4 c \left (2 c d^2 f-a e (13 e f-7 d g)\right ) b+16 a c \left (2 a g e^2+c d (5 e f-3 d g)\right )+2 c \left (8 c^2 f d^2+b e (5 b e f-b d g-4 a e g)-4 c (a e (3 e f-5 d g)+b d (2 e f+d g))\right ) x}{2 (d+e x)^2 \sqrt {c x^2+b x+a}}dx}{2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {e \left (\frac {\int \frac {3 e (5 e f-d g) b^3-4 \left (3 a g e^2+c d (7 e f-2 d g)\right ) b^2+4 c \left (2 c d^2 f-a e (13 e f-7 d g)\right ) b+16 a c \left (2 a g e^2+c d (5 e f-3 d g)\right )+2 c \left (8 c^2 f d^2+b e (5 b e f-b d g-4 a e g)-4 c (a e (3 e f-5 d g)+b d (2 e f+d g))\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c (a e (3 e f-5 d g)+b d (d g+2 e f))+b e (-4 a e g-b d g+5 b e f)+8 c^2 d^2 f\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {e \left (\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c e \left (16 a^2 e^2 g-2 a b e (13 e f-9 d g)+b^2 (-d) (19 e f-5 d g)\right )-3 b^2 e^2 (-4 a e g-b d g+5 b e f)-8 c^2 d (a e (13 e f-11 d g)+b d (d g+3 e f))+16 c^3 d^3 f\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \left (-4 c e (a e (e f-3 d g)+b d (4 e f-d g))+b e^2 (-4 a e g-b d g+5 b e f)+8 c^2 d^2 (2 e f-d g)\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 \left (a e^2-b d e+c d^2\right )}}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c (a e (3 e f-5 d g)+b d (d g+2 e f))+b e (-4 a e g-b d g+5 b e f)+8 c^2 d^2 f\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {e \left (\frac {\frac {3 \left (b^2-4 a c\right ) \left (-4 c e (a e (e f-3 d g)+b d (4 e f-d g))+b e^2 (-4 a e g-b d g+5 b e f)+8 c^2 d^2 (2 e f-d g)\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{a e^2-b d e+c d^2}+\frac {\sqrt {a+b x+c x^2} \left (-2 c e \left (16 a^2 e^2 g-2 a b e (13 e f-9 d g)+b^2 (-d) (19 e f-5 d g)\right )-3 b^2 e^2 (-4 a e g-b d g+5 b e f)-8 c^2 d (a e (13 e f-11 d g)+b d (d g+3 e f))+16 c^3 d^3 f\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c (a e (3 e f-5 d g)+b d (d g+2 e f))+b e (-4 a e g-b d g+5 b e f)+8 c^2 d^2 f\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {e \left (\frac {\frac {\sqrt {a+b x+c x^2} \left (-2 c e \left (16 a^2 e^2 g-2 a b e (13 e f-9 d g)+b^2 (-d) (19 e f-5 d g)\right )-3 b^2 e^2 (-4 a e g-b d g+5 b e f)-8 c^2 d (a e (13 e f-11 d g)+b d (d g+3 e f))+16 c^3 d^3 f\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right ) \left (-4 c e (a e (e f-3 d g)+b d (4 e f-d g))+b e^2 (-4 a e g-b d g+5 b e f)+8 c^2 d^2 (2 e f-d g)\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c (a e (3 e f-5 d g)+b d (d g+2 e f))+b e (-4 a e g-b d g+5 b e f)+8 c^2 d^2 f\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

(-2*(b*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + c*(2*c*d*f + 2* 
a*e*g - b*(e*f + d*g))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x 
)^2*Sqrt[a + b*x + c*x^2]) - (e*(((8*c^2*d^2*f + b*e*(5*b*e*f - b*d*g - 4* 
a*e*g) - 4*c*(a*e*(3*e*f - 5*d*g) + b*d*(2*e*f + d*g)))*Sqrt[a + b*x + c*x 
^2])/(2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) + (((16*c^3*d^3*f - 3*b^2*e^2 
*(5*b*e*f - b*d*g - 4*a*e*g) - 2*c*e*(16*a^2*e^2*g - 2*a*b*e*(13*e*f - 9*d 
*g) - b^2*d*(19*e*f - 5*d*g)) - 8*c^2*d*(a*e*(13*e*f - 11*d*g) + b*d*(3*e* 
f + d*g)))*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*(d + e*x)) - (3 
*(b^2 - 4*a*c)*(8*c^2*d^2*(2*e*f - d*g) + b*e^2*(5*b*e*f - b*d*g - 4*a*e*g 
) - 4*c*e*(a*e*(e*f - 3*d*g) + b*d*(4*e*f - d*g)))*ArcTanh[(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])])/( 
2*(c*d^2 - b*d*e + a*e^2)^(3/2)))/(4*(c*d^2 - b*d*e + a*e^2))))/((b^2 - 4* 
a*c)*(c*d^2 - b*d*e + a*e^2))
 

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 1228
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 1235
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 1237
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* 
x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) 
*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ 
(c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m 
+ 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 
] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 
Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1853\) vs. \(2(534)=1068\).

