\(\int \frac {(f+g x) (a+b x+c x^2)^{3/2}}{(d+e x)^3} \, dx\) [886]

Optimal result

Integrand size = 27, antiderivative size = 353 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {3 (4 c d (e f-2 d g)-e (b e f-4 b d g+2 a e g)+e (2 c e f-4 c d g+b e g) x) \sqrt {a+b x+c x^2}}{4 e^4 (d+e x)}-\frac {(e f-2 d g-e g x) \left (a+b x+c x^2\right )^{3/2}}{2 e^2 (d+e x)^2}+\frac {3 \left (b^2 e^2 g-8 c^2 d (e f-2 d g)+4 c e (b e f-3 b d g+a e g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c} e^5}+\frac {3 \left (8 c^2 d^2 (e f-2 d g)+b e^2 (b e f-5 b d g+4 a e g)-4 c e (b d (2 e f-5 d g)-a e (e f-3 d g))\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 e^5 \sqrt {c d^2-b d e+a e^2}} \] Output:

3/4*(4*c*d*(-2*d*g+e*f)-e*(2*a*e*g-4*b*d*g+b*e*f)+e*(b*e*g-4*c*d*g+2*c*e*f 
)*x)*(c*x^2+b*x+a)^(1/2)/e^4/(e*x+d)-1/2*(-e*g*x-2*d*g+e*f)*(c*x^2+b*x+a)^ 
(3/2)/e^2/(e*x+d)^2+3/8*(b^2*e^2*g-8*c^2*d*(-2*d*g+e*f)+4*c*e*(a*e*g-3*b*d 
*g+b*e*f))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(1/2)/e^5+ 
3/8*(8*c^2*d^2*(-2*d*g+e*f)+b*e^2*(4*a*e*g-5*b*d*g+b*e*f)-4*c*e*(b*d*(-5*d 
*g+2*e*f)-a*e*(-3*d*g+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))/e^5/(a*e^2-b*d*e+c*d^2)^(1/2)
 

Mathematica [A] (verified)

Time = 13.02 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.81 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\frac {\frac {(-e f+d g) (a+x (b+c x))^{5/2}}{(d+e x)^2}-\frac {(2 c d (-e f+3 d g)+e (b e f-5 b d g+4 a e g)) (a+x (b+c x))^{5/2}}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)}-\frac {\frac {(a+x (b+c x))^{3/2} \left (b e^2 (b e f-5 b d g+4 a e g)-2 c^2 d \left (4 d^2 g+e^2 f x-d e (2 f+3 g x)\right )+c e \left (b \left (13 d^2 g+e^2 f x-5 d e (f+g x)\right )+2 a e (-3 d g+e (f+2 g x))\right )\right )}{2 e^2}+\frac {3 \left (2 c^2 e \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)} \left (b e^2 (b e f-4 b d g+3 a e g)-2 c^2 d (e f-2 d g) (-2 d+e x)+c e \left (2 a e (e f-3 d g+e g x)+b \left (-5 d e f+12 d^2 g+e^2 f x-3 d e g x\right )\right )\right )+c^{3/2} \left (c d^2+e (-b d+a e)\right )^2 \left (b^2 e^2 g+8 c^2 d (-e f+2 d g)+4 c e (b e f-3 b d g+a e g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )+c^2 \left (c d^2+e (-b d+a e)\right )^{3/2} \left (8 c^2 d^2 (-e f+2 d g)-b e^2 (b e f-5 b d g+4 a e g)-4 c e (a e (e f-3 d g)+b d (-2 e f+5 d g))\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )\right )}{4 c^2 e^5}}{-c d^2+e (b d-a e)}}{2 \left (c d^2+e (-b d+a e)\right )} \] Input:

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

Output:

(((-(e*f) + d*g)*(a + x*(b + c*x))^(5/2))/(d + e*x)^2 - ((2*c*d*(-(e*f) + 
3*d*g) + e*(b*e*f - 5*b*d*g + 4*a*e*g))*(a + x*(b + c*x))^(5/2))/(2*(c*d^2 
 + e*(-(b*d) + a*e))*(d + e*x)) - (((a + x*(b + c*x))^(3/2)*(b*e^2*(b*e*f 
- 5*b*d*g + 4*a*e*g) - 2*c^2*d*(4*d^2*g + e^2*f*x - d*e*(2*f + 3*g*x)) + c 
*e*(b*(13*d^2*g + e^2*f*x - 5*d*e*(f + g*x)) + 2*a*e*(-3*d*g + e*(f + 2*g* 
x)))))/(2*e^2) + (3*(2*c^2*e*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c* 
x)]*(b*e^2*(b*e*f - 4*b*d*g + 3*a*e*g) - 2*c^2*d*(e*f - 2*d*g)*(-2*d + e*x 
) + c*e*(2*a*e*(e*f - 3*d*g + e*g*x) + b*(-5*d*e*f + 12*d^2*g + e^2*f*x - 
3*d*e*g*x))) + c^(3/2)*(c*d^2 + e*(-(b*d) + a*e))^2*(b^2*e^2*g + 8*c^2*d*( 
-(e*f) + 2*d*g) + 4*c*e*(b*e*f - 3*b*d*g + a*e*g))*ArcTanh[(b + 2*c*x)/(2* 
Sqrt[c]*Sqrt[a + x*(b + c*x)])] + c^2*(c*d^2 + e*(-(b*d) + a*e))^(3/2)*(8* 
c^2*d^2*(-(e*f) + 2*d*g) - b*e^2*(b*e*f - 5*b*d*g + 4*a*e*g) - 4*c*e*(a*e* 
(e*f - 3*d*g) + b*d*(-2*e*f + 5*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)])]))/(4*c^2 
*e^5))/(-(c*d^2) + e*(b*d - a*e)))/(2*(c*d^2 + e*(-(b*d) + a*e)))
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1230, 27, 1230, 25, 1269, 1092, 219, 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.

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {3 \left (\frac {\frac {\left (-4 c e (b d (2 e f-5 d g)-a e (e f-3 d g))+b e^2 (4 a e g-5 b d g+b e f)+8 c^2 d^2 (e f-2 d g)\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c e (a e g-3 b d g+b e f)+b^2 e^2 g-8 c^2 d (e f-2 d g)\right )}{\sqrt {c} e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} (-e (2 a e g-4 b d g+b e f)+e x (b e g-4 c d g+2 c e f)+4 c d (e f-2 d g))}{e^2 (d+e x)}\right )}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-2 d g+e f-e g x)}{2 e^2 (d+e x)^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {3 \left (\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c e (a e g-3 b d g+b e f)+b^2 e^2 g-8 c^2 d (e f-2 d g)\right )}{\sqrt {c} e}-\frac {2 \left (-4 c e (b d (2 e f-5 d g)-a e (e f-3 d g))+b e^2 (4 a e g-5 b d g+b e f)+8 c^2 d^2 (e f-2 d g)\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}}{2 e^2}+\frac {\sqrt {a+b x+c x^2} (-e (2 a e g-4 b d g+b e f)+e x (b e g-4 c d g+2 c e f)+4 c d (e f-2 d g))}{e^2 (d+e x)}\right )}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/2} (-2 d g+e f-e g x)}{2 e^2 (d+e x)^2}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*((e*f - 2*d*g - e*g*x)*(a + b*x + c*x^2)^(3/2))/(e^2*(d + e*x)^2) + ( 
3*(((4*c*d*(e*f - 2*d*g) - e*(b*e*f - 4*b*d*g + 2*a*e*g) + e*(2*c*e*f - 4* 
c*d*g + b*e*g)*x)*Sqrt[a + b*x + c*x^2])/(e^2*(d + e*x)) + (((b^2*e^2*g - 
8*c^2*d*(e*f - 2*d*g) + 4*c*e*(b*e*f - 3*b*d*g + a*e*g))*ArcTanh[(b + 2*c* 
x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*e) + ((8*c^2*d^2*(e*f - 2* 
d*g) + b*e^2*(b*e*f - 5*b*d*g + 4*a*e*g) - 4*c*e*(b*d*(2*e*f - 5*d*g) - a* 
e*(e*f - 3*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])])/(e*Sqrt[c*d^2 - b*d*e + a*e^2]))/( 
2*e^2)))/(4*e^2)
 

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

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 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1426\) vs. \(2(325)=650\).