Time = 2.72 (sec) , antiderivative size = 1854, normalized size of antiderivative = 3.33

method result size
default \(\text {Expression too large to display}\) \(1854\)

Input:

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

Output:

g/e^3*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)/(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)-3/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1 
/(a*e^2-b*d*e+c*d^2)*e^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)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d 
)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^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)-1/(a*e^2-b*d*e+c*d^2)*e^2/((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)))-4*c/(a*e^2-b*d*e+c*d^2)*e^2*(2 
*c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^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))-(d*g-e* 
f)/e^4*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^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)-5/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^ 
2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)/(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)-3/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1/(a 
*e^2-b*d*e+c*d^2)*e^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)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e 
)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^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)-1/(a*e^2-b*d*e+c*d^2)*e^2/((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*(...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4117 vs. \(2 (534) = 1068\).

Time = 36.83 (sec) , antiderivative size = 8276, normalized size of antiderivative = 14.88 \[ \int \frac {f+g x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4204 vs. \(2 (534) = 1068\).

Time = 0.48 (sec) , antiderivative size = 4204, normalized size of antiderivative = 7.56 \[ \int \frac {f+g x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-2*((2*c^7*d^9*f - 9*b*c^6*d^8*e*f + 18*b^2*c^5*d^7*e^2*f - 21*b^3*c^4*d^6 
*e^3*f + 15*b^4*c^3*d^5*e^4*f + 6*a*b^2*c^4*d^5*e^4*f - 12*a^2*c^5*d^5*e^4 
*f - 6*b^5*c^2*d^4*e^5*f - 15*a*b^3*c^3*d^4*e^5*f + 30*a^2*b*c^4*d^4*e^5*f 
 + b^6*c*d^3*e^6*f + 12*a*b^4*c^2*d^3*e^6*f - 18*a^2*b^2*c^3*d^3*e^6*f - 1 
6*a^3*c^4*d^3*e^6*f - 3*a*b^5*c*d^2*e^7*f - 3*a^2*b^3*c^2*d^2*e^7*f + 24*a 
^3*b*c^3*d^2*e^7*f + 3*a^2*b^4*c*d*e^8*f - 6*a^3*b^2*c^2*d*e^8*f - 6*a^4*c 
^3*d*e^8*f - a^3*b^3*c*e^9*f + 3*a^4*b*c^2*e^9*f - b*c^6*d^9*g + 3*b^2*c^5 
*d^8*e*g + 6*a*c^6*d^8*e*g - 3*b^3*c^4*d^7*e^2*g - 24*a*b*c^5*d^7*e^2*g + 
b^4*c^3*d^6*e^3*g + 34*a*b^2*c^4*d^6*e^3*g + 16*a^2*c^5*d^6*e^3*g - 21*a*b 
^3*c^3*d^5*e^4*g - 42*a^2*b*c^4*d^5*e^4*g + 6*a*b^4*c^2*d^4*e^5*g + 36*a^2 
*b^2*c^3*d^4*e^5*g + 12*a^3*c^4*d^4*e^5*g - a*b^5*c*d^3*e^6*g - 13*a^2*b^3 
*c^2*d^3*e^6*g - 16*a^3*b*c^3*d^3*e^6*g + 3*a^2*b^4*c*d^2*e^7*g + 6*a^3*b^ 
2*c^2*d^2*e^7*g - 3*a^3*b^3*c*d*e^8*g + 3*a^4*b*c^2*d*e^8*g + a^4*b^2*c*e^ 
9*g - 2*a^5*c^2*e^9*g)*x/(b^2*c^6*d^12 - 4*a*c^7*d^12 - 6*b^3*c^5*d^11*e + 
 24*a*b*c^6*d^11*e + 15*b^4*c^4*d^10*e^2 - 54*a*b^2*c^5*d^10*e^2 - 24*a^2* 
c^6*d^10*e^2 - 20*b^5*c^3*d^9*e^3 + 50*a*b^3*c^4*d^9*e^3 + 120*a^2*b*c^5*d 
^9*e^3 + 15*b^6*c^2*d^8*e^4 - 225*a^2*b^2*c^4*d^8*e^4 - 60*a^3*c^5*d^8*e^4 
 - 6*b^7*c*d^7*e^5 - 36*a*b^5*c^2*d^7*e^5 + 180*a^2*b^3*c^3*d^7*e^5 + 240* 
a^3*b*c^4*d^7*e^5 + b^8*d^6*e^6 + 26*a*b^6*c*d^6*e^6 - 30*a^2*b^4*c^2*d^6* 
e^6 - 340*a^3*b^2*c^3*d^6*e^6 - 80*a^4*c^4*d^6*e^6 - 6*a*b^7*d^5*e^7 - ...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 17811, normalized size of antiderivative = 32.03 \[ \int \frac {f+g x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 48*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**3*b*c*d**2*e**4* 
g - 96*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**3*b*c*d*e**5*g*x 
 - 48*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**3*b*c*e**6*g*x**2 
 + 144*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**3*c**2*d**3*e**3 
*g - 48*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**3*c**2*d**2*e** 
4*f + 288*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**3*c**2*d**2*e 
**4*g*x - 96*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sq 
rt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**2*d*e 
**5*f*x + 144*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*s 
qrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**3*c**2*d* 
e**5*g*x**2 - 48*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**3*c**2 
*e**6*f*x**2 + 12*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**2*...