Time = 3.02 (sec) , antiderivative size = 1427, normalized size of antiderivative = 4.04

method	result	size
risch	\(\text {Expression too large to display}\)	\(1427\)
default	\(\text {Expression too large to display}\)	\(2961\)

Input:

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

Output:

1/4*(2*c*e*g*x+5*b*e*g-12*c*d*g+4*c*e*f)*(c*x^2+b*x+a)^(1/2)/e^4+1/8/e^4*( 
-8/e^2*(2*a*b*e^3*g-6*a*c*d*e^2*g+2*a*c*e^3*f-3*b^2*d*e^2*g+b^2*e^3*f+12*b 
*c*d^2*e*g-6*b*c*d*e^2*f-10*c^2*d^3*g+6*c^2*d^2*e*f)/((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))+8/e^3*(a^2*e^4*g-4*a*b*d*e^3*g+2*a*b*e^4*f+6*a*c* 
d^2*e^2*g-4*a*c*d*e^3*f+3*b^2*d^2*e^2*g-2*b^2*d*e^3*f-8*b*c*d^3*e*g+6*b*c* 
d^2*e^2*f+5*c^2*d^4*g-4*c^2*d^3*e*f)*(-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)+1/2*(b*e- 
2*c*d)*e/(a*e^2-b*d*e+c*d^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)) 
)-8*(a^2*d*e^4*g-a^2*e^5*f-2*a*b*d^2*e^3*g+2*a*b*d*e^4*f+2*a*c*d^3*e^2*g-2 
*a*c*d^2*e^3*f+b^2*d^3*e^2*g-b^2*d^2*e^3*f-2*b*c*d^4*e*g+2*b*c*d^3*e^2*f+c 
^2*d^5*g-c^2*d^4*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)-3/4*(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)+1/2*(b*e-2*c*d)*e/(a*e^2 
-b*d*e+c*d^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+...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((g*x+f)*(c*x^2+b*x+a)^(3/2)/(e*x+d)^3,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. 1666 vs. \(2 (325) = 650\).

Time = 0.42 (sec) , antiderivative size = 1666, normalized size of antiderivative = 4.72 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*sqrt(c*x^2 + b*x + a)*(2*c*g*x/e^3 + (4*c^2*e^9*f - 12*c^2*d*e^8*g + 5 
*b*c*e^9*g)/(c*e^12)) + 3/4*(8*c^2*d^2*e*f - 8*b*c*d*e^2*f + b^2*e^3*f + 4 
*a*c*e^3*f - 16*c^2*d^3*g + 20*b*c*d^2*e*g - 5*b^2*d*e^2*g - 12*a*c*d*e^2* 
g + 4*a*b*e^3*g)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c)* 
d)/sqrt(-c*d^2 + b*d*e - a*e^2))/(sqrt(-c*d^2 + b*d*e - a*e^2)*e^5) + 3/8* 
(8*c^2*d*e*f - 4*b*c*e^2*f - 16*c^2*d^2*g + 12*b*c*d*e*g - b^2*e^2*g - 4*a 
*c*e^2*g)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/(sq 
rt(c)*e^5) + 1/4*(24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*d^2*e^2*f - 
 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*d*e^3*f + 5*(sqrt(c)*x - sqr 
t(c*x^2 + b*x + a))^3*b^2*e^4*f + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3* 
a*c*e^4*f - 32*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*d^3*e*g + 36*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*d^2*e^2*g - 9*(sqrt(c)*x - sqrt(c*x^ 
2 + b*x + a))^3*b^2*d*e^3*g - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c 
*d*e^3*g + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*e^4*g + 40*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*d^3*e*f - 24*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^2*b*c^(3/2)*d^2*e^2*f - (sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 
*b^2*sqrt(c)*d*e^3*f - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2)* 
d*e^3*f + 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*sqrt(c)*e^4*f - 56* 
(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*d^4*g + 44*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^2*b*c^(3/2)*d^3*e*g - 3*(sqrt(c)*x - sqrt(c*x^2 + b*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 2.71 (sec) , antiderivative size = 5522, normalized size of antiderivative = 15.64 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 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*b*c*d**2*e**3* 
g - 24*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*b*c*d*e**4*g*x 
 - 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*b*c*e**5*g*x**2 
 + 36*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*c**2*d**3*e**2* 
g - 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*c**2*d**2*e**3 
*f + 72*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*c**2*d**2*e** 
3*g*x - 24*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*c**2*d*e** 
4*f*x + 36*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*c**2*d*e** 
4*g*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*c**2*e* 
*5*f*x**2 + 15*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)*b**2*...